Natural Convection, Dissipation & Mathematical

¨ Gudrun Thater

Natural Convection, Dissipation & Power−law Rheology: Mathematical Models & Results

Contents

Introduction

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A Mathematical Modelling 1 Aspects of physics

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1.1 The outer earth mantle . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Thermodynamic concepts . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Bénard problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 18 27

2 Derivation of Approximations

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2.1 2.2 2.3 2.4

The Oberbeck-Boussinesq approximation . Natural convection with dissipative heating Power-law rheology . . . . . . . . . . . . A modified power-law . . . . . . . . . . .

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B Mathematical Treatment 3 Solvability & Stability 3.1 3.2 3.3 3.4

Generalities . . . . . . . . . . . . . . . . . . . . System with dissipation - constant heat capacity System with dissipation - variable heat capacity . Power-law rheology . . . . . . . . . . . . . . .

4 The Boussinesq limit

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64 78 91 97 114

4.1 Prerequisites and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2 Proof of Theorem 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.3 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Appendix

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C References & Tables References

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Glossary

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Tables

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List of Figures

 List of Figures 1.1 1.2 1.3 1.4 1.5 2.1 2.2

Map of earthquake and volcanic regions . . . . . . . . . . . . . Characteristic time scales of geodynamic phenomena . . . . . . Model of the interior of the earth . . . . . . . . . . . . . . . . . Numerical simulation of cold downwellings in the mantle . . . . Convection rolls and rectangular cells . . . . . . . . . . . . . . Approximation scheme . . . . . . . . . . . . . . . . . . . . . . Sketch of relations between generalised viscosity and shear rate

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11 12 12 16 30 37 58

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Introduction The equations of fluid dynamics represent one of the most intensively studied systems of equations in classical physics. In my opinion there are three main reasons for their popularity: At first everyday life and technique are full of situations and applications where we wish to control the flow of liquids and gases: construction of pipeline systems, combustion engines, water-, steam- or gas-turbines, airplanes, and ships. We would like to understand problems in human blood vessels, and to fill yogurt tubs and honey glasses with machines in the most efficient way. We pump oil out of the earth and investigate how our environment is effected by dirty industry and acid rain. We try to predict the weather and the development of the hole in the ozone layer. And - what is less known - even the processes in the outer earth mantle, which lead to earth crust wandering, volcanic activity and hot springs are flow processes if considered in a large time scale. The second reason is that mostly it is possible to simplify the full system of equations which govern processes, such that engineers and mathematicians have a chance to treat solvability, stability and uniqueness questions and to find out if the dependence on certain parameters is continuous. The more difficult the problem the more tools are developed or adapted and the full nice machinery of mathematics and physics can show its usefulness. The more difficult the problem the more interesting and controversial is the way towards results and their discussion in the scientific community - of course, up to the critical point where for the moment the problems become too difficult. The third reason that after about 150 years of related research fluid dynamics are still attractive to scientists is that for a successful project a lot of different specialists are involved: physicists, engineers and chemists create more or less simplifying models and find possibilities to validate them by measuring relevant properties of materials. Numerical computations are performed if experiments are expensive or impossible. Thus, computer specialists and developers of appropriate numerical schemes have to join the group together with numerical analysts (working on the borderline between mathematics and informatics) who show that the schemes work properly. And last but not least mathematicians find a wide field of open problems concerning the structure of solutions and their stability.

Introduction



The most interesting point is, when specialists interact. For instance as mathematicians we benefit from learning how thermodynamic principles fix the frame for the possible dependence of quantities and how they influence stability. Moreover, as profession mostly using paper and pencil we have to adapt to the uncertainties in all measurements and accept approximate models. On the other hand engineers and physicists must be sure about the meaning of uniqueness, bifurcation and stability results provided by mathematicians to ask the right questions to their experiments. Moreover, they should have good reasons to trust in numerical results while people responsible for computational experiments need data from real life and experiments to justify their output. Not to forget that numerical schemes can be improved if mathematicians provide properties of the solutions as expected growth, asymptotic behaviour, regions of possible bifurcation and singular points. Within that frame this book is a contribution to the wide field of fluid dynamics and the work of different specialists from the point of view of a mathematician. Namely - as one expects - Solvability & Stability of certain initial-boundary value problems for partial differential equations are investigated in Chapter 3. Yet, in this treatise mathematics starts already at an earlier - “non-standard” - level since it includes also the Derivation of Approximations in Chapter 2, where the equations which are investigated and which shall approximate fluid flow under certain conditions are derived from basic principles of Continuum Mechanics. Furthermore, Chapter 4 is devoted to the so-called Boussinesq limit. Namely, if the perturbation parameter used in our approximation scheme becomes smaller and smaller and tends to zero we make a first step to control the limit of our system in a precise way. These three chapters together permit a detailed mathematically picture from different points of view. Besides also some Aspects of Physics and facts about models and ↑ rheology1 are included as first chapter. This is necessary to furnish a precise idea about . the frame in which the approximation schemes are developed later on, . possible applications of the equations, . criteria for choosing one system. This shall provide a sound basis for the whole book - not only for mathematicians but for other scientists working in the field of fluid dynamics. A very prominent and useful model to interpret the flow patterns of a fluid layer under the influence of a gravitational field and heating from below is the Oberbeck-Boussinesq approximation. It was designed as the simplest model for the thermomechanical response of linear2 viscous fluids undergoing ↑ isochoric motions in isothermal processes but not necessarily ↑ isochoric ones in non-isothermal processes and stems from the end of the 19 th century [17, 102]. In short “The central physical idea is that typical accelerations promoted in the fluid by variations in the density are always much less than the acceleration of gravity” [51]. However, there has been a long discussion how to justify and delineate this approximation in some appropriate physical frame: only in 1996 Rajagopal/R˚uzˇ iˇcka/Srinivasa derived it from the full thermodynamical theory for Navier-Stokes-Fourier fluids, [114]. 1 2

The symbol ↑ means that the notion is explained in the glossary. All notions are defined in Section 1.2

Introduction One crucial idea was to adapt a new method of treating the constraint of mechanical incompressibility introduced by Hills/Roberts [51]. At the same time the computations in [114] clearly fix the validity of the approximation in terms of quantities which can be measured for concrete configurations (for details see Section 2.1). Nonetheless, there are important examples for which the Oberbeck-Boussinesq approximation does not seem to be appropriate. The rapid technological advances lead to new problems where the three primary factors governing natural convection, namely, body force, fluid volumetric expansion coefficient and the temperature differences all can greatly exceed previously considered bounds. For example when . analysing convection in the earth’s upper mantle [19, 50, 94, 57, 108], . in astrophysics and glaciology [9, 118], . within fast rotating configurations [12, 18]. In this context the simplification which is mainly criticised is that the Oberbeck-Boussinesq approximation neglects dissipative heating [42, 43, 103]. On the other hand the linear stress-strain relation (i.e. the Newtonian fluid model) apparently misses main features of the fluid flow while investigating, e.g., . solid-liquid suspensions occurring as solutions of macromolecules, . blood, . glaciers, . molten plastic. A lot of modern applications are concerned by these arguments and the situation leads to two notable questions: . What are better (and still simple enough) models? . What are measurable quantities which qualify models to be suitable or not? One aim of this book is to provide answers to these questions. Generalisations of the Oberbeck-Boussinesq approximation are no new idea but have a long history. Moreover, this is a wide field and the two questions above are only the starting point for more and more difficult problems to be solved and decisions to be made. More specifically, in Chapter 2 our interest and main new aspect compared to available results is to adapt and transfer the ideas of [114] to other applications where the OberbeckBoussinesq approximation is not suitable but incompressibility with respect to mechanical forces and compressibility with respect to temperature changes is observed. Our aim is to derive the most elementary schemes. We present several new systems together with the information if for concrete applications they are suitable or not. As first relevant problem and useful example we discuss how and when to include the influence of dissipative heating to the usual Oberbeck-Boussinesq approximation in Section 2.2 along the results of [61] (in collaboration with Kagei/R˚uzˇ iˇcka) and arrive at system (2.40). Then we turn our interest to a new and up to the moment completely open field and consider nonlinear stress-strain relations. One new difficulty in comparison to the linear Newtonian model used in Sections 2.1 and 2.2 is a sound nondimensionalisation of the (generalised) ↑ viscosity. Precisely, in Sections 2.3 and 2.4, respectively, we derive four new systems applicable to material with power-law (namely, (2.49) and (2.52)), and with generalised power-law rheology (precisely, (2.57) and (2.59)).

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Introduction

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These problems are closely related to the modelling of flow processes in the earth’s upper mantle but concern a bunch of other applications as well. Moreover, along the lines given in Section 2 one can adapt the scheme to other processes as well and in this sense our treatise is to be considered as a prototype ready to be re-applied. After the derivation and justification of new models the “standard” mathematical questions have to be answered. Thus, in Chapter 3 we turn our interest to solvability and stability. Here first we collect tools, useful examples as well as techniques and known results in Section 3.1. After that in Sections 3.2 and 3.3 we present the knowledge about system (2.40). This is the generalisation of the Oberbeck-Boussinesq approximation which includes dissipative heating for Newtonian fluids. First we study the standard situation with constant heat ˜ and collect the results in Theorem 3.7. Then - as a prototype for further capacity C ˜ may become a function of the temperature (see Theorem 3.14). generalisation - C These two sections are mainly based on the results in [61]. In both cases we construct Galerkin-like approximations, apply energy techniques and - for the stability analysis - results about the linearised operator at the motionless state (which are derived in the appendix). Due to the nonlinearity introduced by the main dissipation term kD(v)k2 the class for possible weak solutions is smaller than for the Oberbeck-Boussinesq approximation since the velocity field v has to be more regular. Namely, instead of the category (0.1) for p = 2 below we must take v ∈ C([0, T ]; V) ∩ L2(0, T ; D(A)) ,

θ ∈ C([0, T ]; L2(Ω)) ∩ L2(0, T ; V) .

˜ the following class of strong solutions provides an appropriate frame For non-constant C v ∈ C([0, T ]; D(A)) ,

∂tv ∈ C([0, T ]; H) ∩ L2(0, T ; V) ,

2

θ ∈ C([0, T ]; H (Ω) ∩ V) ,

∂tθ ∈ C([0, T ]; L2(Ω)) ∩ L2(0, T ; V) .

For the stability analysis a smallness conditions to the data must be assumed and in addition to the well-known critical Rayleigh number also a critical Dissipation number enters. In the last section of Chapter 3 we treat the new systems (2.49) and (2.57) for the powerlaw and generalised power-law model without dissipation. Here the constitutive relation ^ is characterised by properties of the following type of the extra stress tensor T ^ T(D(v)) · D(v) ≥ c1kD(v)kp ^ kT(D(v))k ≤ c2(1 + kD(v)k)p−1

coercivity , polynomial growth

for some constants c1 and c2 and a parameter 0 < p < ∞. In particular, for p = 2 system (2.49) turns into the Oberbeck-Boussinesq approximation. In Theorem 3.27 we prove for p >

9 5

the existence of weak solutions to system (2.49), i.e.

v ∈ L∞ (0, T ; H) ∩ Lp(0, T ; Vp) ,

If p ≥

11 5

θ ∈ L∞ (0, T ; L2(Ω)) ∩ L2(0, T ; V) .

for the appropriate initial values they turn into (unique) strong ones v ∈ C([0, T ]; H) ∩ L∞ (0, T ; V2) ∩ L2(0, T ; D(A)) ,

θ ∈ L∞ (0, T ; V) ∩ L2(0, T ; H2(Ω)) .

(0.1)

Introduction



In addition, Theorem 3.32 shows the same for (2.57). These solutions exist on any finite time interval without smallness conditions to the data. Due to the polynomial growth property the problems have an Lp-structure in the velocity equation while the heat equation carries an L2-structure and we have to manage this interplay. Nevertheless, we can adapt ideas of [84] and elaborate on the sketch [89]. The main tools are the Korn inequality and, again, apriori (i.e. energy) estimates for an approximate (Galerkin-like) solution. Then the Aubin/Lions Lemma and parabolic embeddings help to justify the limits in all terms. While deriving the approximation in Chapter 2 we expand the physical quantities into power series with respect to a small parameter ε and silently assume that the first εpower-terms indeed are the leading terms. Thus, the investigation of the limit process ε → 0 in an appropriate mathematical frame is interesting and necessary to be put into practice to show that, indeed we arrive at the usual Oberbeck-Boussinesq approximation for the incompressible case, i.e. for ε = 0. We call it the Boussinesq limit. Since the corresponding systems are highly nonlinear in Chapter 4 for the moment we can only treat the simplified model problem div v = ε3∂tθ , 1 ρ 1 ρ(∂tv + v · ∇v) − 2µ div D(v) − ∇ div v + 3 ∇p = 3 ∇f , 3 ε ε cv∂tθ − ∆θ = 0 , ∂tρ + div (ρv) = 0 . The main difficulty which remains is that we cannot use tools which fit for solenoidal vector fields developed in the context of Navier-Stokes theory. Even for the linear insteady Stokes problem (4.8) this leads to a bunch of very delicate and complicated compatibility conditions for weak solvability [131]. These results have been obtained in cooperation with Kagei/R˚uzˇ iˇcka [62]. Aside from the new results filling unsatisfactory gaps in the theory of convective flow processes one second important aim of the book is to provide a kaleidoscopic “overall” picture which includes physics, approximation and mathematics in direct relation to the problem under consideration in a unified presentation and language. The subjacent intention is that the topic in general and especially the treatise will find the interest of all parties which work near the borderline between engineering and physical sciences on the one side and mathematics on the other side. Above all there is also some hope to convince the reader (from whatever side) that ”Convection can be entertaining.” Bejan [11]

Part A Mathematical Modelling

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1 Aspects of physics In this first chapter we establish physical terms which play a role in the mathematical treatment later on. Partly this is accomplished for the convenience of the reader but mostly for two reasons: on the one hand to introduce the language, notation, and relations for the physical quantities and on the other hand to show the place where the results presented in this book have their meaning in a physical, technical and everyday frame. Particularly we are concerned with the motion of different types of viscous, heat conductive fluids without memory. In general the notion fluid stands for gas as well as for liquid. By definition gas is material that fills any volume while liquid is characterised by the ability to assume the form of its container after a certain time. It should be noted, that the same material can behave as liquid, elastic or plastic solid depending on the time it is given to act. The concept ↑ viscosity expresses the experimental observation that a fluid packet in general easily changes its shape (filling any volume or taking the shape of the container) but may resist the time rate of such modification due to molecule interactions. The stronger the internal bonds the more viscous a substance behaves - gas being an almost inviscid material. We begin with the modelling of the outer earth mantle. It could be themed by Heraclitus’ (∼500 BC.) famous statement “Παντα ρι”, i.e. everything flows - at least if one waits long enough (see Figure 1.2). This is the section most distant from mathematics and most uncertain in any respect. Then we provide a systematical survey on Thermodynamic concepts. There are several possibilities to proceed. Here we adapt the first chapter of [99] in which exemplarily viscous, heat conductive fluids are handled. We close Chapter 1 with a section on the classical Bénard problem i.e. Rayleigh-Bénard convection in a plane, infinite horizontal fluid layer heated from below inside a gravitational field.

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Aspects of physics

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1.1 The outer earth mantle As far as we know the earth was formed in an enormous accumulation and high-velocity bombardment of meteorites and comets five milliards (i.e. 5 × 109) years ago. The huge amount of discharged heat energy melted the entire planet, and it is still cooling off today. Denser materials like iron sank into the core of the earth, while lighter silicates, other oxygen compounds, and water rose near the surface. The past and present sea-level record shows that the earth is still changing its shape. This is for example correlated to the melting of the Pleistocene ice sheets thousands of years ago. But also today the whole hydrological system - oceans, lakes, ice sheets, and ground-water - is a source of an immense mass redistribution at the earth’s surface. Consequences are elastic deformations of the crust, a mass redistribution within the earth, and hence, changes in the earth’s gravity field. Satellites orbiting the earth are subject to its gravity field and by observing their motion, we are able to measure the height of the equipotential surface (↑ geoid) relative to a reference ellipsoid and the rate of change of the ↑ geoid. On the other hand satellite techniques can also be used to measure changes in the position of the earth’s rotation axis. Of course, longer (less detailed) time series of such measurements are available through astrometric observations. ”Now is an exciting time to be a geophysicist. Advances in instrumentation used for observing Earth (for example, GPS arrays and portable digital seismic networks), the opening up of regions previously inaccessible to Western scientists, as well as a new generation of experimental capabilities, are providing data sets covering spatial, temporal, pressure, and temperature ranges previously unavailable. Computer power has leapfrogged to the point where it is now possible to carry out numerical experiments in three dimensions and with realistic physical properties, such that comparison of predictions of computer models and observations is providing major advances in our understanding of Earth processes. At the same time, society needs more than ever the insights that geophysicists can provide into such important geosystems as petroleum reservoirs, groundwater flow, atmospheres and climate, and active fault systems.” Geophysics Program at the Massachusetts Institute for Technology 3

Earth model. Although in principle we suppose that earth ground beneath our feet is hard and solid, ↑ geodynamic observations show that the surface of the earth is just a thin crust of rock, which is “easily” broken and already fractured like the cracked shell of an egg. The pieces are the earth’s tectonic plates and they move across a layer of fluid rock, their motions driven by forces generated deep inside the earth. The plates spread apart, 3

see http://www-eaps.mit.edu/ep3.html

1.1

The outer earth mantle

Figure 1.1. Map of earthquake and volcanic regions

converge, and slide past one another at their boundaries. These boundaries painted black in Figure4 1.1 are the most geologically active regions on earth. Here, land is created or destroyed. Hot springs spew out mineral-rich waters, volcanoes erupt, and earthquakes occur more or less frequently. On the other hand there are volcanic islands forming huge lines, such as the HawaiianEmperor chain far from the boundaries in the middle of tectonic plates. It is thought, that they are formed while the plate moves over relatively stationary hot regions, called hot spots, stemming from deep mantle plumes. Since that line of volcanic isles extends up to the Aleutian islands, the corresponding plume must have been active for the very long time of about 80 million years. Nowadays it is widely accepted that thermal convection in the mantle is the most important dynamic process in the earths interior. It is the driving force for the movement of the tectonic plates and the cause of ocean basins, continental mountain belts, long-term sea-level fluctuations, and worldwide earthquake and volcanic activity. Nonetheless, only forty years ago plate tectonics were established and the idea of mantle convection was introduced. It was noticed that there are several nearly rigid plates, each moving a few centimetres per year. Since then measurements have clearly demonstrated the existence of convective motion inside the earth mantle but its precise structure is under controversial discussion till today. Through decades of work with data and experiments, scientists have proposed different models of the earth’s interior and have tried to estimate the forces that produce the activity we observe on the surface. But the ↑ rheology of the material inside the earth is the main unknown. In short the non-controversial facts are the following: The earth consists of polycrystalline material which shows a variety of rheological behaviour depending on several inner parameters and outer conditions. In Figure 1.2 the characteristic time scales of various ↑ geodynamic processes are illustrated (from [16, p.18]). Namely, rock is elastic at low temperature and pressure. This is observed near 4

courtesy of National Geographic Data, ftp://ftp.ngdc.noaa.gov/GLOBE DEM/pictures/

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1

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Aspects of physics

Maxwell time steady−state rheology

postseismic deformation

seismology

postglacial polar wander mantle rebound convection

time in s 0

10

10

4

8

10

12

10

16

10

Figure 1.2. Characteristic time scales of geodynamic phenomena

the earth’s surface, even under loads of long duration (with respect to thousands of years under the weight of an ice layer, e.g.). As in the bulk of the earth, it is still elastic at high temperature and pressure, if the load is of short duration. Nevertheless, it is fluid at high pressure and temperature and under loads of long duration. A heavy load on the crust, like an ice cap, large lake, or mountain range, can bend it down into the next layer, the asthenosphere, which can flow out of the way. The load sinks until it is held by ↑ buoyancy force. If the conditions alter, for instance ice caps melt or lakes dry up due to climatic changes, or a mountain range erodes away, the surface of the earth will buoyantly rise back up over thousands of years. This process is called postglacial or isostatic rebound. Available data suggest the following (simplified) model of the earth’s interior: The outer layer of the earth is the crust. At the thinnest parts in the oceans, where new crust is created, it is only a few kilometres thick while on the continents, the average is about 30 km - up to a maximum of 100 km. The crust and, immediately below it, the solid upper part of the mantle (down to about 150 km) together form the earth’s lithosphere.

Figure 1.3. Model of the interior of the earth

1.1


Below the lithosphere is a region of still solid, but softer and weaker, rock called the asthenosphere – which is already considered as part of the mantle. The plates slide over this hot, weak layer. The earth’s entire mantle is a region of rock which is hot and under immense pressure. For that over geological timescales it behaves like a fluid. It is almost 2, 800 km thick (i.e. extends up to a depth of about 2, 900 km). Beneath it lies the outer core, which is thought to consist of liquid iron. Finally, at the centre of the earth, is the solid iron core, a sphere with a diameter of 2, 300 km. This scheme is summarised in Figure 1.3 (courtesy of J. Louie5 ). Mantle convection. Mantle convection is the process that carries heat from the core and up to the crust. To have an idea about the amount of heat which is transferred: during one year, earthquakes alone release 1016 kJ of energy which corresponds to 100, 000 Hiroshima-sized nuclear bombs. And that is just one percent of the total amount of energy that reaches the surface from earth’s innards. The general procedure of thermal convection is well-understood - it’s the method of heat transfer in action when a pot simmers on the stove. Namely, when temperature differences are present in a body, heat is transferred from the hotter to the colder parts. There are several mechanisms of heat transfer. Since iron-bearing minerals have mostly sunken to the core, radiative transfer of heat is almost negligible and the two most important possibilities in the mantle are conduction and convection. In conduction, heat travels by interaction between adjoining atoms or molecules without transfer of mass. In convection heat is carried by moving material. Currents of heated, less-dense material rise to the surface, cool off, and sink cycle by cycle. Of course, the earth’s mantle is no “ordinary” stew, and modelling the mantle’s convection process is extremely challenging. A serious model should deal with complex material properties, including solid-state phase transitions, chemical composition differences, and thermodynamic quantities that depend on both pressure and temperature. Unfortunately, these complex flow laws of the earths mantle have not been well understood up to today due to the inherent difficulties in the studies of rheological properties. There are four main methods to derive information about the flow behaviour i.e. the ↑ rheology of material inside the earth: . interpretation of postglacial isostatic rebound data, . laboratory rock experiments, . seismic measurements, . numerical modelling. Postglacial rebound. During the last glacial cycles, global sea level dropped several times by about 120 m and large ice sheets covered North America, northern Europe and Antarctica during the glacial stages. Later the melted water entered the oceans and loaded the ocean basins. These changes in surface load caused the vertical movement of the crust, which is recorded as the relative sea level data. This record shows temporal variation due to the change in surface load and the mass of the seawater. Traditionally, relative sea-level data at sites near the ↑ Laurentide 5

see http://www.seismo.unr.edu/ftp/pub/louie/class/100/interior.html

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1

Aspects of physics

(North America) and ↑ Fennoscandian (NW Europe) ice sheets are interpreted.

”Of all the available geodynamic signatures, the phenomenon of glacial isostacy clearly has provided the geophysicists with the most information concerning the viscosity structure of the mantle, as the deformation field associated with the process of massive continental deglaciation contains a spectrum rich with information ranging from wave-lengths girdling around the globe, such as the degreetwo spherical harmonic, to horizontal variations of the order of a few hundred kilometres.... Clearly, by conducting a detailed investigation of the history of the relative sea-level one can resolve the variation of mantle viscosity with depth. ” Boschi, Sabadini & Yuen [16, p.25]

Theoretical analysis of this relaxation process, namely inversion analysis, yields information about the rheological properties of the earth’s interior. Usually the earth is modelled as a stratified ↑ viscoelastic material with linear ↑ rheology. This approach provides a value of average ↑ viscosity of the earth of 1021 Pa s. However, some details remain controversial such as the rheological constitutive relation, namely the stress dependence of strain rate. Because nonlinear dislocation ↑ creep has been observed in most of the laboratory studies, some theoretical studies have been made to test the validity of nonlinear ↑ rheology in the analysis of postglacial rebound. Moreover, the changes in the ice-ocean mass balance have displaced mantle material mainly via viscous flow. The perturbation of the equilibrium figure of the earth by glacial isostatic adjustment is still visible today in gravitational and rotational changes. Observations of these time-dependent variations in the earth’s gravity field and rotation therefore provide some constraint on the ↑ rheology of the mantle as well. One example is the ↑ free-air anomaly over the Laurentide region which was covered with about three kilometres of ice during the last glacial maximum some 12, 000 years ago. The negative gravity anomaly, indicative of a current isostatic disequilibrium, provides a measure of the remaining uplift. In the near future, several satellite gravity missions will significantly improve the accuracy of these observations [73]. Another signature connected with glacial forcings are changes in the earth’s rotation. Namely, internal mass redistribution causes ↑ true polar wander, i.e. a shift of the earth’s rotation axis with respect to a mantle-fixed reference frame. In order to match the observation concerning ↑ true polar wander the ↑ viscosity of the lower mantle must be at least one order of magnitude higher than the ↑ viscosity of the upper mantle. Nevertheless, one is still far away from having a definite knowledge about the ↑ rheology of the material in the earth mantle: ”The study of geophysical processes connected with glacial loading appears to be in a paradoxical state. It provides a very important tool in the analysis of the rheology of the mantle. It has firmly established that the mantle is a fluid even at deviatoric stresses as low as 106 Pa and that the evidence for characteristic times longer than about 103 − 104 years is compatible with a Newtonian, steady-state rheology. However, it has been less effective in excluding plausible alternatives.” Ranalli [116, p.225]

1.1


Laboratory experiments. The most direct estimation of the rheological properties of the earth comes from deformation experiments on rocks in laboratories at high temperatures and pressures. There are two complementary approaches to the study of ↑ creep in the laboratory: the experiments may be performed on a single crystal or on polycrystals. In both cases, the temperature is high (θ > 1300 K) in order to obtain strain rate of about 10−8 s−1. The stress difference of ↑ geodynamic interest is in the range of 106 − 108 Pa. At these stresses, and at high temperatures most geological materials under laboratory conditions flow by power-law ↑ creep and the flow is strongly dependent on temperature, pressure, and stress (see [71] for a summary). ↑ Olivine is one of the best-studied materials and is the main constituent of the upper mantle. Various types of mechanical tests are possible in the ductile regime, according to the geometry of the sample and the load. Usually a cylindrical sample of size large relative to that of individual grains is loaded axially in compression. Laboratory results must be extrapolated to shorter time scales or to lower stresses when applied to the earth, and possible changes in the deformation mechanisms at lower stress levels raise questions as to the validity of extrapolation. The impossibility to duplicate in the laboratory the deformation environments of the interior of the earth makes it necessary to strengthen experimental results by means of theoretical conclusions on ↑ creep which are based on microphysical models and therefore are not limited by experiment. Seismic tomography. Measurements of the velocities of seismic waves generated by earthquakes and underground nuclear explosions are the most precisely determined material properties of the mantle (review in [21]). They provide information about the structure of mantle convection since through the hot and cold regions seismic waves travel at different rates due to the different material density. Namely, lateral heterogeneity in seismic velocity is mostly the result of lateral temperature variation (higher temperatures mean lower seismic wave velocities). Global seismology makes use essentially of the coherent part of the seismograms: arrival times of various phases of surface and body waves, differential time arrivals, eigenfrequencies of normal modes of the earth, and more recently waveforms. Seismologists try to characterise the perturbations with respect to these mean reference models, i.e. long wavelengths lateral variations, and to correlate them with plate tectonics and mantle convection. All these studies suggest 3D structures over a wide range of scales in the form of deviations from the spherically averaged profiles. Numerical modelling. Numerical models have proved to be useful for interpreting geophysical and geochemical observations in terms of mantle convection processes. Spherical geometry simulations yield data sets that can be compared statistically to geophysical observations, such as 3D maps of long wavelength seismic velocity variations in the earth’s interior obtained by seismic tomography. One approach is to derive the pattern of flow in the mantle from the tomographic images and to determine whether this flow pattern is capable of explaining surface observables. Tomographic images are converted to density anomalies and then entered as body forces into the equation of motion. Important progress in this direction has been made recently using three-dimensional computations of thermal convection in viscous spherical shells. The most significant shortcoming of

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1



Aspects of physics

present-day models is their treatment of lithospheric plates. ”Indeed, geophysical modelling, a discipline in which numerical experiments are conducted in order to study in a methodical way the effects of varying the essential physical parameters of a particular dynamic process, has in recent years led to an improved understanding of the rheological properties of the mantle by comparisons between observations and theoretical predictions. ” Boschi, Sabadini & Yuen [16, p.2]

Another field of numerical simulation is to develop direct methods for wavefield propagation, in space and time, at global scales to improve our physical understanding of wave propagation in realistic earth models and of the signatures of lateral heterogeneities in the seismograms recorded at the surface. This approach is complementary to traditional inversion approaches, and should allow the quantification of the actual limits of the various approximations used in the direct problem of the inversion methods and suggest new approximations. Main difficulties are: . the geometrical discretisation, . the accurate computation of surface and interface waves, . the solid-liquid coupling at the core-mantle interface, . the computational cost. Another aim of numerical simulations is to explore the possible variations of the mantle ↑ viscosity and its consequences to the form of convection. In Figure6 1.4 one sees cold downwellings from a mantle simulation with a phase change at 670 kilometres depth. Downwellings are deflected at the phase boundary, forming pools of cold material that grow, destabilise, and eventually fall into the lower mantle in cylindrical “avalanches”.

Figure 1.4. Numerical simulation of cold downwellings in the mantle

Rheology of the mantle. The ↑ rheology of the mantle is a primary factor affecting convection. Unfortunately, unique conclusions as to the predominant rheological equation and the variations for ↑ viscosity with depth are not available. Most convection models require some increase of ↑ viscosity with depth in the mantle. The main source of geophys6

courtesy of P. Tackley from http://www.npaci.edu/enVision/v14.2/tackley.html

1.1


ical evidence for the ↑ rheology of the mantle lies in relative sea-level data, gravitational, and rotational data. To the first order it is compatible with a mantle with ↑ Newtonian viscosity. A spherically symmetric inversion of geoidal anomalies, dynamic topography, and plate velocities related to mantle circulation gives a ↑ viscosity of 2 × 1020 Pa s in the asthenosphere, increasing to 7 × 1022 Pa s in the middle part of the lower mantle, with no sharp discontinuities in the mantle transition zone at a depth of 670 km. A non-Newtonian or transitional ↑ rheology is usually predicted on the basis of considerations on deformation mechanisms in silicates. The debate concerning the ↑ viscosity of the mantle of the earth has come to be somewhat polarised by an apparent disagreement between results obtained through inferences based upon data pertaining to the postglacial rebound or glacial isostatic adjustment process and results derived on the basis of large scale nonhydrostatic geodic anomalies which are supported by mantle convection. However, problems exist with the resolving power of geophysical inversion due to the limited amount of observed data. Furthermore, the identification of steady-state is not a simple problem as well. What may appear as a straight line over an interval of time may not be a true straight line when viewed over a larger interval. Numerical simulations however show, that non-Newtonian ↑ rheology has some severe differences compared to the simpler ↑ Newtonian model. The ability of very hot thermal anomalies to reach the surface is greatly increased by nonNewtonian ↑ rheology, and flow pattern seem to be less stable. Whole or layered mantle convection. The observations at the surface of the earth are proof enough to take convection in the upper mantle for granted. But there is no direct evidence for convection in the lower mantle and for some time it was thought that the ↑ viscosity increase with growing pressure could be so high as to prevent convection. Nevertheless, the Rayleigh number7 for the lower mantle is supercritical and thus, also the lower mantle should be convecting. The controversy centers on whether it convects together with the upper mantle or separately with a thermal boundary layer at 670 km. In this context two important sets of data are the velocity anomalies as revealed by seismic tomography and the geochemical evidence for the existence of long-lived separate geochemical reservoirs in the mantle [21, p.210]. The question what is the right model is not easy to decide since all arguments are partly based on unverifiable assumptions and if results are consistent with one style of convection they usually do not prove that the other style is excluded (see [109, p.208f]). The idea of two-layer convection is mainly based on the following arguments:

. The seismic waves of earthquakes stop at a depth of 670 km and this discontinuity seems to separate reservoirs of different chemical composition. . There exist clearly recognisable geochemical differences between mid-ocean ridge basalts and oceanic island basalts. If this would be explained in terms of different depth of origin, then the geochemical evidence requires that the mantle contains inhomogeneities as revealed by trace elements and isotopic data - that persists over timescales of the order of 109 years. Several studies of the mixing time of inhomogeneities in the mantle conclude that such time is comparable to the age of the earth. 7

Definition at the end of Section 1.3.



1



Aspects of physics

Whole mantle convection is supported by: . Why to choose a complicated two-layer model if the simple whole mantle convection model is not clearly ruled out? . Geochemical differences can also be explained by the flow pattern, which moves the upper parts of the mantle more than the lower. This reduces the time necessary to mix considerably. . A thermal boundary at 670 km implies a high temperature gradient, namely, a temperature in the lower mantle about 1000 K higher than in case of whole mantle convection. The consequence is a ↑ viscosity of 6 × 1016 Pa for the lower mantle which contradicts values found by postglacial rebound. The true picture seems to lie between the extremes, and would be provided by a way to reconcile various pieces of seemingly contradictory geophysical evidence. Obviously, there are many open questions and we are far from understanding the behaviour of the earth. Nonetheless a better understanding would help us to protect ourselves with respect to volcanic activity and earthquakes. Hopefully the near (in which timescale so ever) future will see us much wiser.

"

# % (

1.2 Thermodynamic concepts We are interested in the behaviour of viscous fluids which conduct heat. In this section we investigate their motion and laws governing the temperature and density distribution in a three-dimensional region, i.e. all quantities are considered at points x = (x1, x2, x3) ∈ R3 and at a certain time t. In short the main tools for our analysis are . Balance laws of mass, linear momentum and energy and the . application of universal principles of physics. The final aim is to derive the field equations for . density ρ (t, x) or pressure p(t, x), respectively, . velocity v(t, x) = (v1(t, x), v2(t, x), v3(t, x)), and . (absolute) temperature θ(t, x) of the fluid.

1.2

Thermodynamic concepts



First we introduce some notation8. In what follows boldfaced minuscules stand for vectors and vector valued functions whereas boldfaced capital letters represent tensor valued functions, i.e. v = (v1, v2, v3) (row or column depending on the context), T = (Tkl)3k,l=1. We set T> := (Tlk)3k,l=1. The dot between two quantities denotes the corresponding scalar product between vectors or tensors, respectively: v · w :=

3 X

vkwk ,

k=1

T · L :=

3 X

TklLkl ,

while

(T : L)il :=

k,l=1

3 X

TikLkl ,

k=1

i.e. the colon product, the result of which is a tensor. Besides, P3 stands for the usual2 tensor k tr D := k=1 Dkk, D : D =: D , D := D : Dk−1 (k ∈ N, k > 2), kDk2 := D · D, and I is the identity. Note, that tr (D2) = D · D = kDk2. For partial derivatives we use two types of the abbreviations: ∂ else. Moreover, ∂2τσ := ∂τ∂σ, ∂2τ := ∂2ττ. ∂k := ∂x∂k if k ∈ {1, 2, 3} and ∂ξ := ∂ξ As usual ∆ := ∂211 + ∂222 + ∂233 and ∇ := (∂1, ∂2 , ∂3)> . We write div v := ∇> · v = ∂1v1 + ∂2v2 + ∂3v3 ,

div T := ∇> : T = (∂1T1k + ∂2T2k + ∂3T3k)3k=1 ,

while ∇ρ := (∂1ρ , ∂2ρ , ∂3ρ )> , ∇v := (∂ivk)3i,k=1, and the superposed dot is the material time derivative, i.e. scalar quantities:

3 P vk∂kθ , θ˙ := ∂tθ + v · ∇θ = ∂tθ + k=1

vector fields:

3 P b˙ := ∂tb + (b · ∇)v = ∂tb + bk∂kv . k=1

For simplicity we always consider a single body and assume enough smoothness of all considered quantities. Henceforth let L := ∇v, T denote the Cauchy stress tensor, b the density of external body forces, e the specific internal energy, r the radiant heating and q the heat flux vector. All these quantities vary with space and time, i.e. depend on x ∈ R3 and t > 0. For the reader interested in a detailed physical background we refer to [25, 99, 115, 135]. Under smoothness assumptions for the pertinent fields the Balance laws can be formulated as local differential equations. Namely, with our notation mass: momentum: energy:

m := ρ˙ + ρ div v = 0, M := ρ v˙ − div T − ρ b = 0, E := ρ e˙ − T · L − div q + ρ r = 0 .

(1.1)

We consider b and r to be given functions. Nevertheless, in this setting equations (1.1) have too many unknowns to be a closed system. In addition we have to provide sensible9 constitutive relations for T, q, and e. 8 9

See also Part C. i.e. reflecting the properties of the material and in accordance with universal physical principles

1



Aspects of physics

Our experience with viscous, heat conductive fluids points out that T, q, and e at some point depend on the values of ρ, v, and θ in its spatial neighbourhood (Principles of local action and determinism [115, p.32]) and we prescribe the general relations T = T (ρ, θ, v, ∇v, ∇θ) , e = E(ρ, v, θ, ∇v, ∇θ) .

q = Q(ρ, θ, v, ∇v, ∇θ) ,

(1.2)

If concrete functions T , Q, and E, respectively, are fixed on the one hand they define a particular fluid via its material properties and on the other hand we can eliminate T, q, and e from (1.1). Then we can determine the five unknown functions ρ, v = (v1, v2, v3), and θ. However, this is theory. In reality there is no material for which T , Q, and E are known functions. The thermodynamic constitutive theory tries to restrict the generality of the constitutive relations (1.2) in a way that at the end there remains a number of coefficients the values of which can be measured in experiments. Its main tools are the following universal physical principles . Principle of material frame indifference, . Entropy principle, . Thermodynamic stability. Material frame indifference. The Principle of material frame indifference states that the functions T , Q, and E (which reflect material properties) must be the the same for observers in an ↑ inertial (moving with the body) or non-inertial frame (cf. [25, § 6]). To understand the consequences of this postulate we study the effect of ↑ Euclidean transformations: We choose an ↑ inertial frame with coordinates x and a non-inertial frame with coordinates x∗ , specify the relative rotation of the axes by the time-dependent, proper orthogonal10 3 × 3-tensor O and the relative motion of the origins by the time-dependent vector o = (o1, o2, o3). The (antisymmetric) relative angular velocity of the frames is ˙ : O−1 = O ˙ : O> . W := O Any ↑ Euclidean transformation of frame x into frame x∗ and its reverse satisfy x∗ = O(t) : x + o(t)

x = O(t)> : (x∗ − o(t)) .

⇔

(1.3)

Quantities which are frame indifferent are called ↑ objective. To be ↑ objective the transformation of any scalar s, measured in x to its counterpart s∗ in the coordinates x∗ and of any vector or tensor v and T, respectively, to v∗ and T∗ , respectively, must obey s∗ = s ,

v∗ = O : v ,

T ∗ = O : T : O> .

(1.4)

In our calculations we have to treat the scalar quantities ρ, θ, the vectors q, ∇θ, v, and the tensors T and ∇v. We start with considering v and ∇v. For this we differentiate x∗ in (1.3) with respect to time and after that with respect to the space variables and obtain ˙ : x + o˙ , x˙ ∗ = v∗ = O : v + O 10

∇∗ v∗ = O : (∇v) : O> + W ,

i.e. for all t it holds O−1 (t) = O> (t) and moreover det O(t) = 1

(1.5)

1.2




where ∇∗ := ( ∂x∂∗ , ∂x∂∗ , ∂x∂∗ ). The coordinate transformation relating the two observa1 2 3 tional frames to one another transforms the functions T , Q, and E say to T ∗ , Q∗ , and E ∗ depending on ρ∗ , v∗ , θ∗ , ∇∗v∗ , and ∇∗ θ∗ (in particular, (1.4)1 yields ρ = ρ∗ and θ = θ∗ ). On the one hand the Principle of material frame indifference reveals that the functions with and without star applied to the “star-variables” should be the same. Moreover, if we replace the “star-variables” by the formulas (1.3), (1.4)1 and (1.5) expressing them in the non-star-variables the relations should be ↑ objective, i.e they must obey (1.4), namely T ∗ (ρ∗ , v∗ , θ∗ ,∇∗ v∗ , ∇∗θ∗ ) =

= T (ρ∗, v∗ , θ∗ , ∇∗ v∗ , ∇∗ θ∗ ) = O : T (ρ, v, θ, ∇v, ∇θ) : O> ,

Q∗ (ρ∗ , v∗ , θ∗ ,∇∗ v∗ , ∇∗θ∗ ) = = Q(ρ∗ , v∗ , θ∗ , ∇∗ v∗ , ∇∗θ∗ ) = O : Q (ρ, v, θ, ∇v, ∇θ) ,

(1.6)

E ∗ (ρ∗ , v∗ , θ∗ ,∇∗ v∗ , ∇∗θ∗ ) = = E(ρ∗, v∗ , θ∗ , ∇∗ v∗ , ∇∗ θ∗ ) = E (ρ, v, θ, ∇v, ∇θ) . The relations (1.6) must hold for any O(t) and o(t). This restricts the possible form of our constitutive relations. In particular we can choose an instant of time where O = I,

˙ = 1 (L> − L) , O 2

˙ : x. o˙ = −v − O

After a short calculation with this O and o one detects that none of the functions in (1.6) can depend on v and the dependency on ∇v concerns only its symmetric part, namely D := 12 (L + L> ). This simplifies (1.6). In fact, now indeed inserting relations (1.3), (1.4)1 and (1.5) into (1.6) we see that for any proper orthogonal O it holds T (ρ, θ, O : D : O> , O : ∇θ) = O : T (ρ, θ, D, ∇θ) : O> ,

Q(ρ, θ, O : D : O> , O : ∇θ) = O : Q (ρ, θ, D, ∇θ) ,

(1.7)

>

E(ρ, θ, O : D : O , O : ∇θ) = E (ρ, θ, D, ∇θ) .

In particular, (1.7) states that only ↑ objective scalar combinations of D and ∇θ can occur as variables. All these combinations can be reduced to a finite number of independent variables for the scalar, vectorial and tensorial case. Namely, scalars depend on ρ , θ , tr D , tr (D2) , tr (D3) , (∇θ)2 , ∇θ : D : ∇θ , ∇θ : D2 : ∇θ ,

(1.8)

and the vector q is prescribed through q = −κ ∇θ + a D : ∇θ + b D2 : ∇θ .

(1.9)

The scalars κ, a, b may depend on the items in (1.8). Now we assume, that the moment of the momentum is conserved. Then the tensor T is symmetric. Still for T there is a very complicated pattern (cf. [99, formula (1.17)3]) but if we neglect possible dependencies11 11

Fluids for which T does not depend on θ and ρ are called Stokesian fluids.

1



Aspects of physics

of T on the temperature gradient the constitutive relation becomes T = −pI + ω1D + ω2D2.

(1.10)

Here p and ωi (i = 1, 2) may depend on ρ, θ and the invariants12 of D, namely tr D, tr (D2) and tr (D3) (as stated in (1.8)). Often the changes in velocity and temperature are small and thus, all nonlinear terms in the velocity and temperature gradients can be dropped. This defines the so-called NavierStokes-Fourier fluids. In particular, T = −pI + ω1D

with

p = p(ρ, θ, tr D) , ω1 = ω1(ρ, θ) ,

and the most general constitutive relations compatible with the Principle of material frame indifference, simplify as follows (see also [126, §7]) e = e|E(ρ, θ) + β(ρ, θ) tr D , q = −κ(ρ, θ)∇θ ,

(1.11)

T = − p|E(ρ, θ) + λ(ρ, θ) tr D I + 2µ(ρ, θ) D . Here the subscript |E stands for “in equilibrium” (a state with uniform temperature and velocity), κ for the heat conductivity. The coefficients µ and λ are shear and ↑ bulk13 viscosity. The factor β will turn out to be zero and gets no name. Later on we will consider fluids which are mechanical incompressible, i.e. ρ does not depend on p. Under these conditions one defines p|E as mechanical or mean pressure (see [126, §7], [25, §7]) i.e. 1 p|E := − tr T . (1.12) 3 ^ := T + p|EI and T from (1.11) it must hold This means that for T ^ = 0 = (3λ + 2µ) tr D , tr T which is ensured by choosing λ = − 32 µ. Entropy principle. The entropy principle expresses our experience that thermodynamic processes are dissipative and irreversible: We call something dissipative if it transfers energy to waste-heat. In all motion some of the energy involved unalterably loses its ability to do work and is degraded in quality. The famous Second Law of Thermodynamics amounts to saying that, if something is isolated from the rest of the world, it will dissipate all the free energy it has. Equivalently, it maximises its ↑ entropy η and thermal equilibrium is the state of maximum ↑ entropy. More strictly spoken this means: 12

Often (tr D)2 − tr (D2 ) and det D are chosen instead of tr (D2 ) and tr (D3 ). Bulk viscosity stands for different concrete compressibility effects. A typical mix is from [92, p.70]“The equations expressing the continuity of mass and momentum for fluids of zero bulk viscosity ... are ..., where η > 0 and ζ > 0 denote the coefficients of viscosity. ... The dynamic and bulk viscosities η and ζ are assumed to be constant.” (η is our µ and ζ our λ). 13

1.2




(1) There exists an additive ↑ objective scalar quantity η called ↑ entropy. A body possesses ↑ entropy as it possesses mass and energy and the corresponding balance law reads ∂t(ρ η) + div (ρ η v + Φ) = σ . Here Φ = (Φi)3i=1 is the entropy flux, an ↑ objective vector and σ is the density of entropy production, an ↑ objective scalar.

(2) The specific entropy, the entropy flux and the density of entropy production are given by constitutive equations which fulfil the requirements of material frame indifference. For Navier-Stokes-Fourier fluids this means that (compare the scalar and vector valued case in formula (1.11)) η = η|E(ρ, θ) + h(ρ, θ) tr D ,

Φ = ϕ(ρ, θ)∇θ .

(3) The entropy production is non-negative for all processes, i.e. there holds the entropy inequality σ = ∂t(ρ η) + div (ρ η v + Φ) ≥ 0 . (1.13) (4) The temperature is continuous across walls not producing ↑ entropy.

We intend to clarify the possible form of constitutive relations via the entropy principle. First we exploit the entropy inequality (1.13). It has to hold for solutions to the field equations, i.e. the field equations are constraints for the fields satisfying the entropy inequality. A standard procedure to deal with constraints is to define Lagrange multipliers, namely Λρ, Λvi , and Λe, respectively, and treat ∂t(ρ η) + div (ρ ηv + Φ) − Λρm − Λvj Mj − ΛeE ≥ 0 ,

(1.14)

where m, M, and E are defined in (1.1). Thus, to simplify the constitutive relations first we complicate things by introducing the new quantities η, σ, Φ, Λρ, Λvi , and Λe. Let us focus our interest to Navier-Stokes-Fourier fluids. We insert their constitutive relations in (1.14) and carry out all differentiations. Then on the left-hand side of (1.14) there occur terms which are explicitly linear in certain derivatives. Since that inequality has to hold in any point for arbitrary values of the derivatives the factors of these derivatives must vanish. Otherwise it would be possible to violate (1.14). We omit the calculations and state the results 1 Λρ = Λρ1(ρ, θ) + Λe |v|2 , Λe = Λe(θ) , 2 Φ = Λeq , e = e|E(ρ, θ) , η = η|E(ρ, θ) . Thus, inequality (1.14) is reduced to (we set dθ :=

d ) dθ

p|E ρ e σ = −κ |∇θ| dθΛ + ρ η|E − Λ e|E + − Λ1 tr D+ ρ + Λe λ (tr D)2 + 2µ D · D ≥ 0 . 2

(1.15)

e

(1.16)

1



Aspects of physics

Now we think about the equilibrium. We notice, that σ is a function of ρ, θ, |∇θ|2, and D. Of course, σ shall be minimal for equilibrium. Necessary conditions for this are on the one hand ∂∂i θσ|E = 0, ∂Dij σ|E = 0. The first requirement is satisfied identically, the second implies Λρ1 = η|E + Λe(e|E + p|ρE ) such that (1.16) simplifies as follows σ = −κ |∇θ|2dθΛe + Λe λ (tr D)2 + 2µ D · D . (1.17)

On the other hand the second order derivatives of σ must form a positive definite tensor which has the following consequences: κ dθΛe ≤ 0 ,

Λeµ ≥ 0 ,

Λe(λ + 32 µ) ≥ 0 .

(1.18)

Still we have one Lagrange multiplier in (1.17). Now we utilise the forth statement of the entropy principle: the jump of the temperature across an ideal wall (i.e. a wall not producing ↑ entropy) is zero. This does not seem very useful to get rid of Λe. But after some calculation (see [99, Section 1.3.2.4]) one finds that on the contrary this condition implies the fact that Λe is always the same on both sides of an ideal wall - this means for any material it is a universal function of θ. Thus, we can choose one material to evaluate Λe, for example an ideal gas, and find Λe = θ1 . Then (1.15)2 and (1.18) yield Φ=

1 q θ

while

µ,κ ≥ 0,

λ + 32 µ ≥ 0 ,

and moreover that the functions in equilibrium are connected via the Gibbs equation dη|E = θ1 de|E + p|Ed( ρ1 ) , which implies ∂1/ρe|E = θ ∂θp|E − p|E (1.19)

as integrability condition.

Thermodynamic Stability. Thermodynamic stability stands for the experience that the ↑ entropy tends to a maximum as the body tends to equilibrium, i.e. to a configuration where all fields become steady-state. This is a consequence of the nonnegative ↑ entropy production stated by the entropy principle. In general a stability criterion determines how a process develops from a given initial state under certain constitutive equations. Nevertheless, in nature - namely, if forces do not depend on time - we observe that processes tend to equilibrium with constant and uniform fields whatever initial condition we start with. Thus, we exclude constitutive relations for which equilibrium cannot be established from arbitrary initial conditions. Let V be a control volume and n the outer unit normal vector on the boundary ∂V of V. We formulate the balance of ↑ entropy in integral form as follows Z I d ρ η dx + Φ · n ds ≥ 0 . (1.20) dt V ∂V Inequality (1.20) has to hold together with the balance laws (1.1) in integral form: Z Z I d d ρ dx = 0 , ρ v dx − T : n ds = 0 , dt V dt V ∂V Z I (1.21) d 2 1 ρ(e + 2 |v| ) dx + (q − T : v) · n ds = 0 . dt V ∂V

1.2




The first aim is to eliminate the surface integral terms between the relations (1.20) and (1.21) if this is possible. For example, we prescribe Φ = q/θ as for Navier-Stokes-Fourier fluids. Then we assume that the surface of V is ↑ adiabatic, i.e. the normal component of the heat flux vanishes. If in addition ∂V does not move (1.20) and (1.21)1,3 reduce to Z Z Z d d d ρη dx ≥ 0 , ρ dx = 0 , ρ (e + 21 |v|2) dx = 0 . (1.22) dt V dt V dt V Relations (1.22) expressed in words mean that given a body with Φ = q/θ and an ↑ adiabatic, fixed boundary its ↑ entropy tends to a maximum under the constraints of constant mass and energy. Since a force must act to fix the surface, in general the momentum is not constant. Note, that any stability criterion depends on the chosen environment. Now, let us consider a Navier-Stokes-Fourier fluid inside a cylinder. We imagine that a . movable heat-conducting piston is the boundary between two fractions of equal mass m 2 Initially everything is at rest, the specific internal energies at this moment in both parts are e1, e2, and the densities ρ1, ρ2, respectively. All these values shall be uniform but may be different in the two parts. Our experience and the entropy principle suggest that the fluid should tend to an equilibrium in which the corresponding specific internal energy eE and density ρE are uniform throughout the cylinder. Due to conservation of mass, energy, and volume we have e1 − eE = −(e2 − eE) and V1 − VE = −(V2 − VE). Thus, the entropy inequality becomes m η(eE, vE) ≥

m 2

η(e1, V1) +

m η(e2, V2) . 2

(1.23)

If the initial state is already near equilibrium it is useful to expand the right-hand side of (1.23) into a Taylor series about the equilibrium. To be clear we subscript quantities which remain fixed during derivation, i.e. we write ∂2e η V for the second (partial) derivative of η with respect to e for constant V. The result then reads 0 ≥ ∂2e η V (e1 − eE)2 + 2 ∂2eV η (e1 − eE)(V1 − VE) + ∂2V η e (V1 − VE)2 .

To ensure this inequality for all values of (e1 −eE) and (V1 −VE) the tensor of coefficients in this quadratic form must be negative semi-definite. This shall lead to restrictions for the constitutive relations for p and e. Via the Gibbs equation (1.19) we can reformulate this thermodynamic stability condition in terms of θ as follows: the tensor ! − θ12 (∂eθ)V − θ12 (∂V θ)e − θ12 (∂V θ)e ∂V ( pθ ) e

must be negative semi-definite. Yet, it is not very clear what this requirement means. For that it is again restated with the help of the Gibbs equation (1.19) and some calculations to end up with the following relation − (∂θ e)V (θ1 − θE)2 + θ (∂V p)θ (V1 − VE)2 ≤ 0 .

1



Aspects of physics

This inequality is valid if the following two conditions with clear physical meaning are fulfilled 1 cV := (∂θe)V ≥ 0 , βθ := − (∂pV)θ ≥ 0 . V The coefficient cV is called ↑ specific heat at constant volume while βθ is the isothermal compressibility. As well the specific heat at constant pressure cp := (∂θe)p may be used. With some more calculations we find that due to our stability condition the thermal expansion coefficient α := V1 (∂θV)p has the upper bound α2 ≤

1 cp , V θβθ

i.e. without compressibility there is no thermal expansion. Incompressibility. There are different meanings of the notion incompressibility. If we choose p instead of ρ to be an independent variable for Navier-Stokes-Fourier fluids we may resume the following constitutive relations ρ = ρ(p, θ) ,

q = −κ(p, θ)∇θ ,

η = η(p, θ) ,

e = e(p, θ) ,

T = −pI + λ(p, θ)(tr D) I + 2µ(p, θ)D ,

Φ=

1 q, θ

(1.24)

and the corresponding Gibbs equation reads 1 dη = θ

p 1 p (∂θ e)p − 2 (∂θ ρ)p dθ + (∂p e)θ − 2 (∂p ρ)θ dp , ρ θ ρ

(1.25)

which implies the integrability condition p θ (∂θ ρ)p + 2 (∂p ρ)θ . 2 ρ ρ

(∂p e)θ =

(1.26)

Let us define incompressibility as: no constitutive relation depends on p. Then obviously, (1.26) reduces to ∂θ ρ = 0 and thus, ρ is constant. The conservation of mass leads to the relation div v = 0, i.e. tr D = 0, and summarising we find ρ = const. ,

e = e(θ) ,

η = η(θ) ,

T + pI = 2µ(θ)D ,

1 q, θ q = −κ(θ)∇θ .

Φ=

(1.27)

Due to equation (1.25) we have to fulfil dθ η = θ1 dθ e. The system of partial differential equations which is generated by inserting (1.27) into the Balance laws does not contain ∂tp. Thus, one must solve the initial value problems for v and T and the pressure field is adjusted so as to fit the boundary values of the stress components. Later on we will see, that the right way to express compressibility/incompressibility of a given material is a central question for modelling fluid flow and it has different answers depending on the situation which is to be treated.

1.3

The Bénard problem



% (

1.3 The Bénard problem Nature of the problem. Consider an infinite horizontal layer of viscous fluid inside a gravitational field. The fluid shall initially be at rest. Its underside is heated. This yields a temperature difference between the top and the bottom of the fluid. The corresponding thermal expansion makes the fluid at the bottom lighter than the fluid at the top. This top-heavy arrangement is potentially unstable and there will be a tendency to redistribute itself to come to a more stable instant. Our everyday experience says that sooner of later convection will do this job. Nevertheless, first one has to take into account two more possibilities of energy dissipation in the fluid, namely the ↑ viscosity of the fluid which prevents fast motion (more precisely it prevents the change of momentum, i.e. acceleration of the fluid packets is made difficult) and heat conduction. Only if the temperature difference is big enough to rule out ↑ viscosity, and conduction processes, convection will occur. This point is measured by the so called critical Rayleigh number defined in (1.28) below. Then, after the onset of convection, the complexity of the flow is enhanced further by the fact that the temperature distribution is affected to a large extent by the convective flow itself, which carries heat from the bottom to the relatively colder top portion of the fluid. In other words the driving force which causes the flow itself is driven to modifications by the flow. The earliest experiments to demonstrate in a definitive manner the onset of thermal instability in fluids are those of Bénard in 1900. He studied a seemingly simple convective system which he never knew was so complicated that the real physics behind it was uncovered only recently. For an encyclopaedic survey of Bénard’s experiments and results we refer to [75, Chapter 1]. The phenomenon of thermal convection itself had been recognised even earlier by Count Rumford (1797) and James Thomson (1882), [126]. One can even go back to 1790 when Count Rumford used convection to account for the transfer of heat in an apple pie. Only in the 20th century mathematical rigorous investigations were undertaken. Still most famous are the results of John William Strutt, i.e. Lord Rayleigh, [75, Section 2.1]. In one of his last articles, [117], published in 1916, he tried to explain what is called Rayleigh-Bénard convection nowadays. The main assumption in Rayleigh’s theory [117] are: . The fluid itself is incompressible. . The density of the fluid is the only property that gets affected by the change in the temperature across it. . The material experiences uniform gravitational force on the entire volume. . The fluid has a constant temperature at the top and the bottom. The experimental conditions Bénard employed were different in the sense that the fluid layer is not fully confined between two horizontal rigid plates, but is open to air in the

1



Aspects of physics

upper surface. Since the surface is free, surface tension forces affect the flow, which can dominate over even the ↑ buoyancy force. ”As concerns the physics of phenomena, however, combining both names in one term reflects a longstanding confusion in the comprehension of the mechanism of convection, which has not yet been completely overcome: Bénard observed a phenomenon in which instability due to the temperature dependence of the surface-tension coefficient played a substantial role whereas Rayleigh studied convection caused by an instability of another type, which arises from the temperature and density nonuniformity of the fluid layer. The difference between these two mechanisms manifests itself in differences between the flow patterns.”

Getling [44, p.1]

Though in this sense Lord Rayleigh’s explanation was superceded in later years, his work remains as the starting point for most of the modern theories of convection. Convection.14 Consider a fluid packet at the bottom of the layer with heat supplied to it from below. This packet has a higher temperature and thus, less density when compared to the average density of the entire layer. A similar packet at the top side will have relatively higher density due to its lower temperature. In the bottom side, as long as the fluid packet remains in its position it is surrounded by fluid of the same average density and so maintains its static equilibrium with the surrounding. Suppose now due to some random fluctuation a displacement is given to the packet in the upward direction. This will result in an imbalance in the forces acting on the packet. The packet which is originally of less density than the surrounding average density due to its higher temperature now is pushed up into a region of higher density. This creates a positive ↑ buoyancy which causes the packet to raise. The raise will be sustained till the density of the fluid packet while raising equals that of the surrounding. At this point it will simply float as the static equilibrium is restored. The upward force is proportional to the density difference and volume of the packet. As the fluid packet raises through regions of relatively colder fluid whose average density progressively increases due to the lack of additional heat, it results in an increased density gradient between the packet and the surrounding which accelerates the raise. On similar analysis the downward push of a packet of fluid makes it enter a region of less average density resulting in the “heaviness” of the packet thus propelling it down. It would sink getting its initial disturbance enhanced. Thus the whole of the fluid layer is eventually overturned resulting in a circulation of the fluid between the hot and cold ends. It seems from this analysis that convection will be observed in a fluid region whenever there is a temperature gradient, however small it may be. But such sensitive dependence of the initiation of the flow on the temperature gradient is not observed in actual circumstances. There seems to be a cut-off value, of some variable which governs the phenomenon, beyond which only convective flow results. This was, characteristically, explained by Lord Rayleigh. The onset of convection has to take into account two more modes of energy dissipation in the fluid, namely two more forces. One of our initial assumptions is that the fluid is not 14

This explanation follows the line of [100]. See also [11, 44, 59, 112, 126].

1.3




subjected to any external influence which might induce motion. So when the fluid tries to move, or circulate, it does so with minimum velocity. When the fluid packet moves, its motion is impeded by the “viscous drag” between it and the surrounding fluid. Viscosity, as we know, is internal fluid resistance offered to a change in the momentum. It is like a frictional force acting in the opposite direction of motion. In our fluid packet, this acts against the ↑ buoyancy force and tries to impede motion. If the magnitude of the viscous drag force equals the ↑ buoyancy force, motion will cease. The second dissipative effect is conduction. In a real non-adiabatic situation the fluid packet is displaced into a cooler surrounding due to the ↑ buoyancy force. This immediately causes the packet to diffuse out the heat energy to the surrounding fluid as a temperature difference prevails. So if the local temperature difference is reduced by heat diffusion it results in a reduction in the ↑ buoyancy force. It is necessary that the ↑ buoyancy force, which is the result of the temperature gradient, must exceed the dissipative forces of viscous drag and heat diffusion to ensure the onset of convective flow. To realise better the influence of these forces on the onset of convection, these forces are expressed as a nondimensional number called the Rayleigh number which is the buoyant force divided by the product of the viscous drag and the rate of heat diffusion: Ra :=

gαρϑL3 , κµ

(1.28)

where g is the acceleration due to gravity, α is the coefficient of thermal expansion, ϑ the temperature difference between hot and cold end, L the width (vertical distance between the walls), κ the thermal diffusivity, and µ the ↑ viscosity of the fluid. Convection sets in when the Rayleigh number exceeds a certain critical value. When instability sets in, the hot layer tries to go up simultaneously when the cold upper layer tries to come down. Both things will not happen at the same time and the fluid avoids this by separating itself into a pattern of convective cells. In each cell the fluid rotates in a closed orbit and the direction of rotation alternates with successive cells. This roll when viewed in cross section resembles a bloated square the height of which is being determined by the width of the fluid layer. The situation is different if only conduction is present. The corresponding so-called hydrostatic solution (vh, ph, θh) is easily calculated. By definition it has a zero velocity field while the temperature decreases linear from the value θb at the bottom x3 = 0 to the top x3 = L, i.e. vh = (0, 0, 0) ,

θh := θb − Lϑ x3 ,

∇ph = ρα(θb − Lϑ x3)(0, 0, −g)> .

(1.29)

Of course this conduction solution exists for all values of the temperature gradient. Thus, the problem of describing convective motion mathematically formulated is one of non-uniqueness. Namely, one must show that on exceeding a critical temperature gradient new steady solutions of the equations corresponding to convection branch or bifurcate from the conduction solution. For three space dimensions theory provides a lot of possibilities to choose different patterns [19, Fig.2.2], the easiest being rectangular cells as sketched in the figure below.

1



Aspects of physics

Nevertheless, what one can really observe in experiments are two-dimensional rolls.

Figure 1.5. Convection rolls and rectangular cells

Oberbeck-Boussinesq approximation. The exact equations governing ↑ buoyancy driven flows are complicated and difficult to study. The best reputed way to simplify the balance equations (1.1) valid for convective flow processes is the Oberbeck-Boussinesq approximation, which in principle15 makes the following assumptions: . Incompressibility: div v = 0 . . ∇v is sufficiently small in a suitable norm, such that the effect of internal friction (or dissipative heating) D : D can be neglected. . The density is linear in the following sense: There is a constant ρ0 > 0 such that ρ = ρ0 (1 + α(θb − θ))

with

1 0 < α = − ∂θ ρ . ρ0 θ=θb

. ρ is almost constant around ρ0, except in the term with the weight; only in this term the density ρ is maintained, since it originates ↑ buoyancy. . All other fluid properties are constant. These assumptions are embodied by the treatment of Rayleigh as well. Applying the Oberbeck-Boussinesq approximation one usually asks that L is small in comparison to the horizontal dimension of the flow, i.e. the layer shall be thin. A sound argument for this is given by means of the hydrostatic solution: R ρθ , where R is the Let us insert the equation of state for perfect gas, namely, p = M Boltzmann constant and M molar mass, into the mass balance. One finds d Rϑ − MgL ln ρ = dx3 R(Lθb − x3ϑ)

⇒

ρ(x3) = ρ0 1 −

x3ϑ λ Lθb

with λ = −(1 − MgL ). Thus, for gravity-buoyancy-driven convection the density must Rϑ be growing with the height and decreasing with the temperature. But on the contrary our experience is that air is colder but less dense on the mountains. This confirms that the model is appropriate only for bounded layers under certain conditions. 15

There are several possibilities to formulate these assumptions, some formulations having slightly different effects to the equations of motion. We will return to this subject in more detail in Section 2.1.

1.3




Stability. The classical stability analysis helps to find candidates for convective solutions. The starting point is the Oberbeck-Boussinesq approximation as system approximately prescribing the flow in the Bénard layer, namely [44, Sect. 2.1], [112, §2], [126, §8] (for a different formulation see [57]) ρ0 v˙ − µ ∆v + ∇p = ρ0 g (1 + α(θb − θ))(0, 0, 1)> , ρ0 cp θ˙ − κ∆θ = 0 ,

(1.30)

div v = 0 . Since we are looking for steady solutions, the time dependence is cancelled and then new variables (with the same notation) are introduced which stand for the deviation of the convective solutions from the trivial conduction solution (1.29). In particular the boundary conditions for the new quantities are v = 0,

θ = 0 at top and bottom, periodic elsewhere with periodic cell C.

(1.31)

Then one chooses a dimensionless form for all quantities, linearises the equations about the trivial solution (1.29) and finds the system (see also [23, 59, 112]) div v = 0 ,

∆v − ∇p + Ra θ (0, 0, 1)> = 0 , ∆θ + v3 = 0 .

(1.32)

In (1.32) the Rayleigh number is an eigenvalue. Thus, the problem to find convective solutions is transformed into the question for the smallest Ra to which there exist nontrivial solutions to (1.32), the critical Rayleigh number Rac. One can easily reformulate this question in terms of a minimisation problem (cf. Lemma 1 in [112]). 1.1 Lemma. Equations (1.32) are the Euler equations of the variational problem R |∇v|2 dx RC = min (1.33) |∇θ|2 dx C

for the side conditions div v = 0, ∆θ + v3 = 0 and the boundary conditions (1.31).

Proof. Let Rac be the minimum in (1.33), realised by the functions v and θ. Then let vδ and θδ be admissible variations of v and θ. In particular this means that div vδ = 0 and ∆θδ + (vδ)3 = 0. Starting with the condition that the calculus of variations provides we calculate by partial integration and using the side conditions Z Z 0 = (∇v · ∇vδ − Rac∇θ · ∇θδ) dx = (−∆v · vδ + Racθ · ∆θδ) dx CZ ZC = − (∆v · vδ + Racθ(vδ)3) dx = − (∆v + Rac(0, 0, θ)>) · vδ dx . C

C

Since vδ is arbitrary, ∆v + Rac(0, 0, θ)> is orthogonal to all solenoidal functions, and hence it must be a gradient [40, Thm. III.1.1]. For that we end our proof by obtaining ∆v + Rac(0, 0, θ)> = ∇p .

1



Aspects of physics

Another way to understand how Rac enters the stability analysis is to consider the socalled first energy estimates (more details follow in Section 3.1). Namely, the first two equations in (1.32) are multiplied by v and θ, respectively. Then one integrates over the periodic cell C and partial integration provides Z Z 2 2 k∇vk2 := |∇v| dx = Ra θ v3 dx , C C Z Z 2 2 k∇θk2 := |∇θ| dx = θ v3 dx . C C Z 2 2 ⇒ k∇vk2 + k∇θk2 = (1 + Ra ) θ v3 dx ≤ c( Ra )(k∇vk22 + k∇θk22) . C

The last estimate follows via Hölder, Young and Poincaré inequality (see Section 3.1). If Ra is small enough such that c( Ra ) < 1 we find (1 − c( Ra ))(k∇vk22 + k∇θk22) ≤ 0

⇒

k∇vk22 + k∇θk22 = 0 .

Together with the boundary conditions this leads to the only possible solution v = 0, θ = 0, which is the hydrostatic one. Note, that the core of the procedure is the constraint c( Ra ) < 1. Consequently, it works whenever Ra < Rac if Rac is defined through c( Rac) = 1. This definition for Rac and the one given in Lemma 1.1 coincide. Obviously, Rac is the point where the first bifurcation from the hydrostatic solution may occur, i.e. it concerns the most simple convective solution. For the more complicated patterns and the transition to turbulence it is of no use. Final Remarks. Though the study of convection seems to be a very traditional, not to say old-fashioned subject till this day the emergence of new techniques has continued to lead to new and powerful insights. In particular, lasers produce precise local velocity data, computing power is greatly increased, and recent theoretical studies of nonlinear systems have provided an important framework for understanding. Many interesting processes involve thermally driven flows. This concerns such diverse phenomena as convection in stars, the ocean and the atmosphere, and the production of pure semiconductors. Moreover, from a fundamental point of view, convective flows provide experimentally realisable systems for very precise studies of nonlinear phenomena in dissipative systems: onset of turbulence and nonlinear pattern formation. Pure hydrodynamics apart, formation of patterns close to spatially periodic ones can be observed in crystal growth, propagation of solidification fronts, electro-hydrodynamic instabilities of nematic liquid crystals, chemical reaction-diffusion processes, autocatalytic reactions, buckling of thin plates and shells, morphogenesis of plants and animals, etc. and the Oberbeck-Boussinesq approximation forms the springboard for all related further discussion. Namely, the presence of nonlinear terms is responsible for a number of interesting effects. In particular, they allow the possibility of multiple solutions, in an analogous way to the multiple roots which occur in nonlinear algebraic equations. Finally, one should always keep in mind that apart from all mathematics the stability of a particular steady flow allowed by the equations is related to the realisability of this flow. Very special initial conditions may be needed for some stable final state to be reached, and it may turn out that such conditions do not exist at all.

 $ %&& (

2 Derivation of Approximations In Section 1.3 already some simplifications of the general equations governing flow processes have been introduced together with the Oberbeck-Boussinesq approximation. Now our aim is to derive systems of partial differential equations which work in the same spirit as the approximation used by Oberbeck and Boussinesq. More precisely, they describe flow processes, where changes in the volume occur due to temperature changes or changes in the specific entropy but not due to applied pressure or body forces. Sometimes this behaviour is called incompressible in the generalised sense [51]. Moreover, effects of ↑ buoyancy shall be important. Namely, in his attempts to explain the motion of the light in the aether Boussinesq (1903) [17] opened a wide perspective of mechanics and thermodynamics. With a theory of heat convection in fluids and of propagation of heat in deforming or vibrating solids he showed that density fluctuations are of minor importance in the conservation of mass. The motion of a fluid initiated by heat results mostly in an excess of buoyancy and is not due to internal waves excited by density variations. In other words, variations of the density can be neglected in the inertial accelerations but not in the buoyancy term. In the rough presentation of Section 1.3 and in most textbooks this idea is reduced to: “The fluid shall be incompressible, i.e. div v = 0, except in the buoyancy term”. Note, that the incompressibility expressed by div v = 0 means that the fluid shall be ↑ isochoric in any process while the temperature dependence of the density is introduced through the “back-door” in the ↑ buoyancy term without any other justification than that it works in applications. Thus, the derivation of conditions for the validity of the Oberbeck-Boussinesq approximation is not as straightforward as one would presume at first glance. Yet, it is the basis for most of what is known about natural convection and other processes where acoustic phenomena as sound propagation can be neglected. In the literature, a variety of sets of conditions have been assumed which, if satisfied, allow application of approximations. In fact it seems much more sound to model convection as an ↑ adiabatic and non-isochoric process (together with an isothermal process on the boundary) and not to start a priori with div v = 0 as a constitutive relation. Of course, hundred years application teach that often to assume a solenoidal velocity field together with the other simplifications is “good enough” and one should try to justify the corresponding equations in some way - also to delimit the range of its validity as most welcome byproduct.

2



Derivation of Approximations

Admittedly, there have been numerous attempts and we refer only to some of them [9, 23, 46, 51, 91, 92, 95, 108, 114, 117, 128]. To receive an impression about typical ideas let us have a closer look to [128, 95, 46, 51] (in chronological order). (1) One classical idea is the perturbation scheme in Spiegel/Veronis (1960) [128]. Here the state variables are split into three parts, namely . a mean value, . a variation in the absence of motion, and . a fluctuation resulting from the motion. The small perturbation parameter ε is chosen as ratio of the variation of density without motion across the layer to the space average value for the density. Nevertheless, in [128] the order of magnitude analysis is not consistent in the whole, i.e. during the calculations (which finally reproduce the Oberbeck-Boussinesq approximation) not always terms of the same order are kept or neglected. Some choices merely seem motivated by knowing what equation should be the result. Another shortcoming is that the thickness of the layer must be very small. (2) Mihaljan (1962) [95] uses an expansion in terms of two perturbation parameters. The background for such arguments is to assume that if the parameters tend to zero one arrives at the usual incompressible case. Nevertheless, as one of the parameters chosen in [95] tends to zero, the other one becomes infinite (see explanation in [114]) which is at odds with the idea of perturbation schemes in general. Cordon and Velarde (1975) [107] tried to overcome this problem by choosing a suitable ↑ adiabatic hydrostatic reference field to end up with two parameters of the same order of magnitude. (3) Another idea has been put into practice by Gray and Giorgini (1975) [46]. There all quantities (including the material parameters as ↑ viscosity, thermal conductivity etc.) are replaced by their first order Taylor expansion which allows to represent their variation with temperature and pressure in a first order approximation. Then nondimensionalisation and an order of magnitude analysis give reasons to neglect certain terms in an ad hoc manner. At the same moment this procedure fixes the range for which the approximate equations hold. (4) Hills and Roberts (1991) [51] gave a new approach in the context of generalised incompressible fluids. These fluids are characterised by the inner constraint that their density does not depend on the pressure but is changed only due to temperature or specific entropy changes, i.e. div v = αθ˙ . (2.1) The constraint (2.1) is also used in the approach of [114] and here as well. In [51] then the double limit g → ∞, α → 0 is taken while the Rayleigh number shall be of order 1. This procedure seems to be equivocal and has been a serious drawback of earlier contributions, which has not been overcome either in [51]. To arrive at the Boussinesq approximation in addition the layer must be assumed to be thin and the dissipation to be small a priori. The Boussinesq approximation has proved to be suitable if in the mean accelerations promoted in the fluid by variations in the density are much less than the acceleration of gravity. Moreover,

 ”... observational evidence in laboratory systems and in astrophysics and space plasmas points to the potential importance of certain compressibility effects, especially density fluctuations, even when incompressibility appears to be an otherwise good approximation.” Matthaeus & Zank [92]

The first and main question for the derivation of approximations is, how to introduce constitutive assumptions which reflect these typical properties in the most simple and clear way. In this chapter first we reveal the approach developed in [114]. The general outline is fixed in the next paragraph with Figure 2.1 and related explanations, while Section 2.1 provides all details of the procedure. It yields a rigorous justification of the Oberbeck-Boussinesq approximation together with measurable quantities providing bounds for its applicability. The key is to express the incompressibility by (2.1) and to find the right perturbation parameter. Our main new contribution which is performed in Sections 2.2 – 2.4 is to extend and apply this procedure to problems, for which the assumptions of the Oberbeck-Boussinesq approximation are fulfilled only partly, namely the manner of incompressibility shall be the same but . dissipative heating might not be negligible, . the stress-strain relation is not linear but obeys a power-law. More precisely, we measure the influence of dissipative heating and deduce generalisations of the Oberbeck-Boussinesq approximation model for ↑ Newtonian flow in Section 2.2 and fluids with power-law ↑ rheology in Section 2.3 and 2.4, respectively.

In any case, one should not forget that ambiguities are inherent to the problem since we want to take into account Thermodynamics when studying a macroscopic fluid model, i.e. any particle of fluid, which should be a point in space, is actually a box of gas or liquid, with its thermodynamic evolution. There are some very severe conceptual difficulties in reconciling statistical mechanics with classical Thermodynamics. Shortly after Boltzmann claimed to have reduced Thermodynamics to the statistics of molecules, Josef Loschmidt and Ernst Zermelo (among others) pointed out several paradoxes (some claim that the famous Recurrence Theorem16 proven in 1898 by Henri Poincare resolved these paradoxes). Maxwell himself (who was the first to suggest that Thermodynamics might simply be the macroscopic effect of the statistics of molecules) introduced his infamous ↑ Demon, who continues to bedevil physicists today. In 1929, Leo Szilard claimed to have resolved this paradox using Heisenberg’s Uncertainty Principle. In 1982, new insights due to R. Landauer and C. Bennett overturned this paradigm and gave rise to a quite different resolution of Maxwell’s paradox. Finally, according to Landauer, the erasure of a stored bit inevitably dissipates heat and thus increases entropy. In the following we will not contribute to this debate on principles. 16

If a system has a fixed total energy that restricts its dynamics to bounded subsets of its phase space, the system will eventually return as closely as you like to any given initial set of molecular positions and velocities. This apparent contradiction between the behaviour of a deterministic mechanical system of particles and the Second Law of Thermodynamics became known as the ”Recurrence Paradox.”



2


Approximation procedure. We close our introduction to the second chapter with the briefing of the main steps which will be applied in all sections of this chapter (see also Figure 2.1). . Starting point: Balance laws (2.2) and the Second law of Thermodynamics in the form of the Clausius-Duhem inequality (2.3). . To close the system we need constitutive relations for a couple of quantities which obey some universe physical principles introduced in Section 1.2. . The key is to express the incompressibility. . If inserted into the mass balance this directly leads to an equation for the density. . As independent variables we choose v, θ and p. . The next step is to supply constitutive relations for T and q. . Then we express the energy balance in terms of the Helmholtz free energy. . Now one has to choose the appropriate stress-strain relation. For the usual OberbeckBoussinesq approximation it is linear (Newtonian), but in modelling, e.g., the outer earth mantle often a power-law ↑ rheology is chosen (non-Newtonian model). . At this point we arrive at a highly nonlinear system in which no term is neglected. It governs the flow of material which obeys the constitutive assumptions which were introduced. . Now we start the approximation procedure by a nondimensionalisation. The problem is to choose representative values for velocity, time and pressure since they are not obvious from the configuration. . The next non-trivial choice is the small parameter ε which shall serve as perturbation parameter. In accordance with the idea of the Oberbeck-Boussinesq approximation it shall measure compressibility. . With this nondimensionalisation we arrive at a system including dimensionless numbers as the Rayleigh, Grashof, Dissipation and ↑ Prandtl number.

. Now we express all independent variables as power series of ε, insert these series into our nondimensional system and read off a first approximation by balancing the first ε-levels of the series. The approximation shall be a closed system for the partial sums of a certain order of the quantities.

One precisely knows, that terms of the order εk for some k ∈ N are neglected if working with this approximation in place of the full system. This is a known value for the configuration. All calculations in the second chapter are animated by the motto ”Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from use of knowledge gathered in a few observations to predict by logical process the outcomes in many other circumstances.” Truesdell & Noll [101, p.2]



Balance laws

+

Clausius−Duhem inequality

constitutive relations ?

incompressibility

.

div v = α θ

stress

internal energy

T = −p I + T

p e = Ψ − θΨ − α /ρ

T?

linear

nonlinear

equations of motion

nondimensionalisation?

nondimensional numbers

scales

nondimensional system

perturbation parameter ε ?

series in ε

Figure 2.1. Approximation scheme

approximation

2




! " ! ! "

% (

2.1 The Oberbeck-Boussinesq approximation In this section we introduce the framework in which the Oberbeck-Boussinesq approximation is deduced from the full thermodynamic theory for Navier-Stokes-Fourier fluids by means of a perturbation technique proposed by Rajagopal, R˚uzˇ iˇcka and Srinivasa [114]. We present their arguments to provide a basis for the next sections in which this procedure will be adapted and generalised. Constitutive assumptions. Let us repeat the Balance laws which have already been formulated in equations (1.1). Since we are not interested in radiative heating, we drop the corresponding term from the energy balance and use the system17 ρ˙ + ρ div v = 0 , ρ v˙ − div T = ρ b , ρ e˙ − div q = T · L .

(2.2)

It is handy to apply the entropy inequality in the form of the Clausius-Duhem inequality q ≥ 0. (2.3) ρ η˙ + div θ

This is a local form to apply the Second Law of Thermodynamics for simple bodies and ↑ simple materials. To deduce the Oberbeck-Boussinesq approximation one has to provide (2.2) and (2.3) with the characteristic material properties of Navier-Stokes-Fourier fluids (see also Section 1.2). Especially the incompressibility constraint implies several simplifications to the full system (2.2). Precisely, we make the following assumptions. (1) The fluid shall be incompressible in the generalised sense [51]. To understand relation (2.1) we need some more detailed considerations: One can prescribe fluid motion in two ways: firstly in the particle frame, so-called ↑ Lagrangean coordinates, which we denote by X or secondly in the ↑ Eulerian frame x = x(t, X) (details in [25, § 4], [115, Ch. 1]). The velocity field is given by the time derivative of the particle trajectories ˙ X)|X . v(t, X, x) = x(t, The ↑ deformation gradient is the tensor F := ∂Xx. A simple but fundamental identity that can be verified by the definitions and the chain rule is the relation F˙ = LF. Conser17

If T is symmetric, then T · L = T · D.

2.1

The Oberbeck-Boussinesq approximation



vation of mass in the particle frame relates det F to the changes in density. Namely, for incompressible material one has det F = 1 which directly leads to div v = 0. For fluids which are incompressible in the generalised sense det F must be a function ˙ [51, 114]. As usual for our of the temperature and we arrive at the relation div v = αθ, simple approximation scheme we will set α to be constant (though it could depend on the temperature as well). Equation (2.1) and (2.2)1 directly lead to the differential equation αθ˙ = −

ρ˙ . ρ

Integrating this equation one finds an expression for ρ in terms of θ, namely ρ = ρr e−α(θ−θr ) ,

(2.4)

where the constants ρr and θr correspond to some convenient reference state where density and temperature are known. (2) Mechanical pressure. For the stress T we apply (1.12), i.e. ^ := T + pI T

with p := − 31 tr T .

(2.5)

^ is zero. Note, that p in (2.5) is not the thermodynamic Due to this definition the trace of T pressure which would be fixed by an equation of state but stands for the mechanical or hydrostatic pressure and is connected to the (hydrostatic) equilibrium state. Since the density is already determined by (2.4) we treat the mechanical pressure p as the third independent variable, which together with the velocity v and the temperature θ has to be determined from the final system of equations. Namely, inserting (2.5) in (2.2)2 and taking the divergence one arrives at

⇒

^ = div (ρ b) , ˙ + div (∇p − div T) div (ρ v) ^ = ρ div b + ∇ρ · b . ∇ρ · v˙ + ρ div v˙ + ∆p − div ( div T)

^ − ∇p) and In the last term on the right-hand side we replace b via (2.2)2 by v˙ − ρ1 ( div T use (2.1) to calculate div v˙ = αθ¨ + L · L> . Finally we find the following equation for p 1 1 ^ − ραθ¨ − ρL · L> + div ( div T) ^ . ∆p − ∇ρ · ∇p = ρ div b − ∇ρ · div T ρ ρ

(2.6)

(3) For the heat flux we suppose the Fourier law q = −κ ∇θ

(κ is const.).

(2.7)

(4) Now we express e. For this we introduce the specific Helmholtz free energy by Ψ := e − θη .

(2.8)

2




The definition of Ψ combines e, which is the energy required to create a system in the absence of changes in temperature or volume and −θη, the energy one obtains from the system’s environment by heating. Thus, the Helmholtz free energy measures the amount of energy one has to put in to establish the system while the spontaneous energy transfer to the system from the environment is taken into account. A system at equilibrium with fixed external parameters in thermal contact with a heat reservoir has minimal Helmholtz free energy (see also Gibbs equation (1.19)). Using (2.8) we can rewrite (2.3). First we calculate with (2.7) |∇θ|2 1 q − ∆θ . div = κ θ θ2 θ ˙ and find that (2.3) transforms to From (2.8) we deduce that η˙ = − θ1 (Ψ˙ − e˙ + θη) κ |∇θ|2 ρ ˙ ˙ ˙ − ∆θ ≥ 0 . − (Ψ − e + θη) + θ θ θ Multiplying this relation by θ and replacing ρ e˙ via (2.2)3 by T · L + κ ∆θ we gain ˙ +T·L+κ −ρ(η θ˙ + Ψ)

|∇θ|2 ≥ 0. θ

(2.9)

We see that Ψ = Ψ(θ) which means in particular that Ψ˙ = θ˙ ∂θΨ. Moreover, (2.5) leads ^ · L = −pαθ˙ + T ^ · L. Inserting this into (2.9) we obtain to T · L = −p div v + T

2 αp ^ · L + κ |∇θ| ≥ 0 . −ρ η + + ∂θΨ θ˙ + T ρ θ The linearity of this inequality (which must hold for all processes) in θ˙ implies (compare the arguments below (1.14) with respect to the entropy inequality) that the coefficient of θ˙ must be zero, i.e. p η = −∂θΨ − α . ρ That together with (2.8) creates e = Ψ − αθ

p − θ∂θΨ . ρ

Utilising this e the energy balance (2.2)3 transforms to ^ · L, −(ρ θ∂θθΨ + α2θp)θ˙ − αθp˙ − κ ∆θ = T

(2.10)

and the Clausius-Duhem inequality has altered to 2

^ · L + κ |∇θ| ≥ 0 . T θ

(2.11)

2.1




^ shall obey the ↑ Newtonian fluid model (1.24). Since we fixed p (5) The extra stress T as mechanical pressure we observe that λ = − 32 µ. Moreover, direct calculation yields ^ is symmetric we see T ^·L=T ^ · D. Hence, div D = ∆v + ∇ div v and since T ^ = T(θ, ^ D) := − 2 µ (trD) I + 2µ D , T 3 1 ^ · L = 2µ kDk2 − 1 |trD|2 . ^ = µ ∆v + ∇( div v) , T div T 3 3

(2.12)

Employing our assumptions (1) – (5) system (2.2) finally transforms to

ρ = ρr e−α(θ−θr ) , div v = αθ˙ , 1 ρ v˙ − µ (∆v + ∇ div v) + ∇p = ρ b , 3 1 (−ρ θ ∂θθΨ − α2θ p) θ˙ − αθp˙ − κ ∆θ = 2µ kDk2 − |trD|2 . 3

(2.13)

System (2.13) governs the flow of Navier-Stokes-Fourier fluids, which are incompressible in the generalised sense. We set C(θ) := −ρ θ ∂θθΨ. After a lengthy calculation with this definition and the Gibbs equation one sees that C is the ↑ specific heat. The Clausius-Duhem inequality has simplified to the so-called reduced (energy) dissipation inequality (2.11). Nondimensionalisation. The typical situation for the Oberbeck-Boussinesq approximation is the heating of a fluid layer from below in a gravitational field with a certain difference between the temperature θt at the top and θb at the bottom. Let U be the representative velocity, L, t0 and π typical length, time and pressure; ρr is the reference density (used in (2.4)) and g the gravitational acceleration or the acceleration due to any applied body force, respectively. Moreover, we set ϑ := θb − θt ,

Θ :=

1 b (θ + θt) , 2ϑ

c0 := C(ϑΘ) .

(2.14)

This means c0 is the ↑ specific heat at the mean temperature level. We introduce the nondimensional variables: x , L p p := , π x :=

t , t0 C C := , c0 t :=

v , U b b := , g v :=

ρ , ρr θ θ := − Θ . ϑ

ρ :=

We will not choose U, L and t0 independent of one another but t0 := θr := ϑΘ. Note that C(θ) = C ϑ(θ + Θ) . We set

1 ˜ C(θ) := C ϑ(θ + Θ) , c0

ν :=

µ , ρr

Re :=

L . U

LU . ν

(2.15)

Moreover we fix (2.16)

(2.17)

2




The term ν is called kinematic viscosity while Re is the ↑ Reynolds number.

The system of equations for the dimensionless quantities and differential operators is ¯ for θ derived replacing each quantity according to (2.15) - (2.17), i.e. for v we write Uv ¯ we set ϑ(θ + Θ) and so on while the derivatives meet ∇=

1 ∇, L

∆=

1 ∆, L2

∂t =

U 1 ∂¯t = ∂¯t . t0 L

Then (2.13) in terms of the new variables becomes (we skip all bars for convenience): ρ = e−αϑθ , div v = αϑ θ˙ , 1 1 π gL ρ v˙ − ∆v + ∇( div v) + ∇p = 2 ρ b , 2 Re 3 ρrU U ˜ 0UL − α2πULϑ(θ + Θ)p θ˙ − απUL(θ + Θ) p˙ − κ ∆θ = Cc =

(2.18)

1 2νρrU2 kDk2 − |trD|2 . ϑ 3

A priori there is no obvious representative velocity. Nonetheless, natural convection processes are reflected if the gravitational potential energy is completely transformed into kinetic energy. In particular, this means that in the body force term we observe the balance18 [42, 46, 50, 103] gL αϑ ∼ 1 U2

motivating

U :=

Now the perturbation parameter is chosen, namely ε :=

p

gαϑL .

U 1, (gν)1/3

(2.19)

i.e. the process ought to be slow and/or involve big ↑ viscosity. To have a balance between inertial and viscous effects the ↑ Reynolds number should be independent of ε. Thus, the value of L must be of order 1/ε (for fixed material, where ν is the same), that is to say L=

ν2 1/3 Re ν Re ν 1 = Re , = U ε(gν)1/3 ε g

and we observe ε3 U2 = αϑ = . gL Re Finally, we choose π, introduce the abbreviation Υ, and define the ↑ Prandtl number Pr , which measures viscous versus inertial forces: π := ρrgL , 18

Υ :=

1 ρr(ν5g2)1/3 , κϑ

Pr :=

For different choices for U see [19, 50, 57, 94, 108] and the tables in Part C.

c0ν . κ

2.1




Dividing now (2.18)4 by κ we calculate for the coefficients in (2.18)4, that ν2 1/3 αϑ 1 3 1 1 = ε2 Re Υ , απUL = ρrgLν Re = ε ρrg Re κ κϑ κϑ ε g 3 1 2 ε 2 α πULϑ = ε Re Υ = ε5Υ , κ Re 1 1 1 νρrU2 = νρrαϑgL = απUL = ε2Υ , κϑ κϑ Re κ and the equations (2.18) become with our abbreviations ε3

ρ = e− Re θ ,

div v =

ε3 ˙ θ, Re

ε3 1 ε3 ρv˙ − ∆v + ∇( div v) + ∇p = ρb , (2.20) 2 Re Re 3 |trD|2 5 2 2 2 ˜ ˙ Pr Re C − ε Υ(θ + Θ)p θ − ε Re Υ(θ + Θ)p˙ − ∆θ = 2Υε kDk − . 3 Approximation scheme. The variables v, θ and p are expanded into power series with respect to the perturbation parameter ε in the form v=

∞ X

k

ε vk ,

θ=

k=0

∞ X

k

ε θk ,

p=

k=0

∞ X

εkpk .

(2.21)

k=0

The boundary conditions are attached to the respective zero order term and consequently, the higher order terms have to fulfil zero boundary conditions. In system (2.20) all quantities are replaced by these power series and it is assumed that all of them are as smooth as necessary. We analyse the first ε-powers (or “ε-levels”) to read off the simplest approximate system. From equation (2.20)2 it is obvious that div v0 = div v1 = div v2 = 0 . Hence, the term ∇ div v in (2.20)3 will give no contribution in our analysis at the levels ε0–ε5. Analogously, in (2.20)4 the terms ε5Υ(θ + Θ)pθ˙ and tr D will not appear at ε0–ε4. For the density ρ we use a Taylor series of relation (2.20)1, namely ρ=1−

ε3 θ + o(ε6) . Re

(2.22)

Inserting (2.21) and (2.22) the velocity equation becomes ∞ ∞ X X ε3 ε3 X k 1− ε θk + o(ε6) εk ∂tvk + (vl · ∇)vm − Re Re k=0 k=0 l+m=k

∞ ∞ ∞ X ε3 X k ε3 X k 6 3 k ε ε θ + o(ε ) b. ∆v + o(ε ) + ε ∇p = 1 − − k k k Re 2 k=0 Re k=0 k=0

2




Namely, at ε0 we note ∇p0 = b . Therefore the body forces must have a potential to ensure the solvability of this equation. Nevertheless, with regard to applications this is no serious restriction. For ε1 and ε2 we see ∇p1 = ∇p2 = 0, which together with the zero boundary conditions means p1 = p2 = 0 . Finally, for ε3 and ε4 we observe 1 ∆v0 + Re ∇p3 = −θ0b , Re 1 ∂tv1 + (v0 · ∇)v1 + (v1 · ∇)v0 − ∆v1 + Re ∇p4 = −θ1b . Re ∂tv0 + (v0 · ∇)v0 −

(2.23)

˜ is smooth enough to use a Taylor series, i.e. Next we analyse (2.20)4. C ˜ ˜ 0) + (θ − θ0)C ˜ 0 (θ0) = C(θ ˜ 0) + εC ˜ 0 (θ0) C(θ) ∼ C(θ

∞ X

εk−1θk ,

k=1

˜ 0 := ∂θC ˜ and we obtain where C ∞ ∞ X X X 0 k−1 2 ˜ ˜ εk ∂tθk + vl · ∇θm − Pr Re C(θ0) + εC (θ0) ε θk + o(ε ) k=0

k=1

l+m=k

∞ ∞ ∞ X X X X 2 k k −ε Re Υ ε θk + Θ ε ∂tpk + vl · ∇pm − εk∆θk = k=0

k=0 ∞ X

= 2Υε2 D(

k=0

l+m=k ∞ X k

εkvk) · D(

k=0

1 εkvk)|2 . ε vk) − |tr D( 3 k=0 k=0 ∞ X

Let θ˙ 0 := ∂tθ0 + v0 · ∇θ0. We see for ε0, that ˜ 0)θ˙ 0 − ∆θ0 = 0 , Pr Re C(θ

(2.24)

and ε1 raises ˜ 0)(∂tθ1 + v0 · ∇θ1 + v1 · ∇θ0) + C ˜ 0 (θ0)θ1θ˙ 0 − ∆θ1 = 0 . Pr Re C(θ

(2.25)

We observe that v1 = 0, θ1 = 0, p4 = 0 fulfil the equations (2.23) and (2.25) and have zero boundary values. Due to our smoothness assumptions these are the unique solutions. Finally, we set v := v0, θ := θ0 and p := p0 + ε3p3 and obtain the Boussinesq system div v = 0 , v˙ −

1 Re ∆v + 3 (∇p − b) = −θ b , Re ε ˜ θ˙ − ∆θ = 0 . Pr Re C(θ)

(2.26)

2.2

Natural convection with dissipative heating



From our procedure it is clear that in (2.26) both v and θ are precise of order ε and p of order ε4 since v1 = 0, θ1 = 0, p4 = 0. More precisely, let (vf, θf, pf) solve the full system (2.13), and (va, θa, pa) the approximation (2.26) then vf = va + o(ε2) ,

θf = θa + o(ε2) ,

and

pf = pa + o(ε5) .

The different coefficients in (2.26) in comparison to (1.30) are mainly due to a different nondimensionalisation (see Part C for the data used in [112]). 2.1 Remark. Let us have a closer look at the velocity equation (2.26)2. We see that the originally dubious assumption of the Oberbeck-Boussinesq approximation: “ρ is constant in all terms except the body force term, where it is linear in θ” has transformed to the fact that we combine two approximation levels. One level is the equilibrium of body forces and ∇p0, the other one is the adjustment between velocity changes (expressed by the differential operators on the left-hand side) and the buoyancy term. One can hide this fact and obscure the way in which the different approximations for ρ enter that equation by setting ∇^ p :=

Re (∇p − b) ε3

⇒

v˙ −

1 ∆v + ∇^ p = −θ b . Re

! " % & (

2.2 Natural convection with dissipative heating One basic assumption in the derivation of the Oberbeck-Boussinesq approximation is that dissipative heating is small enough to be neglected due to small velocity changes. Nevertheless, already for a long time in the literature one has found the supposition that for large domains or other non-standard applications (for example if the material is very different from water) this is not suitable though the processes seem otherwise near to the regime for which the Boussinesq approximation works fine. The crucial idea concerning the effect of dissipation is that ”The effect of viscous dissipation in natural convection is appreciable when the induced kinetic energy becomes appreciable compared to the amount of heat transferred. This occurs when either the equivalent body force is large or when the convection region is extensive.” Gebhart [43, p. 225]

To express this balance Ostrach introduced the so-called Dissipation number in 1957, [103]. It is the ratio of kinetic energy of the fluid to the heat transferred to the fluid and thus, it relates the product of thermal expansion coefficient, acceleration due to the applied body force and length scale to the ↑ specific heat at constant pressure: Di :=

αgL . cp

(2.27)

2




It has been used in a lot of approaches since then [19, 42, 43, 50, 57, 108]. In particular the authors of [19, 50, 57] investigate processes in the earth’s upper mantle, where Di ∼ 0.12 · · · 0.5 (data19 taken from [19, 94], see tables in Part C). This means that dissipative heating influences but does not dominate the corresponding flow processes. For example, local increases in the temperature of the earth’s mantle due to dissipative heating could possibly explain the high heat flow behind island arcs and the variable composition of oceanic basalts erupted at ridges (see [50] and the arguments in Section 1.1). In [9] Bayly, Levermore and Passot follow a very general approach to model the dynamics of interstellar media. Filtering out fast acoustic motions which are evidently compressible, they investigate density variations in weakly compressible flows. A collection of incompressible approximations is derived taking the kinematic limit in compressible fluid dynamics according to two small nondimensional parameters. These schemes model different situations depending on how these parameters are chosen. Especially, dissipative heating can be included (see Eqs. (8) there). Though all results quoted above are partially justified by experimental data, up to [61] the question How to model the situation where dissipative heating has to be taken into account while staying as close as possible to the usual Boussinesq scheme? had been open and a reasonable derivation of such scheme from the point of view of Continuum Mechanics was missing as well as this was true for the Oberbeck-Boussinesq approximation up to [114]. Moreover, it was even not clear how the right system looks in general. On the basis of Section 2.1 one finds a higher order approximation, which includes dissipative heating by taking into account higher order terms in the power series, namely the ε5-level in the velocity approximation and ε2 of the energy equation. To allow the latter in contrast to the last section for the ↑ specific heat we must insert the more precise expression ˜ ˜ 0) + εC ˜ 0 (θ0) C(θ) = C(θ

∞ X

εk−1θk +

k=1

∞ X 2 ε2 ˜ 00 C (θ0) εk−1θk + o(ε3) . 2 k=1

(2.28)

In this way we find the additional equations ∂tv2 + (v0 · ∇)v2 + (v1 · ∇)v1 + (v2 · ∇)v0 −

1 ∆v2 + Re ∇p5 = −θ2b , (2.29) Re

˜ 0)(∂tθ2 + v0 · ∇θ2 + v1 · ∇θ1 + v2 · ∇θ0)+ Pr Re C(θ

19

˜ 0 (θ0) θ2θ˙ 0 + θ1(∂tθ1 + v0 · ∇θ1 + v1 · ∇θ0) + 1 C ˜ 00 (θ0)θ2θ˙ 0 − +C (2.30) 1 2 1 − Re Υ(θ0 + Θ)(∂tp0 + v0 · ∇p0) − ∆θ2 = 2Υ kD(v0)k2 − |tr D(v0)|2 . 3

different values due to an uncertainty with respect to the value of α

2.2




Since v1 = 0, θ1 = 0 and div v0 = 0 these equations reduce to 1 ∆v2 + Re ∇p5 = −θ2b , ∂tv2 + (v0 · ∇)v2 + (v2 · ∇)v0 − Re ˜ 0)(∂tθ2 + v0 · ∇θ2 + v2 · ∇θ0) + C ˜ 0 (θ0)θ2θ˙ 0 − Pr Re C(θ

− Re Υ(θ0 + Θ)(∂tp0 + v0 · ∇p0) − ∆θ2 = 2ΥkD(v0)k2 .

Unfortunately, the relations, which hold on the approximation levels cannot be rewritten as a system for some approximate quantities. For that we cannot just continue the procedure of the previous section but we have to adapt Section 2.1 appropriately. Namely, . We choose the perturbation parameter ε as a measure of incompressibility (slightly different from the one in [114] and Section 2.1). . We introduce the Dissipation number Di into the system as an additional nondimensional number. The consequence is a different balance of the terms. However, of course our small ε (which means small compressibility) shall lead to a solenoidal velocity field to stay close to the usual Oberbeck-Boussinesq approximation. Though it is known that dissipative heating and compressibility are of the same order as long as the ↑ Grüneisen constant is approximately one (which holds for the usual liquids and gases, see [19, 57, 108]) for possible non-standard applications (for example if g or L are very large) we find it more convenient to avoid any coupling between the considerations for Di and ε. Nondimensionalisation. As in the previous section we set U :=

p

gαϑL ,

and

π := ρr g L ,

but our perturbation parameter is different, namely we take ε3 := αϑ .

(2.31)

It is worth noticing that this perturbation parameter was already chosen by Oberbeck [102]. Inserting these definitions for U and ε into (2.18) we observe that from now on only measurable quantities occur. Let us introduce the nondimensional numbers (see [42, 103]), Gr :=

1 3 gL αϑ , ν2

and

Di :=

αgLρr . c0

(2.32)

The Grashof √ number Gr measures the relation of body forces to viscous forces. This means Pr Gr is a measure for conduction over convection of energy. Note that with our representative velocity the ↑ Reynolds number meets Re =

√ 1 UL = Gr . ν

2




From its definition and (2.31) it follows that Gr = ε3

gL3 , ν2

while

Di = ε3

gLρr . c0ϑ

In order to retain on the one hand both inertial and viscous effects and on the other hand effects due to dissipative heating we must ensure that Di and Gr are of order one compared to ε, i.e. Gr ∼ 1 and Di ∼ 1 and thus, gL3 ∼ ε−3 , ν2

gLρr ∼ ε−3 . c0ϑ

and

The validity of our approximation is clearly delineated for concrete applications. One must compute the size of Gr , Di and ε and this is possible using measurable quantities. Examples for which the above requirements are satisfied are quoted above in the introduction of this section, in Part C, and in [12, 19, 20, 50, 57, 108]. If, however, Gr ∼ 1 and Di ∼ ε one obtains the usual Oberbeck-Boussinesq scheme 3 r ∼ ε−3 while gLρ ∼ ε−2. from the previous section. In this situation gL 2 ν c0 ϑ For the new ε and using the numbers Di and Gr our system (2.18) transforms to 3

ρ = e−ε θ ,

div v = ε3θ˙ ,

ε3 1 ε3ρv˙ − √ ∆v + ∇( div v) + ∇p = ρb , (2.33) 3 Gr 2 Di |trD|2 1 2 3 ˜ ˙ √ √ ˙ ∆θ = kDk − C − ε Di (θ + Θ)p θ − Di (θ + Θ)p − 3 Pr Gr Gr

instead of (2.20).

Approximation scheme. We assume that all quantities are as smooth as we want, set v=

∞ X

k

ε vk ,

k=0

θ=

∞ X

k

ε θk ,

p=

k=0

∞ X

εkpk ,

(2.34)

k=0

and attach the boundary conditions of the quantities to the zero order term. We insert (2.34) in the system (2.33) and see from equation (2.33)2 that div v0 = div v1 = div v2 = 0 .

(2.35)

Due to our choice for ε and (2.4) ρ = 1 − ε3θ + o(ε6) , and the velocity equation turns into (we choose apt expressions for ρ) ε3 1 + o(ε3) ∂tv0 + (v0 · ∇)v0 + ε ∂tv1 + (v0 · ∇)v1 + (v1 · ∇)v0 + o(ε2) − ∞ ∞ ∞ P P ε3 P εk∇pk = 1 − ε3 εkθk + o(ε6) b . εk∆vk + o(ε3) + −√ Gr k=0 k=0 k=0

2.2




At ε0, ε1, and ε2 we note as for the Boussinesq approximation ∇p0 = b ,

p1 = p2 = 0 .

For ε3 we observe ∂tv0 + (v0 · ∇)v0 − √

1 ∆v0 + ∇p3 = −θ0b . Gr

The level ε4 yields ∂tv1 + (v0 · ∇)v1 + (v1 · ∇)v0 − √

1 ∆v1 + ∇p4 = −θ1b , Gr

(2.36)

and ε5 finally provides ∂tv2 + (v0 · ∇)v2 + (v1 · ∇)v1 + (v2 · ∇)v0 − √

1 ∆v2 + ∇p5 = −θ2b . Gr (2.37)

˜ Next we examine (2.33)4 and obtain inserting (2.28) for C ∞ 2 X 2 ∞ ˜ 0) + εC ˜ 0 (θ0) P εk−1θk + ε C ˜ 00 (θ0) εk−1θk + o(ε3) × C(θ 2 k=1 k=1 2 × ∂tθ0 + v0 · ∇θ0 + ε(∂tθ1 + v0 · ∇θ1 + v1 · ∇θ0) + o(ε ) − ∞ P k 2 − Di ( ε θk + Θ) ∂tp0 + v0 · ∇p0 + ε(∂tp1 + v1 · ∇p0 + v0 · ∇p1) + o(ε ) − k=0

−

Pr

1 √

∞ ∞ ∞ P P 2 Di P 1 εk∆θk = √ D( εkvk) · D( εkvk) − |trD( εkvk) |2 . 3 Gr k=0 Gr k=0 k=0 k=0 ∞ X

Now we notice, that due to (2.35) tr D(v0) = tr D(v1) = tr D(v2) = 0 , and see at ε0, that ˜ 0)(∂tθ0 + v0 · ∇θ0) − Di (θ0 + Θ)(∂tp0 + v0 · ∇p0)− C(θ 1 2 Di − √ ∆θ0 = √ D(v0) · D(v0) . Pr Gr Gr It is reasonable (though for the following not essential) to assume that b does not depend on the time. Then the relation ∇p0 = b leads to ∂tp0 + v0 · ∇p0 = v0 · b .

2




Anyway, since b = ∇Φ for some scalar field Φ, we always find ∂tp0 + v0 · ∇p0 = ∂tΦ + v0 · ∇Φ , and ε1 furnishes ˜ 0)(∂tθ1 + v0 · ∇θ1 + v1 · ∇θ0) + C ˜ 0 (θ0)θ1(∂tθ0 + θ0 · ∇v0)− C(θ − Di θ1v0 · b − Di (θ0 + Θ)(∂tp1 + v1 · ∇p0 + v0 · ∇p1)− (2.38) 4 Di 1 ∆θ1 = √ D(v0) · D(v1) . − √ Pr Gr Gr At ε2 we finally collect ˜ 0)(∂tθ2 + v0 · ∇θ2 + v1 · ∇θ1 + v2 · ∇θ0)+ C(θ 0 ˜ + C (θ0) θ2(∂tθ0 + v0 · ∇θ0) + θ1(∂tθ1 + v0 · ∇θ1 + v1 · ∇θ0) + +

1 ˜ 00 C (θ0)θ21(∂tθ0 + v0 · ∇θ0)− 2 − Di θ2v0 · b − θ1(∂tp1 + v1 · b + v0 · ∇p1)

(2.39)

− Di (θ0 + Θ)(∂tp2 + v0 · ∇p2 + v1 · ∇p1 + v2 · b)− 2 Di 1 √ √ ∆θ2 = 2 D(v0) · D(v2) + D(v1) · D(v1) . − Pr Gr Gr Since the solutions are smooth we infer from the unique solvability of (2.36), (2.38) that v1 = 0 ,

θ1 = 0 ,

p4 = 0

because these values fulfil the equations and have zero boundary values. Similarly, from (2.37) and (2.39) and the known values for v1, θ1 and p2 we deduce that v2 = 0 ,

θ2 = 0 ,

p5 = 0 .

One computes that the next ε-levels yield non-trivial contributions. Finally, by setting v := v0, θ := θ0, and p := p0 + ε3p3 we obtain the approximate system div v = 0 , 1 ∆v + ∇p = (1 − ε3θ)b , ε3 v˙ − √ Gr 1 2 Di ˜ θ˙ − Di (θ + Θ)v · b − √ C(θ) ∆θ = √ D(v) · D(v) . Pr Gr Gr

(2.40)

On the one hand this looks very similar to (2.26). As in Remark 2.1 yet we may set ∇^ p :=

1 (∇p − b) ε3

⇒

v˙ − √

1 ∆v + ∇^ p = −θ b . Gr

2.3

Power-law rheology



On the other hand we pay for the (compared to the Boussinesq scheme) additional dissipative heating term D · D by another extra term, namely (θ + Θ)v · b. Thus, our approximation scheme shows, that taking into account dissipative effects in natural convection two supplementary addends occur. This was already observed in [9, 57, 103, 108]. Nevertheless, all the systems derived there are distinct from (2.40) due to different nondimensionalisations and various other techniques used there. As in the previous section our approximation uses different levels in the temperature and the velocity equation and div v = 0. However, in contrast to (2.26) here in system (2.40) all terms are more precise: both v and θ are correct of order ε2 and p of order ε5, respectively, since v1 = v2 = 0, θ1 = θ2 = 0, p4 = p5 = 0. Thus, the difference to the solution of the full system is of the order ε3 for v and θ, and ε6 for p, respectively.

% (

2.3 Power-law rheology Macromolecular liquids as well as biological fluids, suspensions, food stuff and paints share the property that their flow behaviour in general is not in accordance with the theory for classical linearly viscous (i.e. ↑ Newtonian) fluid models. For instance, one can observe [7, 15, 25, 101, 113] . ↑ shear thinning or ↑ shear thickening, . ↑ creep, . stress relaxation, . the presence of normal stress differences in ↑ simple shear flows, . the presence of ↑ yield stress. Nevertheless, understanding the fluid dynamics of such materials is important for plastics manufacture (treating the flow of molten plastic), performance of lubricants, applications of paints, processing of food, the control of circulation of blood, in glaciology, and in geology of the earth’s interior. Already in 1835, Weber discovered liquid-like behaviour during experiments with a solid, namely, silk thread. But what makes the crucial difference between liquids and solids? If a small stress is suddenly exerted on a solid, a deformation will occur. The material will be deforming until molecular (or internal) stresses are established which balance the external stresses. On the contrary, in a fluid, when external stress is exerted, deformation occurs and continues indefinitely until the stress is removed. The ↑ Newtonian fluid defined in (2.12) is the simplest example. There the tensor D is directly proportional to the stress applied to the fluid. However, many fluids exhibit a nonlinear response to stress, and are therefore called non-Newtonian fluids. Such fluids fall halfway between being a solid (where the stress depends on the instantaneous deformation) and ↑ Newtonian fluids (where the stress depends on the instantaneous rate of change in time of the deformation).

2




According to (1.10) (i.e. without temperature dependencies and in accordance with the ^ = ω1D + ω2D2 , where the Principle of material frame indifference) in general T coefficients ωi (i = 1, 2) may depend on the invariants of D. But motivated by the following statement, we are interested in a specific type of non-Newtonian material. ”There are examples of many fluids which are different from a Newtonian fluid only in that their viscosity depends on the symmetric part of the velocity gradient so that in a simple shear flow the viscosity depends on the shear rate. These fluids however do not exhibit normal stress differences in simple shear flow, yield or stress relaxation ... or to be more precise such effects are inconsequential, the dominant departure from the Newtonian behavior being shear thinning or shear thickening.” Málek, Rajagopal & R˚uzˇiˇcka [88]

In particular - as in the ↑ Newtonian model - we skip the D2-term and set ^ = ω1 tr D, tr (D2), tr (D3) D . T

(2.41)

These are a so-called generalised Newtonian fluids since one can consider them as generalisation of (1.11), (1.24), or (2.12), in that the constant ↑ viscosity µ becomes a function of the invariants of D. We select a popular special case of (2.41), namely for some material constants r > 0 and µr > 0 in (2.12) we replace 2µ by µr kDkr−1, i.e. we set [24, 84, 88, 89] ^ := µr kDkr−1 D − 1 (tr D) I = µr |tr (D2)| 12 (r−1) D − 1 (tr D) I . T 3 3

(2.42)

These are so-called power-law models. They are en vogue in planetary physics [81, 93, 104, 122], chemical engineering [7] and glaciology [118]. If r < 1 the material is called ↑ shear thinning or ↑ pseudoplastic, and ↑ shear stress decreases with ↑ shear rate. For r > 1 it is ↑ shear thickening or ↑ dilatant, and ↑ shear stress increases with ↑ shear rate (for examples for both types of materials see the glossary). Obviously, for r = 1 model (2.42) is the ↑ Newtonian. We calculate

^ · L = µr kDkr−1(kDk2 − 1 |trD|2) , T 3 and the Balance laws (2.2) for power-law fluids which are incompressible in the generalised sense read (compare (2.13)) ρ = ρr e−α(θ−θr ) ,

div v = α θ˙ ,

(2.43) ρ v˙ − µr div kDkr−1(D − 31 (tr D) I) + ∇p = ρ b , (C − α2 θ p) θ˙ − α θ p˙ − κ ∆θ = µr kDkr−1(kDk2 − 13 |trD|2) . Apparently, the first two equations are the same as in the two previous sections. For that we will mostly skip them in the following considerations.

2.3

Power-law rheology



Performing our standard nondimensionalisation equations (2.43)3&4 become Ur−1 π 1 1 U ¯˙ r−1 U kDk (tr ρrρ¯ Uv − µr div D − D) I p = ρrρ¯ gb , + ∇¯ L L Lr−1 L 3 L U ¯ U ¯˙ 1 ˜ − α2ϑ(θ¯ + Θ)π¯ − κ 2 ϑ ∆θ¯ = c0C p ϑθ˙ − αϑ(θ¯ + Θ) πp L L L r−1 2 U U 1 = µr r−1 kDkr−1 2 kDk2 − |trD|2 . L L 3 Multiplying the first equation by the factor analogue of (2.18)3&4.

L U2 ρr

and the second one by

L2 ϑ

we arrive at the

Now we must think about the nondimensionalisation of the ↑ viscosity, i.e. the quantity µr kDkr−1. We note h

µr kDk

i

r−1

kg , = ms

h i 1 D = s

⇒

h i sr−2kg µr = . m

It seems to us that for power-law fluids the most sensible way to proceed is to define ν as the mean dynamic viscosity by (there seems to be no example in the literature) ν :=

µr Ur−1 . ρr Lr−1

is a mean ↑ Reynolds number and it is not a priori compaConsequently, then Re = UL ν rable to ↑ Reynolds numbers in the ↑ Newtonian case.

With this definition, we find (skipping the bars for convenience)

1 π 1 gL div kDkr−1 D − (tr D) I + ∇p = ρb, Re 3 ρrU2 U2 ˜ 0UL − α2πULϑ(θ + Θ)p θ˙ − α π UL(θ + Θ) p˙ − κ ∆θ = Cc (2.44) 2 νρrU 1 = kDkr−1 kDk2 − |trD|2 . ϑ 3 √ As in the previous two sections, inserting U = gαϑL and π = ρrgL yields ρ v˙ −

1 1 1 1 r−1 kDk D − (tr D) I + ∇p = ρb, Re 3 αϑ αϑ ˜ 0UL − α2ϑρrgUL2(θ + Θ)p θ˙ − αρrgUL2(θ + Θ) p˙ − κ ∆θ = Cc (2.45) 1 = νρrαgLkDkr−1 kDk2 − |trD|2 . 3 ρ v˙ −

Depending on the choice of ε we can proceed in two different directions. Choosing Re αϑ = ε3 leads to a Boussinesq-like approximation while we end up with r . a system including dissipative heating for αϑ = ε3 and Di = αgLρ c0

2




Boussinesq-like approximation. As in Section 2.1 we choose ε according to (2.19) 1 (i.e. we arrive at ε3 = Re αϑ) and use the symbol Υ = κϑ ρr(ν5g2)1/3. Some simple calculations transform (2.45) to the analogue of (2.20), namely to ε3

ρ = e− Re θ ,

div v =

ε3 , Re

ε3 ε3 1 r−1 ρv˙ − div kDk D − (tr D) I + ∇p = ρb , Re Re 2 3

(2.46) |trD|2 5 2 2 r−1 2 ˜ ˙ ˙ . (PrRe C − ε Υ(θ + Θ)p)θ − ε Re Υ(θ + Θ)p − ∆θ = Υε kDk kDk − 3

The next step is to insert the power series (2.21) into system (2.46) (again, the zero order terms carry the boundary conditions). Obviously, the considerations for the mass balance equation are identical to Section 2.1, i.e. div v0 = div v1 = div v2 = 0 . This means, tr D is of order ε3 at most. ε3 In the velocity equation we reuse ρ = 1 − Re θ + o(ε6) and attain ∞ X X ε3 3 (vl · ∇)vm − εk ∂tvk + 1 + o(ε ) Re l+m=k k=0 3 ∞ ∞ P P ε k r−1 k 3 − div kD ε vk k D ε vk + o(ε ) + Re 2 k=0 k=0 ∞ ∞ X ε3 X k ε θk + o(ε6) b . + εk∇pk = 1 − Re k=0 k=0

Equation (2.46)3 now reads ∞ ∞ X X X 0 k−1 2 ˜ ˜ Pr Re C(θ0) + εC (θ0) ε θk + o(ε ) εk ∂tθk + vl · ∇θm − k=1

2

−ε Re Υ(

∞ X

k

ε θk + Θ)

k=0

k=0

∞ X k=0

= Υε2 kD

∞ P

k

ε ∂tpk +

l+m=k

X

l+m=k

vl · ∇pm

−

∞ X

εk∆θk =

k=0

∞ ∞ X r−1 X k k ε vk k D( ε vk) · D( εkvk) + o(ε3) .

k=0

k=0

k=0

There is no difference to the linear case i.e. to Section 2.1 for the first ε-levels, no matter what value r assumes. That is to say in the velocity equation at ε0 − ε2 we note as before ∇p0 = b, while p1 = p2 = 0 , and for ε3 we observe ∂tv0 + (v0 · ∇)v0 −

1 div kD(v0)kr−1D(v0) + Re ∇p3 = −θ0b . Re

(2.47)

In the temperature equation the levels ε0 and ε1 coincide with (2.24) and (2.25), i.e. ˜ 0)θ˙ 0 − ∆θ0 = 0 , Pr Re C(θ ˜ 0)(∂tθ1 + v0 · ∇θ1 + v1 · ∇θ0) + C ˜ 0 (θ0)θ1θ˙ 0 − ∆θ1 = 0 . Pr Re C(θ

2.3

Power-law rheology



But to find the next steps we must investigate the term kDkr−1 more precisely. Anyway, our experiences in the first sections indicate that it will be enough to consider v := v0 + εv1, and we calculate kD(v)k2 = kD(v0)k2 + 2εD(v0) · D(v1) + ε2kD(v1)k2 . Keeping in mind that kDkr−1 = (kDk2)(r−1)/2 we distinguish several cases: . The case r = 1 is not interesting since it is covered by Section 2.1. . Let r > 0 and r ∈ / {3, 5, 7, ...} . Then kDkr−1 includes noninteger ε-power-levels with the factor D(v1), which meet no other term in the equations and thus, must equal zero. This leads to the condition, that D(v1) = 0, which together with the zero boundary condition on v1 ends up with the solution v1 = 0. Then we find also θ1 = 0 and p4 = 0. . In case r = 3 the ε4-level of the velocity equation is 1 ∂tv1 + (v0 · ∇)v1 + (v1 · ∇)v0 − div 2 D(v0) · D(v1) D(v0)+ (2.48) Re +kD(v0)k2D(v1) + Re ∇p4 = −θ1b , and we directly read off that v1 = 0, θ1 = 0 and p4 = 0 is the (unique) solution. . For r ≥ 5 and r ∈ {3, 5, 7, ...} the ε4-level of the velocity equation contains the term 1 div c1(r, kD(v0)k2) D(v0) · D(v1) D(v0) + kD(v0)kc2 (r)D(v1) , Re

and again, v1 = 0, θ1 = 0 and p4 = 0 is the solution. Now, as before in Section 2.1 we set v := v0, θ := θ0 and p := p0 + ε3p3, and ∇^ p := Re (∇p − b). Then we finally have constructed the Boussinesq-like system ε3

v˙ −

div v = 0 ,

1 div kD(v)kr−1D(v) + ∇^ p = −θ b , Re ˜ θ˙ − ∆θ = 0 . Pr Re C(θ)

(2.49)

From our procedure it is clear that (2.49) is a first order approximation. Indeed, it differs from the usual ↑ Newtonian model only in the expression for the stress. No additional terms occur. We justified the use of the Oberbeck-Boussinesq approximation together with a power-law stress [89]. Approximation with Dissipation. As in Section 2.2 we set αϑ = ε3. Again, using the numbers Di and Gr our system (2.45) transforms to div v = ε3θ˙ , 1 ε3 (2.50) div kDkr−1 D − (tr D) I + ∇p = ρb , ε3ρv˙ − √ 3 Gr Di kDkr−1 |trD|2 1 ˜ − ε3 Di (θ + Θ)p θ˙ − Di (θ + Θ)p˙ − √ √ ∆θ = kDk2 − C . 3 Pr Gr Gr

2




Now we insert the power series (2.21) into system (2.50) (with appropriate boundary conditions). Obviously, the considerations for the mass balance equation are identical to the previous sections, i.e. div v0 = div v1 = div v2 = 0, and tr D is of order ε3 at most. Thus, here the velocity equation turns into

3

3

ε 1 + o(ε )

∞ X k=0

3

ε −√ Gr

X (vl · ∇)vm − ε ∂tvk + k

l+m=k

∞ ∞ P P εkvk + o(ε3) + εkvk kr−1 D div kD k=0

k=0

∞ ∞ X ε3 X k k + ε ∇pk = 1 − ε θk + o(ε6) b , Re k=0

k=0

while along the same lines as in Section 2.2 the temperature equation becomes ∞ ∞ X X 2 ε2 ˜ 00 0 k−1 ˜ ˜ εk−1θk + o(ε3) × C(θ0) + εC (θ0) ε θk + C (θ0) 2 k=1 k=1 ∞ X X k × ε ∂tθk + vl · ∇θm − k=0

− Di (

∞ X

εkθk + Θ) ∂tpk +

k=0

X

l+m=k

vl · ∇pm −

1 √

l+m=k ∞ X k

ε ∆θk− Gr k=0 ∞ P Di k r+1 3 √ = kD( ε vk)k + o(ε ) . Gr k=0 Pr

We easily investigate the first levels, i.e. ε0 − ε2 for the velocity and ε0 in the temperature equation in the already well-known fashion, i.e. ∇p0 = b, p1 = p2 = 0, and 1 r−1 div kD(v0)k D(v0) + ∇p3 = −θ0b , Gr 1 Di ˜ 0)θ˙ 0 − Di (θ0 + Θ)p˙ 0 − √ C(θ ∆θ0 = √ kD(v0)kr+1 . Pr Gr Gr

v˙0 − √

(2.51)

The considerations for the level ε4 for velocity and ε1 in the temperature equation include ε-levels inside kDkr±1. In the different cases for r with the same arguments as in the Boussinesq-like case we find that v1 = 0, θ1 = 0 and p4 = 0 in the velocity equation and may check that this solves also the temperature equation. For the respective next levels the situation concerning the ε-levels inside kDkr±1 is similar, thus, we conclude, that it must hold v2 = 0. This solves the temperature equation for ε2 and the velocity equation for ε5 together with θ2 = 0 and p5 = 0. Thus, setting v := v0 ,

θ := θ0 ,

and

p := p0 + ε3p3

2.4

A modified power-law



we obtain the second order approximation

ε3 v˙ − √

1 div kD(v)kr−1D(v) Gr

˜ θ˙ − Di (θ + Θ)v · b − C(θ)

Pr

1 √

div v = 0 , + ∇p = (1 − ε3θ)b ,

(2.52)

Di ∆θ = √ kD(v)kr+1 . Gr Gr

Both v and θ are correct of order ε2 and p of order ε5, respectively, since v1 = v2 = 0, θ1 = θ2 = 0, p4 = p5 = 0. Thus, the discrepancy to the solution of the full system is of the order ε3 for v and θ, and ε6 for p, respectively. Again, the only difference to the linear (↑ Newtonian) case is the different expression for the stress.

% (

2.4 A modified power-law In the last section we have restricted our attention to power-law fluids. Now, let us revisit the more general model ^ = ω1 tr D, tr (D2), tr (D3) D T (2.41) and (see also [88]) turn our interest to the question: ^ (in dependence on D) can be realised with formula (2.41)? What behaviour of T

On the one hand this question is too complex to be answered in full detail. On the other hand it is reasonable to consider configurations which are used to measure material parameters since for some concrete material one has to determine ω1. Namely, let us investigate (2.41) for viscometric flows. These are the classical flows in a viscometer, which is the device used to measure ↑ viscosity. They are useful for measurements since locally, each viscometric flow can be considered as simple shear flow with constant ↑ shear rate γ [25, Ch. III]. Precisely, a simple shear flow is v = (γx2, 0, 0)>

and we calculate for this flow, that    2  0 γ 0 γ 0 0 1 1 D 2 =  0 γ2 0  , D=  γ 0 0 , 2 4 0 0 0 0 0 0



 0 γ3 0 1 D 3 =  γ3 0 0  . 8 0 0 0

Consequently, tr D = tr D3 = 0 while tr D2 = 21 γ2, and (2.41) takes the form   0 γ 0 ^ = ω1(γ2)  γ 0 0  . T 0 0 0

2




µ pl r > 1 shear thickening

r=1

shear thinning r 1, r < 1, r > 1, r < 1,

γ → 0,

γ → ∞.

It does not seem very reasonable to find limγ→0 µ = ∞ as in the case r < 1. Instead we would expect to have a bound limγ→0 µ = µ0 < ∞. For that we modify the power-law model (2.42) as follows: For some constant d, which has the same unit as D (this is 1s ) we set ^mp := µr (d + kDk)r−1 D − 1 (tr D) I . T 3

(2.53)

Now, we can repeat our calculations in Section 2.3 replacing kDkr−1 by (d + kDk)r−1. We can arrange the constants such that after the nondimensionalisation we arrive at (1 + kDk)r−1. Namely, the Balance laws (2.2) for the modified power-law fluids which are incompressible in the generalised sense read (compare (2.13)) div v = α θ˙ , ρ v˙ − µr div (d + kDk)r−1(D − 31 (tr D) I) = ρ b − ∇p , (2.54) (C − α2 θ p) θ˙ − α θ p˙ − κ ∆θ = µr (d + kDk)r−1(kDk2 − 31 |trD|2) . ρ = ρr e−α(θ−θr ) ,

2.4




Again, we use our standard nondimensionalisation (2.15) and define ν as the mean dyr−1 ↑ namic viscosity, i.e. ν = µρrr U r−1 with the stated consequences for the Reynolds number. L √ Inserting U = gαϑL and π = ρrgL then yields the nondimensional system 1 1 1 1 (1 + kDk)r−1 D − (tr D) I + ∇p = ρb, Re 3 αϑ αϑ ˜ 0UL − α2ϑρrgUL2(θ + Θ)p θ˙ − αρrgUL2(θ + Θ) p˙ − κ ∆θ = Cc (2.55) 1 = νρrαgL(1 + kDk)r−1 kDk2 − |trD|2 . 3 ρ v˙ −

The choice of ε decides the further direction.

Boussinesq-like approximation. As done before we insert (2.19), i.e. ε3 = Re αϑ, 1 ρr(ν5g2)1/3, and the power series (2.21) into (2.55). There is no difference to the Υ = κϑ linear and the power-law case for the first ε-levels. Firstly we see ∇p0 = b, while p1 = p2 = 0 . The levels ε0 and ε1 in the temperature equation are the known ones, namely, ˜ 0)θ˙ 0 − ∆θ0 = 0 , Pr Re C(θ ˜ 0)(∂tθ1 + v0 · ∇θ1 + v1 · ∇θ0) + C ˜ 0 (θ0)θ1θ˙ 0 − ∆θ1 = 0 . Pr Re C(θ

The velocity equation for ε3 is a bit different to (2.47), precisely we find ∂tv0 + (v0 · ∇)v0 −

1 div (1 + kD(v0)k)r−1D(v0) + Re ∇p3 = −θ0b . (2.56) Re

Now we investigate the term (1 + kDk)r−1 for v := v0 + εv1. With the same arguments as in the previous section we exclude the possibility that v1 6= 0. Thus, v = v0 and the zero-ε-level of (1 + kDk)r−1 is (1 + kD(v0)k)r−1. With v1 = 0 we again find θ1 = 0 and p4 = 0. Thus, finally we set v := v0

θ := θ0 ,

p := p0 + ε3p3 ,

and

∇^ p :=

Re (∇p − b) , ε3

which leads to the Boussinesq-like system

v˙ −

div v = 0 ,

1 div (1 + kD(v)k)r−1D(v) + ∇^ p = −θ b , Re ˜ θ˙ − ∆θ = 0 . Pr Re C(θ)

(2.57)

System (2.57) is a first order approximation and differs from the usual (↑ Newtonian) Oberbeck-Boussinesq approximation only in the expression for the stress. No additional terms occur. We justified the use of the Oberbeck-Boussinesq approximation together with a modified power-law stress. This is not apriori self evident as Remark 2.3 will show.

2




Approximation with Dissipation. Now we choose αϑ = ε3 and use the numbers Di and Gr . We insert the by now well-known power series. Obviously, the considerations for the mass balance equation are identical to the previous sections, i.e. div v0 = div v1 = div v2 = 0, and tr D is of order ε3 at most. We easily investigate the first levels, i.e. ε0 − ε3 for the velocity and ε0 in the temperature equation in the already well-known fashion, i.e. ∇p0 = b, p1 = p2 = 0, while 1 div (1 + kD(v0)k)r−1D(v0) = −θ0b − ∇p3 , (2.58) Gr 1 Di ˜ 0)θ˙ 0 − Di (θ0 + Θ)p˙ 0 − √ C(θ ∆θ0 = √ (1 + kD(v0)k)r−1kD(v0)k2 . Pr Gr Gr

v˙0 − √

The next levels would include ε-levels inside kDkr±1 which as hitherto leads to the conclusion v1 = 0. Then likewise, θ1 = 0 and p4 = 0. Moreover, in the next step we find v2 = 0. This solves the temperature equation for ε2 and the velocity equation for ε5 together with θ2 = 0 and p5 = 0. For v := v0, θ := θ0 and p := p0 + ε3p3 we establish the second order approximation div v = 0 , 1 ε3 v˙ − √ div (1 + kD(v)k)r−1D(v) + ∇p = (1 − ε3θ)b , (2.59) Gr Di 1 ˜ θ˙ − Di (θ + Θ)v · b − √ ∆θ = √ (1 + kD(v)k)r−1kDk2 . C(θ) Pr Gr Gr Again, only the expression for the stress differs from the linear (↑ Newtonian) case. 2.2 Remark. Iterating the ideas about suitable models instead of ^mp = µr (d + kDk)r−1 D − 1 (tr D) I T 3

one could as well think about a combination of linear and nonlinear effects, namely ^ = (2µ + µrkDkr−1) D − 1 (tr D) I . T 3 For this model we find the obvious approximations.

2.3 Remark. There are a lot of other types of nonlinear fluid models [7, 30, 137]. In contrast to the power-law in general a nonlinear stress-strain relation will be lost under the Oberbeck-Boussinesq approximation and only a linear part of the stress remains. As an example we take second grade fluids. They are characterised by the constitutive relation −

^ := 2µD + 2α1 d D + 4α2D2 , T dt where

d− dt

(2.60)

is the lower convected derivative, namely d− (.) := ∂t(.) + (v · ∇)(.) + ∇v> (.) + (.)∇v , dt

(2.61)

2.4


and µ, α1 and α2 are material constants. Applying the nondimensionalisation (2.15) to second grade fluids, the 2µD-term behaves as shown in Section 2.1. − in comparison to D. At But both, ddt D and D2 are scaled with and additional factor U L 1 ν2 1/3 1/3 first sight, since U = ε(gν) and L = ε Re g , this part of the stress is of order ε2, which means that a second grade fluid under the Oberbeck-Boussinesq approximation20 behaves like a ↑ Newtonian. Moreover, it is not possible to establish a higher order approximation including at least one of the additional terms of a second grade fluid, since we cannot rewrite corresponding ε-level equations as a closed system for some approximate quantities. To understand this better in [105] we define two nondimensional numbers A1 and A2, which depend on the material constants α1, α2 and on the configuration through the spatial i dimension L and the mean density ρr. Namely, Ai := Lα 2 ρ (i = 1, 2). They control the r effect of the non-Newtonian part of the stress tensor in the following sense: if Ai, i = 1, 2, is of order ε, then the non-Newtonian part of the stress tensor is dropped, if it is of order 1 it must be kept. It is very interesting that the number A1 is the same expression (in case ρr = 1) as the one measuring the non-Newtonian behaviour found in [34] by completely different arguments and that already in [35, 134] the absorption number, which is the reciprocal to A1, had been implemented therefor.

20

This is true in our second order scheme including dissipation as well.



Part B Mathematical Treatment

"

% & & (

3 Solvability & Stability In the first part we derived systems of partial differential equations which approximately govern flow processes in natural convection. We resumed and generalised the popular Oberbeck-Boussinesq approximation such that we can consider the influence of dissipation and a power-law rheology. We understood the context in which such approximations are useful and make sense. Nonetheless the mathematics in that part has been restricted to the systematic and axiomatic treatment and some basic principles. In the third chapter we take up our new systems and turn our attention to the more usual questions in the context of mathematical theory for partial differential equations: . existence of weak and strong solutions, . uniqueness, and stability. These questions are interconnected and ask for more involved mathematical techniques. Fortunately, we benefit from results of functional analysis, which means we are standing on the shoulder of giants. Nevertheless, sometimes the answers are not easy to understand even for the mathematician working in the field since precise statements refer to a bunch of interrelated assumptions and need special notions very far from physics or every day experience. Finally, the interpretation of results on a level where experiments could validate or measure bounds often is problematic. The first section provides an overview about related known results and focuses on the various techniques as overall preparation. Then we will exploit adapted energy techniques to study our new systems in the following sections. Namely, Sections 3.2 & 3.3 are based on the results in [61] and devoted to the Newtonian model with dissipation. In the last section of this chapter we derive results for power-law fluids on the basis of [84].

3



Solvability & Stability

# % (

3.1 Generalities This section is a collection of relevant notation, techniques, concepts and universal results. Since the Oberbeck-Boussinesq approximation can be regarded as the root of our problems we would like to provide an idea about known mathematical results and related techniques. Evidently, since the days of Rayleigh, it has attracted a lot of attention and we will not try to give a complete picture (since this is hopeless) but confine ourselves to some aspects which seem representative or especially interesting to us. From the mathematical point of view, the Oberbeck-Boussinesq approximation (2.26) is a decoupled system. Namely, first one solves the heat equation (3.14) for the temperature, which then is inserted at the right-hand side of the velocity equation. This is a Navier-Stokes problem as written down in (3.7) with a special right-hand side. Even the system for the perturbation of the hydrostatic solution formulated in (3.15) below is weakly coupled. Hence, in principle the solvability theory can be found in the textbooks on viscous flow problems e.g. [3, 77, 80, 129, 133, 138] though there is no formulation of results for the Oberbeck-Boussinesq approximation. Leafing through these books one notices that throughout several decades the NavierStokes problem has been and is still challenging and a whole bundle of techniques have been developed to solve it. Depending on the applied ideas there are various types of results since the sort of solutions which fit the schemes are different. To give one prominent example. For a long time already it has been known (see e.g. references in [129]), that locally (in time) there exists a unique strong solution to the NavierStokes problem for reasonable data. But except for small ones (whatever “small” means for concrete applications) it is an open question if this solution exists globally. This is considered as such a central hitch that it became one of the “Millennium Prize Problems” of the Clay Foundation (see http://www.claymath.org/Millennium Prize Problems/). In the context of Mathematical Physics the stability theory and the development of different convection patterns have been discussed in the classical books [23, 59, 126] and in more modern contexts with different aims in [33, 44, 75, 83]. The classical linear stability analysis for Bénard convection is clearly explained in [58, 112], while [74] gives some new impulses. In the 90s several papers exploited semigroup theory and fixed point arguments to treat existence and stability of convection - e.g. [53, 96, 97]. This can also be used to treat convection in exterior domains21 as done in [52, 54]. If the fluid is inhomogeneous the equations become more involved, see e.g. [63]. Temperature dependent 21 i.e. circumfluent to a bounded domain. There occur several difficulties due to the unboundedness of the domain. Especially the behaviour of the functions as |x| → ∞ has to be put into the right frame to find classes of possible solutions. One could criticise that unbounded domains do not exist in nature and on the first sight this might seem an academic problem typical for mathematicians but not really relevant for applications. Nevertheless, while simulating processes like pumping of materials in a bounded region engineers often neglect the influence of boundaries which are “far away” to avoid the prescription of boundary values which are difficult to choose and hence, end up with a problem in an unbounded domain.

3.1

Generalities



viscosity complicates the application of known techniques (cf. [26, 47]). Within the mathematical theory of partial differential equations the properties of solutions are characterised in different standard ways. One possibility is to sort them into hierarchies of functions spaces. Certain properties of these spaces are well-known and the measure for the quality of solutions is its norm in the spaces (details e.g. in the textbooks [3, 6, 40, 129]. For a readable record of calculations and arguments we need appropriate abbreviations. Hence, we supplement the conventions of Section 1.2 by the following notation. Notation of basic function spaces. Let X be a ↑ Banach space. We denote its norm by sup |x∗ (x)|. k . kX while X∗ is its ↑ dual space equipped with the norm kx∗ kX∗ := x∈X: kxkX =1

↑

All spaces hereafter are Banach spaces with respect to the given norm. Assume that D ⊂ R3 is a bounded ↑ domain, D its closure, k ∈ N, and 1 ≤ q ≤ ∞. First solely we consider functions defined in D with values in R. In the manuscript α is used as multiindex : α = (α1 , α2 , α3), |α| := α1 + α2 + α3, j 1 α2 α3 αi ∈ N0, and ∂α := ∂α 1 ∂2 ∂3 . ∇ is the tensor of all derivatives of order j ∈ N. The space C(D) contains the continuous functions in D while functions, that are k times continuously differentiable in D form the space Ck(D) . The smooth functions C∞ (D) play an important role in the theory, particularly the functions with compact ↑ support in k D, denoted by C∞ 0 (D) . Usually one defines C (D) as space of the restrictions of all functions φ ∈ Ck(R3) to D which together with all derivatives up to the order k are bounded in R3. This space is endowed with the norm kφkCk := max sup |∂αφ(x)| . |α|≤k x∈D

Obviously, in a bounded ↑ domain, continuity up to the boundary implies boundedness. For 0 < µ ≤ 1, Ck,µ(D) is the subspace of Ck(D), containing the functions for which all derivatives with |α| ≤ k are µ-Hölder continuous in D, i.e.: |∂αφ(x) − ∂αφ(y)| ≤ c|x − y|µ for x, y ∈ D with x 6= y , where c > 0 is independent of φ. This is a ↑ Banach space with respect to the norm |∂αφ(x) − ∂αφ(y)| |α|≤k x,y∈D |x − y|µ

kφkCk,µ := kφkCk + max sup

while x 6= y .

Often one writes Ck+µ := Ck,µ. Besides BUCµ(D) stands for the space of bounded and uniformly µ-Hölder continuous functions endowed with the Cµ-norm. Most significant in the following are the Lebesgue spaces Lq(D) . For 1 ≤ q < ∞ they are equipped with the norm Z 1/q . kukLq (D) := |u(x)|q dx D

These are integrals in the Lebesgue sense and for that the notion Lq-function always stands for an equivalence class of functions which may differ on a set of zero Lq-measure. In contrast to the “C-spaces”, which measure properties of functions pointwise, characteristics expressed by “L-spaces” and its descendants hold in the integral average.

3




The functions defined in D for which there is c ∈ R such that |u(x)| ≤ c a.e. in D form the space L∞ (D) . It is endowed with the norm kukL∞ (D) := ess supx∈D |u(x)| . For simplicity in the following we write kukLq (D) =: k . kq while hu, viD :=

Z

uv dx ,

D

hu, viD :=

Z

D

u · v dx .

(3.1)

The symbol hu, vi has sense whenever uv ∈ L1(Ω), which is true, e.g. for u ∈ Lq(Ω) if q . Since Lq0 (Ω) can be identified with (Lq(Ω))∗, the brackets v ∈ Lq0 (Ω) and q 0 := q−1 in (3.1) often are called Lq-duality pairing. Moreover, L2(D) and L2(D)3 are ↑ Hilbert spaces with respect to the scalar product Z Z (u, v)D := uv dx , (u, v)D := u · v dx , (3.2) D

D

respectively. If it is clear from the context, on which ↑ domain we calculate, then the subscript D is dropped in (3.1) and (3.2). ForRmatrices we apply the corresponding scalar product inside the integral, i.e., (V, W)D := D V · W dx. Often derivatives are understood in the distributional sense (sometimes they are called weak derivatives), i.e. for 1 ≤ i ≤ 3 and scalar functions u, v defined in D Z Z ∂iu = v ⇔ u∂iφ dx = − vφ dx ∀ φ ∈ C∞ 0 (D) , D D Z Z α α |α| ∂ u=v ⇔ u∂ φ dx = (−1) vφ dx ∀ φ ∈ C∞ 0 (D) . D

D

The Sobolev spaces Wqm(D) , m ∈ N, comprise functions in Lq(D) with all weak deriva-

tives of order |α| ≤ m in Lq(D). They are ↑ Banach spaces with the norm kukWqm (D) :=

X

|α|≤m

k∂

α

1/q

ukqq

=: kukm,q ;

W2m(D) =: Hm(D) .

In fluid dynamics we meet the condition div v = 0, i.e. we restrict the study to solenoidal velocity fields. Then C∞ 0,σ(D) denotes the set of solenoidal vector fields on D whose ∞ components are in C0 (D). In the literature usually H and V (or J and J1) are the closures 3 1 3 of C∞ 0,σ(D) in L2(D) and H (D) , respectively. In our context we modify this to ∞ C∞ 0,per(D) := C (D) ∩ {φ : φ = 0 for x3 = 0, x3 = 1, li-periodic in xi, i = 1, 2} ,

∞ 3 C∞ 0,per,σ(D) := C0,per(D) ∩ {Φ : div Φ = 0} .

3 1 3 Then H and V denote the closures of C∞ 0,per,σ(D) in L2(D) and H (D) , respectively, 3 while P is the orthogonal projector from L2(D) to H. Then the Stokes operator A is defined by A := −P∆ with domain D(A) := H2(D)3 ∩ V. 1 For the temperature we define V as the closure of C∞ 0,per(D) in H (D).

3.1

Generalities



The trace of functions is the generalisation of the boundary value (see e.g. [40, II.3], [129, II.1.2]). Up to now we considered Sobolev spaces with m ∈ N. To characterise traces of functions in Wqm(D) we need Wqm(∂D)-spaces with m ∈ R. Let us consider the example 1−1/q

Wq

(∂D) , the space of functions for which the following norm is finite22 1/q Z Z |u(y) − u(x)|q dsy dsx kukW1−1/q (∂D) := kukLq (∂D) + q |y − x|1+q ∂D ∂D

(ds is the Lebesgue surface measure). If ∂D is regular enough (e.g. ↑ Lipschitz) and 1−1/q 1 < q < ∞, then u ∈ W 1,q(D) implies that the trace of u is in Wq (∂D) and if 1−1/q 1 w ∈ Wq (∂D) there exists a u ∈ Wq(D) with the trace w. Analogously one finds the corresponding trace spaces for functions in Wqm(D), [80, II§2]. The Bochner spaces are useful to express the properties of functions depending on time and space. For example, Lq(0, T ; X) are ↑ Banach spaces of Lq-functions on the interval (0, T ) with values in the ↑ Banach space X. The corresponding norm is defined by ZT 1/q kukLq (0,T;X) := . ku(t)kqX dt 0

In the following, X is a Lebesgue or Sobolev space or the intersection of several of them. Similarly one defines the spaces Wqm(0, T ; X), C([0, T ]; X), BUCµ([0, T ]; X), etc.. Tools & Inequalities. Aubin/Lions, [84, Lem. 1.2.48]: Let X0, X1 be separable and reflexive Banach spaces, for which X0 ⊂ X ⊂ X1 (with continuous injections) and the embedding X0 into X is compact. Y := Lq(0, T ; X0) ∩ {φ : ∂tφ ∈ Lp(0, T ; X1)}, 1 < q < p < ∞. Then also the embedding of Y into Lq(0, T ; X) is compact. Embedding , [40, Th. II.2.2/3]: Let D ⊂ Rn be a bounded ↑ Lipschitz domain. For q ≥ 1, m ≥ 0, assume that u ∈ Wqm(D). nq . If mq < n then u ∈ Lr(D) for all r ∈ [q, n−mq ] and kukr ≤ c kukm,q. . If mq = n then u ∈ Lr(D) for all r ∈ [q, ∞) and kukr ≤ c kukm,q. ˜ a.e., u ˜ ∈ Ck(D), k ∈ [0, m − n . If mq > n then u = u ), and kukCk ≤ c kukm,q. q

Gagliardo/Nirenberg , [38, Th. 10.1]: Let D ⊂ Rn be a bounded domain with ∂D ∈ Cm, and u ∈ Wqm(D) ∩ Lr(D), 1 ≤ r, q ≤ ∞. For any integer j ∈ [0, m), and for any j γ ∈ [m , 1], let p1 := nj + γ q1 − m + (1 − γ) 1r . If m − j − n 6∈ N0, then n q k∇jukp ≤ c kukγm,qkukr1−γ .

If m − j −

n q

∈ N0, then this estimate holds for γ =

j . m

Gronwall, [6, (I.2.18)]: If the function φ(t) ≥ 0 for t ∈ (0, T ) obeys the inequality Zt φ(t) ≤ c + A(s)φ(s) + B(s) ds , 0

22

The precise definition is given in (4.5).

3




where c > 0, A(t), B(t) ≥ 0 and A(t), B(t) ∈ L1(0, T ) then φ(t) ≤ e

Rt

0 A(s)ds

c+

Hölder, [129, (I.3.3.6)]: 1 ≤ s, q, r ≤ ∞,

Zt

B(s)e−

+

1 q

0

1 r

=

Rτ 0

1 s

dτ .

A(s) ds

⇒

kuvks ≤ kukqkvkr .

Interpolation, [129, (I.3.3.7)]: Let 1 ≤ q ≤ s ≤ r ≤ ∞, α ∈ [0, 1],

1 s

=

α q

+

1−α . r

Then

1−α kuks ≤ kukα . qkukr

Laplace estimate, [40, p. 49f]: Let v ∈ D(A) and θ ∈ H2(Ω) ∩ V, then kvkH2 ≤ c(D) kAvk2 ,

kθkH2 ≤ c(D) k∆θk2 .

Parabolic embedding , [84, Lemma 1.2.45]: Let H be a Hilbert space and X a Banach space with X = H. If v ∈ Lp(I; X), ∂tv ∈ Lp0 (I; X∗ ) then v ∈ C(¯I; H). Poincaré, [40, (II.4.2)]: For v ∈ H10(D) it holds kvk2 ≤ c k∇vk2 . Young , [40, (II.1.5)]: γ > 0, 1 < r < ∞, 1r + r10 = 1 ⇒ a b ≤

1 1 1−r γ r0

0

ar + γr br .

Linear evolution equations. We want to solve nonlinear systems of partial differential equations for vector functions depending on the time t and the space variable x. A standard auxiliary step is to study related linear problems. One is the (abstract) evolution problem (e.g. [2, Ch. II/III], [129, IV.1.5]) d u(t) + Lu(t) = f(t) , dt

u(0) = u0 ,

(3.3)

where L is a linear operator (for example the Stokes operator), f and u0 are given and u is the unknown function. In general all functions and the operator will depend on other variables as well, which are not indicated in (3.3). Moreover (in contrast to the rest of the treatise) u, L, and f can also be vectors though we do not use boldface letters. We are interested in the development of the process in time starting from the initial value u0. The (final) aim is to find a formula for solutions to (3.3) which directly leads to estimates and later on to existence results for the nonlinear problems under consideration. The simplest case for (3.3) is L = a for some a ∈ R. Here it is well-known that the general solution to the homogeneous equation (i.e. f = 0) is u = c e−at for all c ∈ R. The constant c is fixed by the initial condition, namely, we obtain c = u0. Then the inhomogeneous problem can be solved by the variation of constant formula u(t) = e

−ta

u0 +

Zt

e−(t−τ)af(τ) dτ .

(3.4)

0

If we treat a system of ordinary differential equations with constant coefficients a is replaced by some matrix A and the expression e−tA substitutes e−ta. Here by definition, P tk k [5], etA := ∞ k=0 k! A , but in general this infinite sum is difficult to calculate.

3.1

Generalities



^ of A and to The standard way to overcome this problem is to find the Jordan form A ^ −tA evaluate e , which is simpler. Thus, if A is positive definite (then the Jordan form is a diagonal matrix formed by the eigenvalues of A) the main work is to find its eigenvalues and eigenvectors, which (after several steps) define e−tA and u(t) via the analog of (3.4). Now, for a bounded linear operator L working from the ↑ Banach space X to X we construct the counterpart of (3.4). We define the operator etL as follows: tL

e g :=

∞ X tk

k! k=0

Lkg ,

g ∈ X,

(3.5)

where L0g := g and Lkg := L(Lk−1g). In case L = A this coincides with the definition above. In this way we can give some meaning to (see [129, IV.1.5/6]) u(t) = e

−tL

u0 +

Zt

e−(t−τ)L f(τ) dτ ,

0

t ≥ 0.

(3.6)

With the help of the definition (3.5) one can show, that e−tL (t ≥ 0) has the properties of a semigroup, namely, e−tL e−sL = e−(t+s)L

for s, t ≥ 0 ,

e0L = I ,

I being the identity. Nevertheless, formula (3.5) is not very suitable to calculate e−tL . Instead if L is positive and selfadjoint one uses a spectral representation (see e.g. [72], [129, II.3.2]). Moreover, for given L one has to investigate under which (smoothness and integrability) conditions to u0 and f the related maps t 7→ e

−tL

u0 ,

and

t 7→

Zt

e−(t−τ)L f(τ) dτ

0

for t ≥ 0 are (strongly) continuous (together with the corresponding estimates), if the function u solves (3.3), and which smoothness and integrability properties of u follow from the chosen setting for u0 and f. In general this is nontrivial but for several classes of operators and especially the Stokes operator (e.g. [129] and references therein) one can benefit from a well-established theory. Energy stability. We want to solve systems of equations which approximate the motion of incompressible fluid. In place we present the crucial ideas of a method which we apply later on, namely energy estimates. In the next sections together with the Galerkin approximation they bear the existence and stability proofs. Here we intend to show the basic steps for uncomplex examples and focus on stability results. A lot of different notions are called stability. The classical [23, 59, 112, 139] and eldest is the so-called linear stability theory (see Paragraph “Stability” in Section 1.3). Therein nonlinear problems must be linearised at the solution, the stability of which is examined (if it is the zero solution this means, the nonlinear terms are skipped). Then the corresponding eigenvalue problem, (see e.g. (3.26) or (1.32)), for a parameter as ↑ Reynolds or Rayleigh number is studied. This analysis yields a bound for instability, i.e. whenever

3




this bound for the related physical parameter is exceeded any perturbation grows in time and the solution is unstable. But this does not guarantee that for values below this bound it remains stable. So-called subcritical instability may occur. A different approach is chosen here. It is called nonlinear energy stability since the full nonlinear system is considered and a scalar function called energy is investigated, which measures the size of perturbations to the solution under consideration as t → ∞. Let us start with a simple example, an initial boundary value problem for the NavierStokes equations in a smooth bounded ↑ domain Ω and the time interval I := (0, ∞) approximately governing the motion of viscous fluid under the action of the given data: 1 v˙ − ν∆v + ∇p = f , ρ

in Ω × I ,

div v = 0

on ∂Ω × I ,

v = vΓ

(3.7) v(0, ·) = v0

in Ω

(in contrast to the systems which were derived in the last chapter here the quantities still have their dimensions). If we shake a container containing water as fiercely as we wish and then leave it undisturbed under the influence of gravity and no-slip boundary condition (i.e. vΓ = 0), we expect, that the motion of the fluid inside slows down and finally is at rest, i.e. v → 0. In other words we expect the zero solution to be stable under arbitrarily large perturbations v0. In a very elegant way this can be shown for any conservative23 body force f = ∇f. We start with a lemma about the nonlinear term. 3.1 Lemma. Let φ ∈ H1(Ω), Φ ∈ H1(Ω)3, v ∈ H1(Ω)3, div v = 0, vΓ = 0. Then (v · ∇φ, φ) = 0

and

(v · ∇Φ, Φ) = 0 .

Proof. First, since div v = 0 we observe div (φv) = v · ∇φ + φ div v = v · ∇φ. Let ⊗ denote the outer product, i.e. Φ ⊗ v := (Φivk)3i,k=1 then

div (Φ ⊗ v) = ( div (Φiv))3i=1 = (Φi div v + v · ∇Φi)3i=1 = v · ∇Φ. Thus, (v · ∇Φ, Φ) = ( div (Φ ⊗ v), Φ) = −(Φ ⊗ v, ∇Φ) + (n · (Φ ⊗ v), Φ)∂Ω 1 = −(Φ ⊗ v, ∇Φ) = − (v, ∇|Φ|2) 2 1 = ( div v, |Φ|2) + (n · v, |Φ|2)∂Ω = 0 , 2

and the analogous calculations for φ instead of Φ hold true as well. 3.2 Remark. From the proof it is clear, that the statements of Lemma 3.1 remain true if v = 0 on ∂Ω is replaced by v ∈ V for the setting in the periodic cell, since then the boundary terms are zero as well. In a similar way one proves (v · ∇v, Φ) = −(v · ∇Φ, v). The L2(Ω) scalar product of (3.7) for f = ∇f and P :=

p ρ

− f with v reads

(∂tv, v) + (v · ∇v, v) − ν(∆v, v) + (∇P, v) = 0 . 23

i.e. f has a potential f with f = ∇f. In particular this means that there are no sources.

(3.8)

3.1

Generalities



We set E(t) := 12 kvk22 and calculate 1 d d E(t) = (∂tv, v) = kvk22 , dt 2 dt −ν(∆v, v) = ν(∇v, ∇v) − ν(n · ∇v, v)∂Ω = νk∇vk22 , (∇P, v) = −(P, div v) + (P, n · v)∂Ω = 0 . Moreover, (v · ∇v, v) = 0 and we can rewrite (3.8) as follows Poincaré d E(t) = −νk∇vk22 ≤ −νcpkvk22 = −2νcpE(t) . dt

(3.9)

Finally, with the help of Gronwall’s inequality we deduce from inequality (3.9) that E(t) ≤ e−2νcp E(0)

⇒

t→∞

E(t) → 0 .

This means, that v → 0 no matter how large E(0) and the zero solution is called unconditionally stable. Thus, we verified our common sense expectation mathematically. The main steps were (1) scalar multiplication of the system by v, (2) definition of an energy, collecting (useful) norms of v, (3) re-writing of the scalar product in terms of the energy and its time derivative (plus possible additional terms), (4) application of the Gronwall lemma. Note, that since (v · ∇v, v) = 0 our analysis indeed has been linear. If we are interested in the stability of a nonzero steady solution this situation changes. More precisely, let (u, P) solve the steady version of (3.7) with f = ∇f, i.e. u · ∇u − ν∆u + ∇P = 0 ,

div u = 0

in Ω × I ,

u = uΓ

on ∂Ω × I ,

˜ solve (3.7) with vΓ = uΓ and the same f. We set ˜, P) and (v ˜ − u, v := v

˜−P p := P

for the unsteady perturbation of the steady solution (u, P). If we insert these relations we find (after some obvious calculations), that the pair (v, p) solves ˜ · ∇v − ν∆v + ∇p = 0, div v = 0 ∂tv + v · ∇u + v v(0, ·) = v0 in Ω , v=0

in Ω × I , on ∂Ω × I .

(3.10)

Treating (3.10) as (3.7) above we want to find a sufficient condition to u, ensuring that ˜ · ∇v, v) = 0 and we are left with one new v → 0 as t → ∞. Lemma 3.1 yields that (v and nonlinear term, namely (v · ∇u, v). Hence, the analog of (3.9) becomes d E(t) = −νk∇vk22 − (v · ∇u, v) . dt

(3.11)

3




Evidently, the outcome of the receipt formulated above depends on the size of (v · ∇u, v) in comparison to νk∇vk22. To calculate in the most clear and simple way we nondimensionalise (3.11). Following [132] we choose24 (see Section 2.1 below formula (2.15) to remember how to perform the respective calculations) x = Lx ,

v = Uv ,

L2 t = t, ν

(3.12)

and the nondimensional form of (3.11) omitting all bars becomes d E(t) = −k∇vk22 − Re (v · ∇u, v) =: −D + Re SU . dt

(3.13)

We set V0 := {Φ ∈ H1(Ω)3 : div Φ = 0, Φ = 0 on ∂Ω , Φ 6= 0}. Then 1 d SU SU −1 E(t) ≤ −D Re − max . , and we set Re c := max V0 D V0 D dt Re Obviously if Re < Re c, then

Poincaré d Re c − Re E(t) ≤ −D =: −Dr ≤ −2rcpE(t) . dt Re c

Note that r > 0 and with Gronwall’s inequality we deduce E(t) ≤ e−2rcp E(0)

⇒

t→∞

E(t) → 0 .

This means that v → 0 as t → ∞ and u is stable for all initial disturbances, i.e. unconditionally stable. Note that it is not possible to deduce the stability behaviour of E(t) as t → ∞ if Re > Re c in this way since if E(t) ≤ ecE(0) for c > 0 it can decay or increase.

We finish this paragraph studying the stability of the hydrostatic solution (0, θh, ph) defined in (1.29) for the Oberbeck-Boussinesq approximation in a layer of thickness L. If we closely tie in with the explanations above this means we choose (3.7) with the special right-hand side f = (0, 0, gαθ)> coupled with the heat equation θ˙ − κ∆θ = 0 .

(3.14)

As before we set v = 0 on the surfaces while θ = θb, θ = θt at the bottom and the top surface, respectively. In x1- and x2-direction both are periodic with periodic cell C, in ˜ θ˜ solve this problem we set ˜, P, particular, the domain Ω above is replaced by C. If v ˜, v := v

˜ − ph , p := P

θ := θ˜ − θh ;

β := L1 (θb − θt) .

This perturbation (v, p, θ) to the hydrostatic solution fulfils (cf. (3.31), (3.33))

div v = 0 , v˙ − ν∆v + ρ10 ∇p = (0, 0, gαθ)> in C × I , θ˙ − κ∆θ = βv3 v = 0 , θ = 0 on x3 = 0 and x3 = L , t ∈ I , 24

The last is compatible with t =

L U

and U =

ν L

as explicitly chosen later on.

3.1

Generalities



plus periodicity and initial conditions. If we nondimensionalise as follows U :=

ν , L

p θ = U (βν)/(καg) θ ,

p = ρ0U2 p ,

x, t, and v as in (3.12), we end up with the problem (bars neglected)

div v = 0 , v˙ − ∆v + ∇p = (0, 0, λθ)> in C × I , Pr θ˙ − ∆θ = λv3 v = 0,

θ=0

on x3 = 0 , x3 = 1 + periodicity , t ∈ I ; v(0, x) = v0(x) , θ(0, x) = θ0(x) x ∈ C .

We already introduced the Rayleigh number Ra in (1.28) and set λ := ↑ Prandtl number Pr is well-known from the previous chapter.

(3.15)

√

Ra . The

3.3 Remark. We also find (3.15) if we take our nondimensional system (2.26), reformulate it for the perturbation of the hydrostatic solution and perform the transformation v 7→ Re v ,

t 7→ Re t ,

θ 7→ Pr 1/2 Re −1θ ,

p 7→ Re −2p .

(3.16)

Obviously, to adapt the procedure from above the velocity and the heat equation have to be treated in a similar but not in the same way. Namely, we choose the factor v for the first one while the heat equation is multiplied by θ, respectively, and by Lemma 3.1 (v · ∇v, v) = 0 = (v · ∇θ, θ). This provides the two linear equations 1 d kvk22 + k∇vk22 = λ(v3, θ) , 2 dt

1 d Pr kθk22 + k∇θk22 = λ(v3, θ) . 2 dt

(3.17)

We set E(t) = 12 (kvk22 + Pr kθk22). Thus, 12 (kvk22 + kθk22) ≤ max{1, Pr −1}E(t). The both equations in (3.17) lead to =:D

=:S }| { 1 z }| { z d S E = λ 2(v3, θ) − (k∇vk22 + k∇θk22) ≤ −Dλ − max dt λ V×V\{(0,0)} D 1

1 Poincaré 1 =: −Dλ =: −Da ≤ −2acp max 1, E(t) =: −cE(t) . − λ λc Pr √ Consequently, if λ < λc = Rac then E(t) ≤ e−cE(0) for some constant c > 0. This means that (v(t), θ(t)) → (0, 0) as t → ∞, i.e. the hydrostatic solution is stable for all initial disturbances provided Ra < Rac.

Let us dwell on λc for a moment. It solves the following maximum problem 2(v3, θ) 1 . := max 2 V×V\{(0,0)} k∇vk2 λc 2 + k∇θk2

The Euler-Lagrange equations for it are ([132, (3.65)]) div v = 0 ,

−∆v + ∇p = λ(0, 0, θ)> ,

−∆θ = λv3 in Ω ,

(3.18)

3




together with the boundary conditions in (3.15). This coincides with the eigenvalue problem for the linear analysis (see (1.32) together with Lemma 1.1 under a different nondimensionalisation and [132, (3.57)]), i.e. remarkably for the Oberbeck-Boussinesq approximation the instability and the stability bound for the hydrostatic solution are the same. Energy methods are popular in the nonlinear stability theory of fluid motion. Nevertheless, they have two main disadvantages, [41]. . The stability bounds can be illogical low and difficult to compare to experimental data. . In certain examples only generalised energy functionals lead to sound results, which have no clear physical meaning. A prominent example for the second fact is Bénard convection between rotating plates. The classical method introduced above fails to include the effect that rotation inhibits convection. A similar problem arises if additional magnetic forces enter [120, 121]. Only somewhat obscure generalised energy functionals (see (II) in [41]) lead to results compatible with experiments. There arises the question if there is a canonical procedure for constructing them. Algorithms for a lot of important cases are provided in [41]. In the same context there emerges the problem that the norms in E are not strong enough to guarantee the global existence of strong solutions to the Oberbeck-Boussinesq problem even for small E(0). Hence, one would like to control the energy stability for higher order norms of (v, θ) as done in [64, 65, 123, 139]. The method used by these authors is based on the decomposition of the velocity field, which is solenoidal and periodic in the plane into the mean flow, the poloidal and the toroidal field as often done by physicists. The investigation of nonlinear aspects of ↑ buoyancy-driven flow already began at the end of the 1950s. At that time the linear theory was essentially complete. The main hope was to solve the question of the preferred convection pattern. Nonetheless, all experimental evidence suggests that the configuration of the lateral walls, rather than nonlinear effects, determines the form of the patterns and there is no adequate theoretical explanation [75]. Moreover, the transition to time dependence is largely an open problem. None of these two questions will be touched in the following sections. Shortly summarising the findings of the last decades in these respects one must confess, that much remains to be done in future. Amann’s results. Here we want to give a short overview25 on the existence and stability results of Amann [1, 2, 4], for viscous, (apriori) incompressible, heat-conducting fluids since they contain our equations as special case. He studies the system div v = 0 , ^ = ρb , ρ v˙ + ∇p − div T ^ ·D+ρr Cρθ˙ + div q = T

(3.19)

for the unknowns v, p and θ in a bounded smooth domain Ω ⊂ R3 for t > 0. Obviously, system (3.19) reflects the conservation laws (1.1) for incompressible fluids and one can normalise the system such that ρ = 1. This is done in [4] and below. Let the boundary consist of two disjoint parts Γ0 ∪ Γ1 = ∂Ω, which are relatively open in ∂Ω. On Γ0 the value of θ is prescribed, on Γ1 the heat flux. 25

He also has a close look on bifurcation - a topic, which we skip here.

3.1

Generalities



More precisely, for all t > 0 the system is provided with the boundary conditions v = vΓ

on ∂Ω ,

θ = θΓ

on Γ0 ,

−q · n = h

on Γ1 ,

(3.20)

(vΓ = 0 in [2]) and the initial conditions v(0, x) = v0(x) ,

θ(0, x) = θ0(x)

for x ∈ Ω .

(3.21)

Here vΓ (t, x), θΓ (t, x), h(t, x, θ), v0(x), and θ0(x) are smooth given functions. ^ shall represent Stokesian fluids, i.e. Amann chooses the following constitutive relations: T ^ := T(D, ^ T θ)

and

^=T ^> . T

(3.22)

In [1] he starts with the case of isothermal Stokesian fluid, i.e. the coefficients do not depend on θ. As shown in our first section (3.22) reduce to (compare (1.10)) ^ = ω1D + ω2D2 , T

with ωi = ωi(θ, tr (D2), tr (D3)) . (3.23)

Item q := −($0I + $1D + $2D2)∇θ , with $k = $k(θ, tr (D2), tr (D3)) , (3.24) i.e. the coefficients $k do not depend on ∇θ, which is another simplifying assumption. It is sound to presume that there exists a uniform temperature distribution θ¯ for which the fluid is at rest if b = 0 and h = r = 0. A conclusion of this behaviour is that for t > 0, θ > 0 there are representations ¯ , b(t, x, θ) = b0(t, x) + b1(x, θ)(θ − θ) ¯ , r(t, x, θ) = r0(t, x) + r1(x, θ)(θ − θ) ¯ , h(t, x, θ) = h0(t, x) + h1(x, θ)(θ − θ)

x ∈ Ω,

x ∈ Ω,

(3.25)

x ∈ Γ1 .

Here the functions b1, r1, h1, ω1, ω2, $1, $2, and $3 shall all be smooth while it is also ¯ > 0, as well as κ := $0(0, 0, θ) ¯ > 0. presumed that ν := 21 ω1(0, 0, θ) Model (3.23) is more general than the power-law fluids in which we are interested even though it misses to capture normal stress differences. We would like to point out that the problem studied in [2, 4] is a complex nonlinear system which includes as special case the well-studied Navier-Stokes equations. The aim of [2, 4] is to find global strong solutions for system (3.19) - (3.25) under a smallness condition for the data. 1−1/q

(∂Ω) As stated before (since ∂Ω is smooth enough) u ∈ Wq1(Ω) has a trace in Wq 1−1/q and conversely, a trace in Wq (∂Ω) can be extended to a function in Wq1(Ω) and the corresponding estimates hold. More generally spoken the trace operator is linear and s−1/q bounded from Wqs (Ω) to Wq (∂Ω) if 1 < q < ∞ and s > 1/q. The Wqs -spaces with s < 0 are defined as the corresponding ↑ dual space (with respect to (3.1)) of Wq−s0 functions with zero trace (if the trace exists). Let us make this more precise: We are especially interested in the trace operator on Γ0 and denote it by τ. s−1/q The corresponding extension operator working from Wq (Γ0) to Wqs (Ω) for s > 1/q

3




is indicated by R. One symbol for all s ∈ R combines

 s 1/q < s < ∞ ,  {u ∈ Wq(Ω) : τu = 0} , s s f Wq(Ω) , −2 + 1/q < s < 1/q , Wq(Ω) :=  f−s ∗ (Wq0 (Ω)) , −∞ < s < −2 + 1/q .

In the classical linear stability analysis [23, 59, 112, 139] for solutions to partial differential equations one investigates the corresponding linear eigenvalue problem. For our nonlinear system we meet the following two problems which are linearisations at the motionless state (see also the appendix for the suitable linearisation of the problem with dissipation): −ν∆v + ∇p = σv , −κ∆θ − ¯r1θ = µθ in Ω ,

div v = 0 θ=0

in Ω ,

v = 0 on ∂Ω .

¯ 1θ = 0 κ∇θ · n − h

on Γ0 ,

(3.26)

on Γ1 .

(3.27)

¯ h ¯ 1 := h1(·, θ). ¯ It is well-known that the least eigenvalue of (3.26) Here ¯r1 := r1(·, θ), σ0 > 0, while the least eigenvalue µ0 of problem (3.27) is characterised by f1(Ω)} , µ0 = min {f(θ) : θ ∈ W 2 kθk2 =1

f(θ) :=

Z

2

2

(κ|∇θ| − ¯r1θ ) dx −

Ω

Z

¯ 1θ2 ds . h

Γ1

There are conditions which ensure µ0 > 0 but in other situations µ0 < 0 is possible. To formulate Amann’s results we need some more notation. Let I := [0, T ) for some time T > 0 and q > 3 fixed. Moreover we set D0 := (Wq2(Ω))3 × Wq1(Ω) , f−1(Ω) , B0 := (Lq(Ω))3 × W q

D := D0 × Wq1(Ω) ,

B := B0 × Wq−1(Γ1) × Wq2−1/q(∂Ω) × Wq1−1/q(Γ0) .

A solution to problem (3.19) - (3.21) is a triple (v, p, θ) ∈ C(¯I ; D) with 1

(v, θ − RθΓ ) ∈ C (I ; B0) ,

Z

Ω

p dx = 0 ∀ t ∈ I .

Such solution fulfils the first two equations of system (3.19) in the strong sense, while the f10 and t ∈ I energy equation holds in the following integral sense for any φ ∈ W q ˙ φi = h 1 q, ∇φi + hq · ∇ 1 + 1 T ^ · D + 1 r, φi + h 1 h, φiΓ0 . hθ, C C C C C

(3.28)

This formulation facilitates the treatment of the nonlinear boundary condition. A solution is global if T = ∞.

µ0 3.4 Theorem. [4, Theorem 2.2] Let δ ∈ (0, 1), µ0 > 0 and fix ω ∈ [0, min{σ0, C( ¯ }]. θ) Then there exist constants Ki > 0 (i = 0, 1) such that for each set of data

((b0, r0, h0, vΓ , θΓ ), (v0, θ0)) ∈ BUCδ(R+, B) × D0

3.1

Generalities



satisfying the compatibility conditions div v0 = 0 ,

v0|∂Ω = vΓ (0, ·) ,

θ0|Γ0 = θΓ (0, ·) ,

Z

Γ

vΓ (t, ·) · n ds = 0 ∀ t ≥ 0 ,

as well as the smallness condition ¯ 1,q + keωt(b0, r0, h0, vΓ , θΓ − θ)k ¯ Cδ (R ;B) ≤ K0 , kv0k2,q + kθ0 − θk + there exists a unique global solution and for t ≥ 0 −ωt ¯ k(v, p, θ − θ)(t)k . D + k∂t(v, θ − RθΓ )(t)kB0 ≤ K1e

¯ and otherwise zero A consequence of Theorem 3.4 is that the for µ0 > 0, θ0 = θΓ = θ, ¯ data the rest state (0, 0, θ) exists and it is stable with respect to small perturbations of all data. Now we turn our interest to the case µ0 ≤ 0. We introduce Z 2−1/q 2−1/q c Wq (∂Ω) := Wq (∂Ω) ∩ {v : v · n ds = 0} , Bb := B0 ×

Wq−1(Γ1)

×

∂Ω 2−1/q c W (∂Ω) q

× Wq1−1/q(Γ0) .

Bb is a closed subspace of B and thus a ↑ Banach space.

3.5 Theorem. [4, Theorem 3.1] Let δ ∈ (0, 1), µ0 ≤ 0 and fix T > 0. Then there exists a constants K > 0 such that for each set of data b × D0 ((b0, r0, h0, vΓ , θΓ ), (v0, θ0)) ∈ Cδ(I, B)

satisfying the compatibility conditions div v0 = 0 ,

v0|∂Ω = vΓ (0, ·) ,

θ0|Γ0 = θΓ (0, ·)

as well as the smallness condition ¯ 1,q + k(b0, r0, h0, vΓ , θΓ )kCδ (I;B) ≤ K , kv0k2,q + kθ0 − θk there exists a unique solution for t ∈ I = [0, T ]. If µ0 < 0 the rest state is unstable. T in Theorem 3.5 may be large but cannot be ∞.

The proofs of Theorems 3.4 & 3.5 are based on the theory of abstract evolution equations as presented in [3]. They apply embedding properties of suitable for the theory but more complicated function spaces, semigroup theory and the Helmholtz projection (here the condition div v = 0 is integrated into the admissible function spaces. As a consequence the pressure drops out and one obtains a new slightly reduced problem). Finally the existence of solutions in a neighbourhood of the rest state follows from the implicit function theorem. This explains the necessity of smallness conditions to the data.

3




% & (

3.2 System with dissipation - constant heat capacity After the general view in Section 3.1 let us turn back our interest to system (2.40) as approximation for Rayleigh-Bénard convection. To obtain a well-posed problem we must choose a suitable domain and prescribe corresponding initial and boundary conditions. Having in mind Figure 1.5 we model convection cells prescribing periodic boundary conditions in direction x1 and x2, respectively. Namely, for some positive constants l1 and l2 we assume that v and θ are lj-periodic in xj for j = 1, 2 and that v=0

on the surfaces

x3 = 0 and

x3 = 1 .

(3.29)

The temperature on the top and the bottom face previously had been set to θt and θb, respectively. Due to the nondimensionalisation these values transformed to θ = − 21

for

x3 = 1 ,

while θ =

1 2

for x3 = 0 .

(3.30)

Then it is enough to consider one periodic cell to know the solution everywhere. The advantage is that the cell is a bounded domain, namely we take Ω := (0, l1) × (0, l2) × (0, 1) especially, when we speak about integrability, though the equations hold in R2 × (0, 1) in a certain time interval I := (0, T ). We are mainly interested in motion under gravity force, i.e. we could replace the nondimensional body force by b = (0, 0, −1). Nevertheless, to avoid possible confusion about signs from now on we write b and mean (0, 0, 1). We already observed that the hydrostatic solution (vh, ph, θh) defined in (1.29) represents the motionless state of pure conduction and solves (2.40), (3.29), and (3.30). Thus, to find the onset of convection we must investigate the deviation from pure conduction. We define ˜ := v − vh = v , v

θ˜ := θ − θh ,

and

ε3∇˜ p := ∇p − ∇ph + θhb .

(3.31)

Clearly, θ˜ is zero at the top and the bottom of the layer and the choice for the pressure is for convenience. Moreover, we abbreviate ^ := Θ

1 2

+ Θ,

and

˜ θ˜ + η := (C(

1 2

− x3))−1 .

(3.32)

^ > 1 and η depends not only on θ˜ but also explicitly on x3. After these changes Note that Θ and skipping of all tildes (2.40) has transformed to the following system for (t, x) ∈ I×Ω

3.2

System with dissipation - constant heat capacity



(compare this to (3.15) for the Boussinesq system) div v = 0 , 1 ∆v + ∇p = θb , Gr η ^ − x3 + θ) v3 − v3 = 2√Di η D(v) · D(v) , √ θ˙ − ∆θ + Di η (Θ Pr Gr Gr v˙ − √

v = 0,

θ=0

on

x3 = 0 and

x3 = 1

(3.33)

(3.34)

plus periodic boundary conditions in x1- and x2-direction, and the initial conditions v(0, x) = v0(x) ,

θ(0, x) = θ0(x) x ∈ Ω .

(3.35)

At the beginning of our calculations we characterise which solutions we want to treat. In general in the context of energy techniques one starts with so-called weak solutions. A characteristic property of weak solutions is, that the L2-norm of their gradients is finite. They are no pointwise solutions to the problem under consideration but solve an integral identity just as in the definition of weak derivatives, where smooth test functions appear26 . One says that they solve it in the weak sense. Namely, for each smooth ϕ with div ϕ = 0 and smooth φ there shall hold27 1 (∇v, ∇ϕ) + (v · ∇v, ϕ) = (θb, ϕ) , Gr η (∂tθ, φ) − ( √ ∆θ, φ) + (v · ∇θ, φ)+ Pr Gr 2 Di ^ − x3 + θ)v3, φ) − (v3, φ) = √ + Di (η(Θ (ηD(v) · D(v), φ) Gr (∂tv, ϕ) + √

(3.36)

on almost all time levels. One sees that system (3.36) makes sense for less regular functions ϕ and φ as long as the integrals exist. Precisely the class in which this scenario for the Navier-Stokes equations works is to take initial values (v0, θ0) ∈ H × L2(Ω) and to construct solutions v ∈ L∞ (I; H) ∩ L2(0, T ; V) , θ ∈ L∞ (I; L2(Ω)) ∩ L2(0, T ; V) .

(3.37)

Note, that the boundary conditions are part of the definition of our spaces. Parabolic embedding then yields that indeed v ∈ C(¯I; H) and θ ∈ C(¯I; L2(Ω)). Thus, the initial conditions are attained. Nevertheless, for our system with Rt dissipation we cannot close the estimates corresponding to (3.37) without controlling 0 kAvk22 ds even if η is constant (see calculations up to (3.47)). For that we make the following agreement about the appropriate spaces for solutions. 26

There are various sound possibilities to choose the integral identity. We will state only the one we use. Remember, that (·, ·)Ω stands for the scalar product in both L2 (Ω) and L2 (Ω)3 , respectively, and we omit the subscript. Since div ϕ = 0 the pressure term drops out. 27

3




3.6 Definition. Let T > 0, I := (0, T ), ¯I := [0, T ]. (i) For (v0, θ0) ∈ V × L2(Ω) a pair (v, θ) is called a weak solution of problem (3.33) - (3.35) on [0, T ] if v ∈ C(¯I; V) ∩ L2(0, T ; D(A)) ,

θ ∈ C(¯I; L2(Ω)) ∩ L2(0, T ; V) ,

and (v, θ) fulfils (3.36) on (0, T ) together with the initial conditions (3.35). (ii) For (v0, θ0) ∈ D(A)×(H2(Ω)∩V) a pair (v, θ) is called a strong solution of problem (3.33) - (3.35) on [0, T ] if v ∈ C(¯I; D(A)) , θ ∈ C(¯I; H2(Ω) ∩ V) ,

∂tv ∈ C(¯I; H) ∩ L2(0, T ; V) , ∂tθ ∈ C(¯I; L2(Ω)) ∩ L2(0, T ; V) ,

and (v, θ) fulfils the equations (3.36) on (0, T ) together with the initial conditions (3.35). In this section we stay close to the Boussinesq approximation, namely we set η = 1. Later on we will treat the more general case η 6= const. in Section 3.3, Theorem 3.14. 3.7 Theorem. Consider problem (3.33) - (3.35) with η = 1. (i) For each pair (v0, θ0) ∈ V × L2(Ω) there exists some T = T (k∇v0k22 + kθ0k22) > 0 and a unique weak solution (v, θ) on [0, T ]. (ii) Moreover, there exist positive constants ζc, Rac(Di), and δ such that the solution (v, θ) exists globally in time if 0 ≤ Di ≤ ζc ,

Ra < Rac(Di) ,

and k∇v0k22 + kθ0k22 ≤ δ .

In this case for some positive constants c and β it holds k∇v(t)k22 + kθ(t)k22 ≤ ce−βt(k∇v0k22 + kθ0k22) . 3.8 Remark. (i) Existence, uniqueness and stability of strong solutions are formulated in Theorem 3.14. They apply to the special case η = 1 as well since all assumptions in Theorem 3.14 are obviously fulfilled in this case. (ii) For each (v0, θ0) ∈ D(A) × V likewise there exist solutions in the class v ∈ C(¯I; D(A)) , θ ∈ C(¯I; V) ∩ L2(0, T ; H2(Ω)) ,

∂tv ∈ C(¯I; H) ∩ L2(0, T ; V) ,

∂tθ ∈ L2(0, T ; L2(Ω))

for some T = T (kAv0k22 + k∇θ0k22) > 0. This solution is unique if η = 1. Moreover, there exist positive constants ζc, Rac(Di ), and δ such that if 0 ≤ Di ≤ ζc and Ra < Rac(Di ), then this solution exists globally in time if kAv0k22 + k∇θ0k22 ≤ δ. In this case (v, θ) satisfies kAv(t)k22 + k∇θ(t)k22 ≤ c e−βt(k∇v0k22 + kθ0k22) . (iii) Theorem 3.13 determines the values of ζc and Rac(Di). They are related to estimate (3.54) which is crucial for global existence and stability results.

3.2




Sketch of the proof of Theorem 3.7. Notice, that in this section always η = 1. To avoid a two-letter symbol throughout all calculations we denote Di by ζ. We apply the energy method28 since it is simple and in some sense near to physics. First the existence of solutions is shown and after that uniqueness. Outline of the existence proof: We construct a solution by the Faedo-Galerkin method. For this we choose suitable smooth orthonormal bases of H and L2(Ω) and build approximations v[n] and θ[n] of v and θ with the corresponding first n basis functions. Due to our construction v[n] and θ[n] are smooth and are determined from a finite system of ordinary differential equations. If we prove that certain norms of v[n] and θ[n] are bounded we finally may (using standard arguments) let n → ∞ finding v and θ. More precisely, let (ωn)n∈N be the orthonormal basis of H consisting of the eigenfunctions of A (the existence of such basis is well-known, see e.g. [6, Lemma 2.14]). In addition this basis is an orthogonal basis in V. Let B be the operator defined by B := −∆ , D(B) := H2(Ω)∩V and (τn)n∈N the orthonormal basis of L2(Ω) consist[n] [n] ing of the eigenfunctions of B. For each n ∈ N let (v0 , θ0 ) be the orthogonal projection in H × L2(Ω) of (v0, θ0) onto the space spanned by (ω1, τ1), · · · , (ωn, τn). We set [n]

v (t, x) :=

n X

[n]

fn,j(t) ωj(x) ,

θ (t, x) :=

j=1

satisfying

n X

gn,j(t) τj(x) ,

(3.38)

j=1

[n]

v[n](0, ·) = v0 ,

[n]

θ[n](0, ·) = θ0 ,

and

1 ∂tv[n], ωj + √ (∇v[n], ∇ωj) + (v[n] · ∇v[n], ωj) = (θ[n]b, ωj) , Gr 1 [n] √ ∂tθ[n], τj + (∇θ[n], ∇τj) + (v[n] · ∇θ[n], τj) = (v3 , τj)− (3.39) Pr Gr ^ − x3 + θ[n])v3[n], τj)+ √2ζ (D(v[n]) · D(v[n]), τj) . −ζ((Θ Gr In contrast to (3.36) (where η = η(θ)) here we applied partial integration to (−∆θ, φ). Inserting (3.38) into (3.39) we find a system of ordinary differential equation for the coefficients fn,j and gn,j. Since we use orthogonal bases most of the terms drop out such that the system becomes decoupled. The apriori estimates below provide uniform in n bounds for |fn,j|2 and |gn,j|2 and the right-hand sides fulfil the ↑ Carathéodory conditions. Thus, the solvability on some time interval is standard. To pass to the limit n → ∞ we must control certain norms. For this we introduce abbreviations for suitable “energies”: Zt 2 2 E0(t) := sup kv(s)k2 + kθ(s)k2 + k∇v(s)k22 + k∇θ(s)k22 ds , 0≤s≤t

2

E1(t) := sup k∇v(s)k2 + 0≤s≤t

E (t) := E0(t) + E1(t) . 28

Zt 0

0

kAv(s)k22 ds ,

(3.40)

See the previous section and [132] for an outline of the method and simple results. For a review of energy methods within the framework of regularity theory for Navier-Stokes equations, see also [49, 127].

3




Note that the norms in E precisely correspond to our class of weak solutions. Obviously, E0(0) = kv0k22 + kθ0k22, E(0) = E0(0) + k∇v0k22. If we insert v[n] and θ[n] into any of the energies we add a superscript [n] to its symbol. This diction applies to [n] [n] E(0), E0(0) as well, e.g. E[n](0) means that E(0) is calculated at (v0 , θ0 ). 3.9 Proposition. There is a constant T0 = T0(E(0)) > 0 such that the approximate solution (v[n], θ[n]) exists on [0, T0], and for all t ∈ [0, T0] it holds the estimate E[n](t) ≤ c E[n](0) , where c and T0 are a positive constants independent of n. [n]

[n]

Since by definition of (v0 , θ0 ) it holds E[n](0) ≤ c1 E(0) = c2 for any n ∈ N Proposition 3.9 shows the boundedness of the norms of (v[n], θ[n]) collected in E, on some time interval [0, T0] independently of n. Thus, we can choose a subsequence of (v[n], θ[n]) (which we denote with the same symbol for convenience) such that for I := (0, T0) v[n] v [n] ∗

v

[n]

θ

v

θ ∗

θ[n] θ

weakly in L2(I; D(A)) , ∗-weakly in L∞ (I; V) , weakly in L2(I; V) ,

∗-weakly in L∞ (I; L2(Ω))

(cf. also the plan of the proof of Theorem 3.27). Finally, the limit in all terms of (3.39) and the denseness of functions fn,j(t) ωj(x) and gn,j(t) τj(x) in the spaces of our testfunctions yield the ↑ convergence of (v[n], θ[n])n∈N to a weak solution (v, θ) on [0, T0], i.e. locally in time. Obviously, (v, θ) obeys the estimate E(t) ≤ c1 E(0) = c2 . To prove global existence, we use the weighted energies Zt βs 2 2 N0(t) := sup e (kv(s)k2 + kθ(s)k2) + β eβs(k∇v(s)k22 + k∇θ(s)k22)ds , 0≤s≤t

2

N1(t) := sup eβsk∇v(s)k2 + β 0≤s≤t

N (t) := N0(t) + N1(t) ,

Zt 0

0

eβskAv(s)k22 ds ,

(3.41)

related to E0(t), E1(t), and E(t), respectively. Note, that N(0) = E(0). However, the first (auxiliary) step here is to estimate the L2-norm of u := (v, θ). For this we apply results about the linearised operator at the motionless state which are derived in the appendix and are formulated in Theorem 3.13. As in the case of the Oberbeck-Boussinesq approximation, we expect that the existence of stable global solutions is connected on the one hand to a “smallness” condition to the data and on the other hand to a critical Rayleigh number. Nevertheless, since there is one additional parameter which characterises our configuration - the Dissipation number - the stability results become more involved here. More precisely, we find that

3.2




. the critical Rayleigh number depends on the Dissipation number: Rac = Rac(ζ), . in addition there is critical Dissipation number ζc > 0. 3.10 Proposition. Let ζc > 0 , Rac(ζ) > 0 be given by Theorem 3.13. Suppose that the solution (v, θ) exists on [0, T ] for some T > 0 and that 0 ≤ ζ ≤ ζc ,

Ra < Rac(ζ) .

as well as

(3.42)

Then there are β0 > 0 and c > 0 such that if 0 < β ≤ β0 it holds for all t ∈ [0, T ] , ku(t)k2H×L2 (Ω)

≤ ce

−2βt

ku(0)k2H×L2 (Ω)

1 1 2 2 + 2 N0(t) + 2 N1(t) . β β

(3.43)

Here the constant c is independent of β and β0 → 0 as Ra → Rac(ζ).

Now we are ready to show the global existence and stability of solutions applying the energy method and Proposition 3.10. 3.11 Proposition. (i) Let β > 0 be fixed. There are constant 0 < T1 = T1(E(0)) ≤ T0, c1 > 0 independent of T1 such that for all t ∈ [0, T1] we observe that N(t) ≤ c1 (E(0) + E(0)2) .

(ii) There exist positive constants β, δ and c2, which do not depend on T , such that if (3.42) holds and N(T ) ≤ δ we find N(T ) ≤ c2(E(0) + E(0)2) .

Again δ → 0 and β → 0 as Ra → Rac(ζ).

Outline of the Uniqueness proof. Let (v1, θ1), (v2, θ2) be two solutions to the same initial value. We set v := v1 − v2, θ := θ1 − θ2, and H(t) := kv(t)k22 + k∇v(t)k22 + kθ(t)k22. The aim is to show, that d H(t) dt

≤ F(t)H(t) ,

where F(t) ∈ L1(0, T ). Since H(0) = 0 we deduce that H(t) = 0 and, thus, v = 0, θ = 0, i.e. the solution is unique and the proof of our theorem is finished. After this short outline we have to elaborate on our proof. We start with: Some nonlinear estimates. In what follows the letter c denotes constants which may vary from line to line and may depend on the periods l1, l2, Pr , Gr , and Di if not specified. Mostly without explicit reference to them we permanently benefit from the Hölder, Poincaré, and Young inequalities, and the Laplace estimates (see Sec. 3.1). 3.12 Lemma. There exist constants c = c(l1, l2) > 0 such that: 1/2

1/2

(i)

kv · ∇θk2 ≤ c k∇vk2k∇θk2 k∆θk2

(ii)

kv · ∇vk2 ≤ c k∇vk2 kAvk2

for v ∈ V, θ ∈ H2(Ω) ∩ V ,

3/2

1/2

for v ∈ D(A) ,

3/2

1/2

for v ∈ D(A) .

(iii) kD(v) · D(v)k2 ≤ c kAvk2 k∇vk2

3




Proof. Clearly, kv · ∇θk2 ≤ kvk6k∇θk3. Then embedding and interpolation yield kvk6 ≤ c k∇vk2 ,

1/2

1/2

1/2

1/2

k∇θk3 ≤ c k∇θk2 kθkH2 ≤ c k∇θk2 k∆θk2

under our assumptions. Inequalities (ii) and (iii) are proved similarly. Proof of Proposition 3.9. Let n ∈ N be fixed. As stated before the smooth approximate solutions (v[n], θ[n]) exist on some time interval, say [0, T ]. The crucial idea is to show that there is c > 0 independent of T such that for t ∈ [0, T ], 2 . (3.44) E[n](t) ≤ E[n](0) + c T E[n](t) + (T 1/2 + T ) E[n](t) The statement of Proposition 3.9 follows from (3.44) by choosing T0 appropriately small at the end of our proof. To show (3.44) we proceed as follows: For j = 1, · · · , n we multiply (3.39)1 by fn,j(t) and add the resulting equations finding 1 1 d [n] 2 [n] k∇v[n]k22 = (v3 , θ[n]) ≤ kv[n]k2kθ[n]k2 . kv k2 + √ 2 dt Gr Now we consider the second equation, i.e. we multiply (3.39)2 by gn,j(t) and add the resulting equations for j = 1, · · · , n in this way obtaining 1 d [n] 2 1 [n] ^ − x3)v[n] √ kθ k2 + k∇θ[n]k22 = −ζ((Θ 3 , θ )+ 2 dt Pr Gr 2ζ [n] [n] + (v3 , θ[n]) − ζ(θ[n]v3 , θ[n]) + √ (D(v[n]) · D(v[n]), θ[n]) . Gr ^ − x3 < c. There are several terms on the Note, (v[n] · ∇θ[n], θ[n]) = 0 and that 0 < Θ right-hand side. The first two terms are bounded by c kv[n]k2kθ[n]k2 while the estimates for the others are similar to the ones in the proof of the nonlinear estimates, namely [n]

| − (v3 θ[n], θ[n])| ≤ kv[n]k3kθ[n]k6kθ[n]k2 1/2

1/2

≤ c k∇v[n]k2 kv[n]k2 k∇θ[n]k2kθ[n]k2 , 3/2

(3.45)

1/2

|(D(v[n]) · D(v[n]), θ[n])| ≤ c k∇v[n]k2 kAv[n]k2 k∇θ[n]k2 . We then find 1 d [n] 2 1 √ k∇θ[n]k22 ≤ F(v[n], θ[n]) , kθ k2 + 2 dt Pr Gr

(3.46)

where 1/2

1/2

1/2

3/2

F(v, θ) := c (kvk2kθk2 + kvk2 k∇vk2 kθk2k∇θk2 + kAvk2 k∇vk2 k∇θk2) . Application of Young’s inequality yields F(v, θ) ≤ c kvk22 + kθk22 + kvk2k∇vk2kθk22 + kAvk2k∇vk32 + γk∇θk22 .

3.2

System with dissipation - constant heat capacity  √ We set γ < 1/( Pr Gr ) and absorb γk∇θk22 at the left-hand side of (3.46). Adding both estimates we finally produce 1 d 1 [n] 2 [n] 2 [n] 2 [n] 2 ˜ [n], θ[n]) , (kv k2 + kθ k2) + √ k∇θ k2 ≤ F(v 2k∇v k2 + dt Pr Gr with ˜ θ) := c (kvk2 + kθk2 + kvk2k∇vk2kθk2 + kAvk2k∇vk3) . F(v, 2 2 2 2

(3.47)

Now for fixed t > 0 we integrate (3.47) over [0, s], where 0 < s ≤ t. Since F˜ is nonnegative for all s ∈ [0, t] we deduce Zs 0

F˜ dτ ≤

Zt

˜F dτ .

0

Moreover, for t < T (skipping for the moment the superscript [n]) we calculate Zt 0

kAvk2k∇vk32 ds

≤ sup

0≤s≤t

≤

Zt 0

≤T

Zt

k∇vk22

0

3/2 Z t

sup k∇vk22

0≤s≤t

1/2

sup 0≤s≤t

1/2 Z t

kAvk22 ds 0

2

k∇vk22

0

1/2 k∇vk22 ds

1/2 2 kAvk2 ds T 1/2 +

Zt 0

2 kAvk22 ds ,

kθk22k∇vk2kvk2 ds ≤ T sup kθk22( sup kvk22 + sup k∇vk22) . 0≤s≤t

0≤s≤t

0≤s≤t

We thus obtain [n] E0 (t)

≤

[n] E0 (0)

2 [n] 1/2 [n] . + c T E0 (t) + (T + T ) E (t)

(3.48)

[n]

Now we need such estimate for E1 (t). Therefor we multiply equation (3.39)1 by λjfn,j(t), where λj is the jth eigenvalue of A, i.e. λr(ωr, v[n]) = (Aωr, v[n]) = (∇ωr, ∇v[n]). We add up obtaining the approximate analog to the choice ϕ = Av in (3.36) and are lead to the equation 1 d 1 [n] k∇v[n]k22 + √ kAv[n]k22 = (θ[n], Av3 ) − (v[n] · ∇v[n], Av[n]) . 2 dt Gr In an obvious way one derives d 1 kAv[n]k22 ≤ c (kAv[n]k2k∇v[n]k32 + kθ[n]k22) , k∇v[n]k22 + √ dt Gr 2 [n] [n] [n] . E1 (t) ≤ E1 (0) + c T E0 (t) + T 1/2 E[n](t)

(3.49)

3




This and (3.48) yield (3.44). 1 Now we choose T0 small enough. Let firstly, T0 ≤ min{1, 2c }. Thus, 1/2

E[n](t) ≤ 2E[n](0) + 4cT0

E[n](t)

2

=: b + a E[n](t)

2

.

(3.50)

√ 1 (1 − 1 − 16ab) solves the Let now 16ab < 1, i.e. T0 ≤ (128cE[n](0))−2. Then X := 4a quadratic equation 12 x = b + ax2. Moreover, we calculate that 2b ≤ X ≤ 4b. Since by definition E[n](t) is continuous in t and 4E[n](0) = 2b ≤ X the relation [n] E (t) ≤ X is true for t in a certain neighbourhood of t = 0. For these t we see E[n](t) ≤ b + a E[n](t)

2

4b 1 = 4E[n](0) . ≤ b + aX2 = X ≤ 2 2

In fact, the relation E[n](t) ≤ 4E[n](0) holds for all t ∈ [0, T0]: To prove this let us assume that on the contrary there is t0 ∈ [0, T0] such that E[n](t0) > 4E[n](0). Then ∃ t1 ∈ [0, T0] such that E[n](t1) = 4E[n](0) and E[n](t) < 4E[n](0) for all 0 ≤ t < t1. If we choose T0 = (128cE[n](0))−2 the estimate (3.50) yields 4E[n](0) = E[n](t1) ≤ 2E[n](0) +

4c(4E[n](0))2 5 [n] = E (0) , 128cE[n](0) 2

which is a contradiction. 1 , (128cc1E(0))−2} Since E[n](0) ≤ c1 E(0) indeed, we can choose 0 < T0 = min{1, 2c and the proof of Proposition 3.9 is complete.

Proof of Proposition 3.10. For simplicity throughout this proof k · k denotes the norm kuk := (kvk22 + kθk22)1/2 for

u := (v, θ) ∈ H × L2(Ω) .

We first re-write our system as evolution problem du + Lu + B1(u) + B2(u) + B3(u) = 0, dt

u(0) = u0 ,

(3.51)

where Lu is the linear and Bi are the nonlinear parts, namely, 

 Lu :=  B1(u) :=

P(v · ∇v) v · ∇θ

−

Pr

1 √

√ Gr

 1 Av − P(θb)  Gr , ^ − x3) − 1 v3 ∆θ + ζ(Θ 

, B2(u) := 

0



 , B3(u) := 2ζ D·D −√ Gr

(3.52)

0 ζθv3

.

3.2




As auxiliary step we consider the eigenvalue problem linearised at the motionless state Lu = σu .

(3.53)

Since Ω is bounded and L is strongly ↑ elliptic29 the spectrum σ(L) of L consists of discrete eigenvalues (σn)n≥1 with Re σ1 ≤ Re σ2 ≤ · · · ≤ Re σn ≤ · · · → +∞ (see [38, Th. 14.6] together with [72, Th. III.6.29]). The eigenvalues of L depend on Ra and ζ and for that we write σj( Ra , ζ). In the appendix we prove 3.13 Theorem. There exist ζc > 0 and Rac(ζ) ≥ Rac such that if 0 ≤ ζ ≤ ζc and Ra < Rac(ζ), then σ1( Ra , ζ) > 0. Moreover, if 0 ≤ ζ ≤ ζc and Ra > Rac(ζ), then σ1( Ra , ζ) < 0. Here the number Rac(ζ) satisfies Rac(0) = Rac

Rac(ζ) > Rac for 0 < ζ ≤ ζc .

and

Theorem 3.13 states that, if 0 ≤ ζ ≤ ζc and Ra < Rac(ζ), then the spectrum σ(L) of the operator L satisfies for some β0 > 0 Re σ(L) ≥ 2β0 > 0 ,

while β0 → 0 as Ra → Rac(ζ) .

From this follows (see [74, p. 289]) that for 0 < β ≤ β0 and t > 0, ke−tL ukH×L2 (Ω) ≤ c e−βtkukH×L2 (Ω) . Since u is the unique solution of (3.51), we can write u as u(t) = e−tL u0 + G1(u) + G2(u) + G3(u) , where for j = 1, 2, 3 Gj(u) := −

Zt 0

e−(t−s)L Bj(u) ds .

By Lemma 3.12 (i), (ii) and the Gagliardo/Nirenberg inequality 3/4

we find

29

1/4

kvk∞ ≤ c kvkH2 kvk2 ≤ c kAvk2 , 3/2 1/2 kB1(u)k ≤ c k∇vk2 kAvk2 + kAvk2k∇θk2 ,

This follows from the estimates in Lemma 3.12.

(3.54)

3




which implies that kG1(u)k ≤ c

Zt

≤ ce

0

3/2 1/2 e−β(t−s) k∇vk2 kAvk2 + kAvk2k∇θk2 ds

−βt

Z t

e

0

+ kG1(u)k2 ≤ c e−2βt

3/4 Z t

βs

k∇vk22 ds

Z t 0

e

1/4

βs

0

kAvk22 ds

+

1/2! 1/2 Z t , eβsk∇θk22 ds eβskAvk22 ds 0

1 1 N0(t)2 + 2 N1(t)2 . 2 β β

Due to the estimate Lemma 3.12 (iii) kG2(u)k2 ≤ c e−2βt ≤ c e−2βt

Z t 0

2 Z t 2! eβsk∇vk22 ds + eβskAvk22 ds

0

1 1 2 N (t) + N1(t)2 . 0 β2 β2

Besides, kB3(u)k ≤ c kθv3k2 ≤ c kθk6kv3k3 ≤ c k∇θk2k∇vk2 ,

e−2βt kG3(u)k ≤ c N0(t)2 , 2 β 2

and the proof is complete.

Proof of Proposition 3.11 We proof (ii). Assertion (i) is proved similarly. First for

1 1 √ 0 < β ≤ min β0, √ , Gr 2 Pr Gr we show N0(t) ≤ E0(0) +

3/2 2 2 c E (0) + N (t) + N (t) + N (t) . 0 0 0 1 β3

(3.55)

Here β0 stems from Proposition 3.10 and c > 0 is independent of β. We follow the line of the proof of Proposition 3.9 up to estimate (3.47). The main difference is that instead of the approximate formulation of Proposition 3.9 here we directly choose ϕ = v and φ = θ in (3.36). Obviously, then we obtain 1 1 d 2 2 2 2 ˜ θ) . kvk2 + kθk2 + √ k∇θk2 ≤ F(v, 2k∇vk2 + dt Pr Gr

(3.56)

3.2




Inequality (3.55) is a consequence of (3.56), Proposition 3.10 and the following standard arguments: (a) On the left-hand side of (3.56) one partially applies the Poincaré inequality, namely 2 2 2 2 2 2 ^ ^ ^ 2β(k∇vk 2 + k∇θk2) ≥ β(k∇vk2 + k∇θk2) + c β(kvk2 + kθk2) .

(b) The resulting inequality is multiplied by eβt and integrated taking into account d βt d e f(t) = βeβtf(t) + eβt f(t) . dt dt (c) Finally, for r ∈ N it holds Zt 0

βs

r

e b(s) ds ≤

βs

sup e b(s) 0≤s≤t

r

Zt

e−β(r−1)s ds =

0

c βr−1

r sup eβsb(s) .

0≤s≤t

In a similar manner to above (see (3.49)), using Proposition 3.10 we deduce for

1 0 < β ≤ min β0, √ 2 Gr and c > 0 independent of β that N1(t) ≤ E1(0) +

3 2 2 c . + N (t) + N (t) E (0) + N (t) 1 1 0 0 β4

(3.57)

It then follows from (3.55) and (3.57) that, N(t) ≤ E(0) +

3 2 3/2 c + N(t) + N(t) E(0) + N(t) β4

(3.58)

with c > 0 independent of β. This yields the assertion of Proposition 3.11 (ii). Uniqueness proof. Recall that v := v1 − v2 and θ := θ1 − θ2. To derive the necessary estimates consider the difference between the equations (3.36) for v1 and v2 and the ones for θ1 and θ2, respectively. After an obvious reformulation of mixed terms we obtain the following weak formulation: 1 (∇v, ∇ϕ) = −(v · ∇v1, ϕ) − (v2 · ∇v, ϕ) + (θb, ϕ) , Gr 1 2ζ √ (∂tθ, φ) + (∇θ, ∇φ) = √ (D(v) · D(v1 + v2), φ) − ζ(θv1 · b, φ)− Pr Gr Gr ^ − x3 − 1 )v3, φ) − (v · ∇θ1, φ) − (v2 · ∇θ, φ) . −ζ(θ2v3, φ) − ζ((Θ ζ (3.59) (∂tv, ϕ) + √

3




First we insert ϕ = v and find since (v2 · ∇v, v) = 0 1 d 1 kvk22 + √ k∇vk22 = −(v · ∇v1, v) + (θ, v3) . 2 dt Gr

(3.60)

To the terms on the right-hand side we apply Lemma 3.12 and Young’s inequality for some small γ > 0. Hence, 1/2

1/2

| − (v · ∇v1, v)| ≤ c k∇vk2k∇v1k2 kAv1k2 kvk2

≤ γk∇vk22 + c kvk22(k∇v1k22 + kAv1k22) ,

|(θ, v3)| ≤ c (kvk22 + kθk22) . Now we set ϕ = Av and obtain 1 1 d k∇vk22 + √ kAvk22 = −(v2 · ∇v, Av) − (v · ∇v1, Av) + (θ, Av3) . 2 dt Gr Analogously we deduce (for the second term we use Young with 1/2

1 4

(3.61)

+ 43 )

1/2

| − (v · ∇v1, Av)| ≤ c k∇vk2k∇v1k2 kAv1k2 kAvk2

≤ γkAvk22 + c k∇vk22(k∇v1k22 + kAv1k22) , 1/2

1/2

| − (v2 · ∇v, Av)| ≤ c k∇v2k2k∇vk2 kAvk2 kAvk2 ≤ γkAvk22 + c k∇vk22k∇v2k42 ,

|(θ, Av3)| ≤ γkAvk22 + c kθk22 . Now we consider equation (3.59)2 with φ = θ. There are a lot of terms on the right-hand side but the estimates are similar to the ones above and we state only two of them: | − (v · ∇θ1, θ)| = |(v · ∇θ, θ1)| ≤ kvk6k∇θk2kθ1k3 1/2

1/2

≤ c k∇vk2k∇θk2k∇θ1k2 kθ1k2

≤ γk∇θk22 + ck∇vk22k∇θ1k2kθ1k2 , 1/2

1/2

|(D(v) · D(v1 + v2), θ)| ≤ c k∇vk2k∇(v1 + v2)k2 kkA(v1 + v2)k2 k∇θk2

≤ γk∇θk2 + c (kA(v1 + v2)k22 + k∇(v1 + v2)k22)k∇vk22 .

All terms with small γ are put to the left-hand side. And we collect F(t) := c(1 + k∇v1k22 + k∇v2k22 + k∇v2k42 + kAv1k22 + kAv2k22 + k∇θ1k22 + k∇θ2k22) . Since (v1, θ1) and (v2, θ2) are weak solutions to our problem, F(t) ∈ L1(0, T ) and the last step, namely the uniqueness proof of our theorem is finished.

3.3

System with dissipation - variable heat capacity



"

3.3 System with dissipation - variable heat capacity As we have seen in Section 1.1 while modelling more complicated systems like the convection of the earth mantle often it is too na¨ıve to suppose that ↑ viscosity, heat conductivity and/or heat capacity are constant. One more artistic way for the ↑ viscosity has been ˜ shown in the context of power-law rheology. Another direction is to let ν, κ and/or C ˜ ˜ vary with the temperature. In our modelling we included the last case, namely C = C(θ). From the mathematical point of view this means that some manipulations of the previous section become impossible. In this section we will show how the results and the proofs have to be modified for variable heat capacity. On the one hand we will see that in this case we need two types of assumptions to the behaviour of η. On the other hand (even with these assumptions) it is not possible to close the estimates for weak solutions. Instead we need several higher order norms which lead to the class of strong solutions as appropriate existence and uniqueness class. Note, that existence can be proved in a slightly less regular class (see Remark 3.8 (ii)), but uniqueness is then open. We start this section by formulating our results. 3.14 Theorem. (i) Assume that η is a smooth function satisfying d 0 < d1 ≤ η(θ, x3) ≤ d2 , η(θ) ≤ d3 for all 0 ≤ x3 ≤ 1 dθ

(3.62)

for some constants dj (j = 1, 2, 3). Then for each (v0, θ0) ∈ D(A) × (H2(Ω) ∩ V), there exist T0 > 0 and a unique strong solution (v, θ) on [0, T0]. Here T0 depends on dj (j = 1, 2, 3), the H2-norm of (v0, θ0), and the physical parameters. (ii) In addition to (3.62), assume also that ˜ h) = η(0, x3) = 1. C(θ

(3.63)

Then there exist ζc > 0, Rac(Di) and δ > 0 such that if 0 ≤ Di ≤ ζc and Ra < Rac(Di), then for each (v0, θ0) with ^ E(0) := kAv0k22 + kθ0k2H2 (Ω) ≤ δ , the strong solution (v, θ) exists on [0, ∞) and satisfies

^ kAv(t)k22 + k∂tv(t)k22 + kθ(t)k2H2 (Ω) + k∂tθ(t)k22 ≤ c1e−βtE(0)

for some positive constants c1 and β, which means that the motionless state is asymptotically stable. Here, δ → 0 and β → 0 as Ra → Rac(Di). Furthermore, Rac(0) = Rac ,

Rac(Di) > Rac

for 0 < Di ≤ ζc .

If Ra > Rac(Di) then the motionless state is unstable.

3




3.15 Remark. (i) Under assumption (3.62) likewise, for each (v0, θ0) ∈ D(A) × V there exists a solution of problem (3.33) - (3.35) in the class of Remark 3.8 (ii) for some T = T (kAv0k22 + k∇θ0k22) > 0. Moreover, if (3.63), 0 ≤ Di ≤ ζc, and Ra < Rac(Di) hold, then this solution exists globally in time provided that kAv0k22 + k∇θ0k22 ≤ δ for some δ > 0. In this case (v, θ) satisfies kAv(t)k22 + k∇θ(t)k22 ≤ c e−βt(k∇v0k22 + kθ0k22) . However, the uniqueness of such solutions in that class is an open problem if η is not constant (cf. also Remark 3.20). (ii) In physical terms condition (3.63) means that the non-constant ↑ specific heat is connected to the deviation from pure conduction. Without (3.63) the analysis of the linearised operator at the motionless state becomes much more complicated since η(0) depends on x3. (iii) Again, Theorem 3.13 determines the values of ζc and Rac(Di). (iv) For the moment let ↑ viscosity and thermal conductivity be non-constant, namely, they depend on θ and thus κ∆θ is replaced by div (κ(θ)∇θ) and ν∆v by div (ν(θ)∇v). Assume that the bounds (3.62) and d d 0 < ν1 ≤ ν(θ) ≤ ν2, | dt ν(θ)| ≤ ν3, 0 < κ1 ≤ κ(θ) ≤ κ2, | dt κ(θ)| ≤ κ3

hold for some constants dj, νj and κj (j = 1, 2, 3). In this case the same assertions as in Theorem 3.14 can be proved in a similar manner provided the analogs of (3.63) hold for κ and ν. To start our proof let us re-consider the weak formulation of the temperature equation: (∂tθ, φ) − (

η(θ) √ ∆θ, φ) + (v · ∇θ, φ)+ Pr Gr

^ − x3 + θ)v3, φ) − (v3, φ) = √2ζ (η(θ)D(v) · D(v), φ) . +ζ(η(θ)(Θ Gr

(3.36)

We immediately see that . the main difference is the estimate for (η∆θ, θ). We note that −(η∆θ, θ) = (η∇θ, ∇θ) + (∇η · ∇θ, θ) . Rt Lemma 3.16 (iii) shows that additionally the term 0 k∆θ(s)k22 ds enters the estimates for E0. Thus, we must ensure that it isRbounded. The corresponding energy estimate t (choosing φ = −∆θ in (3.36)) creates 0 kAvkr2 ds with r > 2 on the right-hand side and thus, we must find a bound for sup kAvk2, too. Through the equation of motion, Av is connected to ∂tv. . For the two other η-terms we can proceed as in Section 3.2 after application of the bounds d1 and/or d2, respectively, for η (see Lemma 3.16 (i), (ii)).

3.3




All the above mentioned higher order derivatives are included into E2(t), namely E2(t) := sup kAv(s)k22 + k∂tv(s)k22 + k∇θ(s)k22 + 0≤s≤t Zt + k∇∂tv(s)k22 + k∆θ(s)k22 + k∂tθ(s)k22 ds , 0 Zt 2 2 (3.64) E3(t) := sup k∆θ(s)k2 + k∂tθ(s)k2 + k∇∂tθ(s)k22 ds , 0≤s≤t

˜ (t) := E(t) + E2(t) , E ˜ + E3(t) , ^ (t) := E(t) E

0

˜ E(0) := E(0) + kAv0k22 + k∇θ0k22 , ˜ ^ E(0) := E(0) + k∆θ0k22 .

The norms in E3(t) enter our scheme in the uniqueness proof (see Remark 3.20). Throughout this section we always silently assume that (3.62) holds. Since in principle the proof of Theorem 3.14 resembles that of Theorem 3.7, it is enough to focus on the differences while using the sketch of the proof in the last section. Some nonlinear estimates. 3.16 Lemma. There exist constants c = c(l1, l2) > 0 such that: (i)

For γ > 0, θ ∈ V, v1 ∈ D(A), v2 ∈ V,

|(η(θ, x3)D(v1) · D(v2), θ)| ≤ γk∇θk22 + c d22k∇v1k2kAv1k2k∇v2k22 .

(ii)

For v = (v1, v2, v3) ∈ V, θ1 ∈ V, θ2 ∈ L2(Ω), 1/2

1/2

(η(θ, x3)θ1v3, θ2) ≤ cd2k∇θ1k2kvk2 k∇vk2 kθ2k2 . (iii) For γ > 0, θ1, θ2 ∈ H2(Ω) ∩ V, 2 −(η(θ1, x3)∆θ2, θ2) ≥ (d1 − γ)k∇θ 2k2− 1/2 3/2 2 − cd3 kθ2k2 k∇θ2k2 k∆θ1k2 + d3kθ2k2 .

Proof. (i) We calculate

|(η(θ, x3)D(v1) · D(v2), θ)| ≤ d2kD(v1)k3kD(v2)k2kθk6 1/2

1/2

≤ cd2k∇v1k2 kAv1k2 k∇v2k2k∇θk2

≤ γk∇θk22 + cd22k∇v1k2kAv1k2k∇v2k22 . The inequality (ii) can be proved similarly. Relation (iii) is shown in the following way: − (η(θ1, x3)∆θ2, θ2) = (η(θ1, x3)∇θ2, ∇θ2) + (∇θ2, θ2∇η(θ1, x3)) ≥ d1k∇θ2k22 − d3k∇θ2k2kθ2k3k∇θ1k6 − d3k∇θ2k2kθ2k2 1/2

3/2

≥ (d1 − γ)k∇θ2k22 − cd3(kθ2k2 k∇θ2k2 k∆θ1k2 + d3kθ2k22)

for arbitrary γ > 0. This completes the proof.

3




3.17 Lemma. Let θ, θ1, ∈ V, θ2 ∈ H2(Ω) ∩ V, v ∈ V, v1, v2 ∈ D(A) while η 0 := d η(θ, x3). Then there exist constants c = c(l1, l2) > 0 such that for γ > 0 dθ |(η 0 θ1∆θ2, θ1)| ≤ γk∇θ1k22 + cd43kθ1k22k∆θ2k42 ,

(i) (ii)

|(η 0 θD(v1) · D(v2), θ)| ≤ γk∇θk22 + cd23kθk22kAv1k22kAv2k22 .

Proof.

|(η 0 θ1∆θ2, θ1)| ≤ d3kθ1k6k∆θ2k2kθ1k3 3/2

1/2

≤ cd3k∇θ1k2 kθ1k2 k∆θ2k2

≤ γk∇θ1k22 + cd43kθ1k22k∆θ2k42 ,

|(η 0 θD(v1) · D(v2), θ)| ≤ d3kθk23kD(v1)k6kD(v2)k6 ≤ cd3k∇θk2kθk2kAv1k2kAv2k2

≤ γk∇θk22 + cd23kθk22kAv1k22kAv2k22 .

Proposition 3.9 must be reformulated as follows. ˜ 3.18 Proposition. There is a constant T0 = T0(E(0)) > 0, such that the approximate [n] [n] solutions (v , θ ) exist on [0, T0], and for all t ∈ [0, T0] it holds ˜ [n](t) ≤ c E ˜ [n](0) E ^ [n](t) with T0 = T0(E(0)) ^ whilst c > 0 is independent of n. The same holds for E > 0. Proof of Proposition 3.18. In the same way as in the Proof of Proposition 3.9 we derive an estimate (3.46) for the norms in E0, namely we show 1 d1 d [n] 2 [n] 2 [n] 2 [n] 2 ˜ [n], θ[n]) . (kv k2 + kθ k2) + √ k∇θ k2 ≤ F(v 2k∇v k2 + dt Pr Gr The estimates for v[n] are identical, while for θ[n] we have one crucial difference for η 6= const., namely (omitting [n] for a while) −(η∆θ, θ) ≥ (d1 − γ)k∇θ2k22 − c (kθk22 + k∇θk62 + k∆θk22) . ˜ θ), i.e., Thus in addition to the terms in (3.47) we include k∇θk62 + k∆θk22 into F(v, ˜F(v, θ) := c (kvk2 + kθk2 + kvk2k∇vk2kθk2 + kAvk2k∇vk3 + k∇θk6 + k∆θk2) . 2 2 2 2 2 2 The calculations for E1 are the same, but we cannot close the estimates at this point since we need a bound for ∆θ[n] as well. Clearly, we choose φ =R −∆θ[n] in the approximate formulation of (3.36) since this t should provide a bound for 0 k∆θ[n](s)k22 ds. Note that here there is no serious difference between the case η = 1 and η 6= const.. The obvious result is 1 d d1 √ k∆θ[n]k22 ≤ F1(v[n], θ[n]) . k∇θ[n]k22 + 2 dt Pr Gr

3.3


To find the precise expression for F1 we apply Lemma 3.12 (i), Lemma 3.16 (ii) with θ2 = −∆θ, Lemma 3.12 (ii), and Young’s inequality to produce γk∆θ[n]k22- terms in all four estimates on the right-hand side. Absorbing these small terms on the left we find d1 d √ k∇θ[n]k22 + k∆θ[n]k22 ≤ F˜1(v[n], θ[n]) , dt Pr Gr where F˜1(v, θ) := c (k∇vk32kAvk2 + kvk2k∇vk2k∇θk22 + kvk22 + k∇vk2kAvk32) . Unfortunately, here we come up with the expression k∇vk2kAvk32. Thus, we have to proceed to find estimates for sup kAvk2, too. This is done in the next step differentiating (3.39)1 with respect to t and choosing the test function ∂tv[n] (note, that (v[n] · ∇∂tv[n], ∂tv[n]) = 0) arriving at 1 1 d [n] k∇∂tv[n]k22 = −(∂tv[n] · ∇v[n], ∂tv[n]) + (∂tθ[n], ∂tv3 ) . k∂tv[n]k22 + √ 2 dt Gr The estimates are obvious and we put γk∇∂tv[n]k22 -terms to the left. Nonetheless, to find ˜ we must replace the quantities ∂tv[n] and ∂tθ[n] through expresour bounds in terms of E sions derived from the equations, namely (we skip all superscripts [n] for the moment) k∂tvk22 ≤ c (kAvk22 + kv · ∇vk22 + kθk22) ≤ c (kAvk22 + kAvk42 + kθk22) ,

k∂tθk22 ≤ c (k∆θk22 + kAvk22k∇θk22 + kvk22 + kAvk42) .

˜ Now we may proceed as in the proof of Proposition 3.9 to finish the demonstration for E since relation ∂tv(0) = − Gr −1/2Av0 − P(v0 · ∇v0) + P(θ0b) ˜ ˜ 2. + E(0) provides k∂tv(0)k2 ≤ E(0) 2

^ the equation (3.39)2 is differentiated with respect to t and To handle E3(t) (and thus, E) [n] then tested by ∂tθ . Analogously to ∂tv(0) one treats ∂tθ(0). The precise estimates for E3(t) are technical, but straightforward applying again our nonlinear estimates, especially Lemma 3.17. This completes the proof of Proposition 3.18 The useful version of Proposition 3.11 is now: ^ 3.19 Proposition. (i) Let β > 0 be fixed. There are constants 0 < T1(E(0)) ≤ T0, c > 0 independent of T1 such that for all t ∈ [0, T1], ^ ^ ^ 2) , N(t) ≤ c (E(0) + E(0)

˜ ˜ ˜ 2) . N(t) ≤ c (E(0) + E(0)

(ii) Let 0 ≤ ζ ≤ ζc, Ra < Rac(ζ) and (3.62) as well as (3.63) hold. Then there exist ^ ) ≤ δ we find positive constants β, δ and c, which do not depend on T , such that if N(T ^ ) ≤ c(E(0) ^ ^ 2) . N(T + E(0) ˜ ˜ The same holds for N(t) and E(0). Again, δ → 0 and β → 0 as Ra → Rac(ζ).



3




Proof of Proposition 3.19. We mainly focus on the differences between the cases η 6= const. and η = 1. The principal line of the proof is the same as for Proposition 3.11, i.e. we may re-write the estimates of Proposition 3.18 for the weak solution and obtain first an estimate of the form (3.56). Of course, in comparison to (3.56) here additionally a term with k∆θk22 appears in ˜F(v, θ). Then we derive the same higher order estimates as in Proposition 3.18 for our weak solution. Besides, Proposition 3.10 has to be changed accordingly, namely (3.43) is replaced by ! 3 1 X 2 2 −2βt 2 ku(0)kH×L2 (Ω) + 2 ku(t)kH×L2 (Ω) ≤ c e (3.65) Ni(t) β i=0 and we modify B3(u) to  B˜3(u) := 



0

^ − x3) + η θ v3 − ζ (η − 1)(Θ

. η−1 √ ∆θ Pr Gr

(3.66)

This change of B3(u) means that the linear part Lu remains the same as in the proof of Proposition 3.10. Here condition (3.63) enters the proof and yields Z1 η − 1 = θ η 0 (τθ)dτ . 0

η−1 √ ∆θ in B˜3(u) may be treated as a perturbation while the estimates Pr Gr for the other new terms in B3 are obvious. ˜ re-written for the weak ^ and E Hence, along the steps (a) – (c) from the estimates for E ˜ respectively, if β is chosen appropri^ and N, solution we obtain an analog of (3.58) for N ately. From this estimate follows statement (ii). (i) is shown similarly.

Thus the term −

Uniqueness. The uniqueness of strong solutions is shown in the same way as the uniqueness of weak solutions in the previous section and we omit most details. Let (v1, θ1), (v2, θ2) be two solutions to the same initial value. We set v := v1 − v2 and θ := θ1 − θ2. With the help of the estimate 3/4

1/4

kθk∞ ≤ c k∆θk2 kθk2

we derive

|((η(θ1) − η(θ2))∆θ3, θ4)| ≤ d3kθ1 − θ2k∞ k∆θ3k2kθ4k2 3/4

1/4

≤ c d3k∆θk2 kθk2 k∆θ3k2kθ4k2

and |(η(θ1)D(v1) · D(v1) − η(θ2)D(v2) · D(v2), θ)| ≤ |((η(θ1) − η(θ2))D(v1) · D(v1), θ)| + |(η(θ2)D(v) · D(v1 + v2), θ)| 3/4

1/4

3/2

1/2

≤ c (d3k∆θk2 kθk2 kAv1k2 k∇v1k2 kθk2+ 3/4

1/4

3/4

1/4

+ kAvk2 k∇vk2 kA(v1 + v2)k2 k∇(v1 + v2)k2 kθk2) .

(3.67)

3.4

Power-law rheology



From these estimates we deduce that d H(t) ≤ F(t)H(t) , dt where H(t) = (kv(t)k22 + k∇vk22 + kθk22 + k∇θk22)(t) with H(0) = 0 and F(t) is some function in L1(0, T ). It is now easy to deduce that H(t) = 0 identically and the proof of our theorem is finished.

3.20 Remark. The reason that we cannot prove uniqueness in the class of Remark 3.8 (ii) is the estimate (3.67). It leads to the term k∆θ2k82 in F(t) for which we need that θ ∈ C(¯I; H2(Ω)) in the existence proof.

% ' (

3.4 Power-law rheology In Sections 2.3 & 2.4 we considered generalised Newtonian fluids and derived two types of approximations for the popular power-law model and a modified power-law. Now we want to provide existence and solvability theory for these equations. Let us consider the simplest case, i.e. the Boussinesq-like approximation for constant heat capacity characterised by the following system div v = 0 , v˙ −

for

1 î + ∇^ div T p = (0, 0, θ)> , Re ˜ θ˙ − ∆θ = 0 Pr Re C

^1 = kD(v)kr−1D(v) , T

and

^2 = 1 + kD(v)k T

(2.49)

r−1 D(v) .

The difference to the equations for the Newtonian case is that in the velocity equation î, i = 1, 2, for some r > 0. Thus, instead of the Laplace operator we come across div T î becomes a gradient . as before, in the weak formulation the divergence applied to T applied to the test function, . the parameter r characterises mainly the behaviour of the system and its solutions. ^ with The latter is a good reason not treat the concrete examples above but any tensor T similar properties. ^ fulfils one of the Precisely, we assume for some constants c1 and c2 that the tensor T conditions (T1)1 or (T1)2, respectively, and (T2) which are formulated below. kD(v)kr+1 ^ (T1) T(D(v)) · D(v) ≥ c1 coercivity , (kD(v)kr+1 − 1)

3

 ^ kT(D(v))k ≤ c2(1 + kD(v)k)r

(T2)


polynomial growth .

As Lemma 3.22 states a sufficient condition for (T1) and (T2) to hold is There is a scalar potential U = U(D) ∈ C2(R9) with the properties: U(0) = 0 . For i, k = 1, 2, 3 it holds ∂Dik U(0) = 0 as well as Tik = ∂Dik U(D) . There are constants ci > 0 such that for all symmetric scalar 3 × 3-matrices F and for the tensor (∂2Dmn DrsU(D))3m,n,r,s=1 =: U 00 (D) we find

(U)

00

(F : U (D)) · F ≥ c2

kDkr−1kFk2 , (1 + kDk)r−1kFk2 ,

(3.68)

kU 00 (D)k ≤ c1(1 + kDk)r−1 .

Note, that in case r ≥ 1 the property (3.68)2 implies (3.68)1. Clearly, examples for extra stress tensors fulfilling the growth condition and having a potential are power-law and modified power-law models (see also [84, 1.1.69, 1.1.73]). ^ = 2µD with µ = µ(kDk2) the related Namely, for any extra stress tensor of the form T potential is Z k Dk2 U(D) = µ(s) ds . (3.69) 0

Then for some constant µ0 > 0 the choice µ1(s) = µ0s(r−1)/2 produces U1 =

2µ0 kDkr+1 and r+1

^1 · D = 2µ0kDkr−1D · D = 2µ0kDkr+1 , T

while µ2(s) = µ0(1 + s)(r−1)/2 leads to 2µ0 2µ0 2µ0 (1 + kDk)r+1 − (1 + kDk)r + , r+1 r (r + 1)r ^2 · D = 2µ0(1 + kDk)r−1kDk2 . T U2 =

^1 fulfils (3.68)1 while T ^2 satisfies (3.68)2. One can check, that T Preparations. In the previous sections we could directly benefit from the identity −(∆v, v) = k∇vk22 for all

v∈V.

^ together Now the situation is more involved. Assumption (T1)1 and the symmetry of T with Korn’s inequality Lemma 3.21 for suitable v lead to Z Z Z Z ^ · v dx = T ^ · ∇v dx = T ^ · D(v) dx ≥ c1 kD(v)kr+1 dx ≥ c2 k∇vkr+1 − div T r+1 . Ω

Ω

Ω

Ω

3.4

Power-law rheology



Thus, for the velocity the L2-setting with the ↑ Hilbert space structure is adequate only in the special case r = 1 and the scalar product, which we used in the weak formulation of the two previous sections, has to be replaced by the duality pairing of Lr+1 and its dual. Furthermore, in this section we prescribe periodic boundary conditions for v (see (3.72) and Remark 3.28). For these two reasons the apt space for the velocity is Vp, defined as follows: Ckper(Ω) := Ck(Ω) ∩ {φ : li-periodic in xi (i = 1, 2) , 1-periodic in x3} , Z 3 ∞ ∞ Cper,σ(Ω) := Cper(Ω) ∩ {Φ : div Φ = 0, Φ dx = 0} ,

k ∈ N,

Ω

H :=

closure of

Vp :=

closure of

C∞ per,σ(Ω) C∞ per,σ(Ω)

in

L2(Ω)3 ,

in

Wp1(Ω)3 ,

p := r + 1 . (3.70)

For θ the heat equation still produces estimates and results in the L2-setting. At the same moment our force term is created by θ. Thus, we cannot directly use results which put the data in the appropriate dual space of v (see e.g. [84, Ch. 5]). Moreover, we cannot split the occurring scalar products of v and θ in the straightforward way applied up to now. The treatment of so-called “modified Navier-Stokes equations” with a nonlinear extra stress tensor was started independently by Ladyˇzenskaya and Lions with different motivation around 1970 (see e.g.[77, 82] and in addition Kaniel [67]). In principle they used the theory of monotone operators together with compactness arguments to show the existence of weak solutions in case p ≥ 11 under the conditions (T1) and (T2). These 5 5 solutions are unique if v0 ∈ H and p ≥ 2 . Their theory is applicable to the Dirichlet problem as well. The obvious unsatisfactory fact is that the case p = 2 for which the existence of weak solutions had been proved before by different methods is excluded. Moreover, a lot of applications in engineering sciences are related to exponents p < 2 (i.e. ↑ shear thinning behaviour). The classical results have been improved and completed in various directions since then. Especially in the 1990th the problem became interesting to Neˇcas and co-workers (in the context of multipolar fluids) and there is a whole bunch of related papers in that period which are summarised in the monograph [84]. The central idea was to introduce a weaker class of solutions, the so-called Young measure valued solutions and then to investigate under which conditions they become weak and strong ones. In particular they could shift the bound in the existence proof for weak solutions with periodic boundary conditions to [13, 85]. Strong solutions for small data 6 ≥ p > 59 and for strong solutions to p ≥ 11 5 together with the stability of the rest state were investigated in [88] for 6 ≥ p ≥ 35 . Since [84] several improvements, generalisations (e.g. electrorheological fluids, pressure & viscosity-dependent shear), and numerical estimates have been provided. Concerning existence of weak and strong solutions and long time behaviour we refer to [24, 27, 29, 36, 37, 78, 86, 87, 110, 118, 119] while regularity questions were discussed in [39, 48, 68–70, 78, 79, 124, 125] and numerical schemes in [28, 111]. All these results concern the velocity equations with shear dependent viscosity. The coupled setting with the heat equation in which we are interested has been studied

3




in [89] using the technique presented in [84] and in [47]. A bit later Amann [1, 2, 4] analysed this subject from a different and more general point of view. His results have been summarised before in Section 3.1. Now let us return to the equations. To simplify the handling and since the size of our ˜ Pr := 1, and b := (0, 0, 1). parameters does not matter we set Re := 1 as well as C To avoid confusion with the parameter p we write π := p ^ for the pressure term. Again, we investigate the deviation from pure conduction introducing the variables declared in (3.31). Precisely, the system under consideration for Ω := (0, l1) × (0, l2) × (0, 1) and I := (0, T ) is (compare this to systems (3.15) and (3.33)) div v = 0 , ^ v˙ − div (T(D(v))) + ∇π = θb , θ˙ − ∆θ − v3 = 0 .

(3.71)

We choose the initial values (3.35), i.e. v(0, x) = v0(x), θ(0, x) = θ0(x), x ∈ Ω. For v we require periodic boundary conditions, precisely, v|x1 =0 = v|x1 =l1 , ∇v|x1 =0 = ∇v|x1 =l1 ,

v|x2 =0 = v|x2 =l2 , ∇v|x2 =0 = ∇v|x2 =l2 ,

v|x3 =0 = v|x3 =1 , ∇v|x3 =0 = ∇v|x3 =1 ,

(3.72)

while for θ we keep the setting of the previous sections, namely θ=0

on x3 = 0

and x3 = 1 ,

(3.73)

and periodic boundary conditions in x1- and x2-direction. Notice, that one may apply the Poincaré inequality to functions in Vp and V and that here D(A) := H2(Ω)3 ∩ V2. Now we collect the auxiliary tools for this section. 3.21 Lemma. Korn [84, 1.2.11]. Let 1 < p < ∞ and v ∈ Vp. Then there is a constant which only depends on geometrical properties of the domain such that c kvk1,p ≤ kD(v)kp .

The aim of the following two lemmata is to show the connection between the conditions (T1), (T2) on the one side and (U) on the other side and to provide crucial estimates for the new nonlinear stress term. The consequences of (3.68)1 are always in the first line behind braces and those of (3.68)2 below. ^ satisfy (U). Then 3.22 Lemma. [84, Lemma 5.1.19]. Let p > 1 and T ^ T(D) · D ≥ c1

kDkp , (kDkp − 1) ,

^ kT(D)k ≤ c2 (1 + kDk)p−1 .

(3.74) (3.75)

3.4

Power-law rheology



In case p ≥ 2 in addition ^ T(D) · D ≥ c3(kDk2 + kDkp) ,

(3.76)

^ ^ hT(D(v)) − T(D(u)), D(v − u)i ≥ c4kD(v − u)kp , ^ ^ hT(D(v)) − T(D(u)), D(v − u)i ≥ c5 kD(v − u)kp + kD(v − u)k2 .

3.23 Remark. An interesting consequence of estimate (3.76) and Korn’s inequality is that for the case p ≥ 2 & (T1)1 in the first energy estimates (compare Remark 3.3) we can resolve the right-hand sides (θ, v3) such that for small enough constants the outcome can be hidden at the left-hand side. More clearly this means that there is a critical Rayleigh number and an unconditionally stable region. We will not dwell on that question. ^ satisfy (U). Then 3.24 Lemma. [84, Lemma 5.1.35]. Let p > 1 and T p

c1(1 + kDk) ≥ U(D) ≥ c2

kDkp , kDkp − pkDk .

(3.77)

To deduce Lemma 3.24 from Lemma 3.22 one directly uses U(D) =

Z1 0

d U(sD) ds = ds

Z1 X 3

∂U(sD) Dij ds = 0 i,j=1 ∂Dij

Z1 0

1^ T(sD) · Ds ds . s

3.25 Lemma. [84, Lemma 5.2.44]. Assume I := (0, T ) for some T < ∞. (i) Let u, v ∈ Vp. Then hu · ∇u, vi is finite for p ≥ 59 . ZT 11 (ii) If u ∈ Lp(I; Vp) ∩ L∞ (I; H), v ∈ Lp(I; Vp), p ≥ 5 then hu · ∇u, vi dt is finite. 0

Proof. (i) By embedding, each function which fulfils our boundary condition and is in 3p . If p ≥ 95 , then we observe Wp1(Ω)3 is contained in Ls(Ω) for s := 3−p 1≥

1 1 3−p 1 +2 = +2 p s p 3p

and Hölder’s inequality yields Z u · ∇uv dx ≤ kuksk∇ukpkvks . Ω

(ii) We define q such that with s from (i) we guarantee 1s + p1 + q1 = 1, i.e. q = p ∈ [ 59 , 12 ] we may use the interpolation inequality 5 kvkq ≤ kvk21−αkvkα s

with α =

12−5p . 5p−6

3p . 4p−6

For

3




Then Hölder’s inequality and that interpolation provide Z T Z ZT ZT 1+α u · ∇u v dx dt ≤ kvksk∇ukpkukq dt ≤ c kvkskuk1−α dt 2 k∇ukp 0 Ω 0 0 ZT ≤ c kukL∞ (I;H) kvksk∇ukp1+α dt 0 p−1 ZT p (1+α) p−1 p . dt k∇ukp ≤ c kukL∞ (I;H)kvkLp (I;Vp ) 0

Between the last two lines we applied Hölder’s inequality with the exponents p and p 0 as well as the embedding kvks ≤ k∇vkp. The terms on the right-hand side are finite if p (1 + α) p−1 ≤ p, i.e. as long as p ≥ 11 . 5

Formulation of the results. Let always I := (0, T ) for some T < ∞. 3.26 Definition. The couple (v, θ) is a weak solution of (3.71), (3.72) if for a.e. t > 0 ^ h∂tv, ϕi + hv · ∇v, ϕi + hT(D(v)), D(ϕ)i = hθb, ϕi , (∂tθ, φ) + (v · ∇θ, φ) + (∇θ, ∇φ) = (v3, φ)

(3.78)

hold for ϕ ∈ C1(¯I; Vp) and φ ∈ C1(¯I; V) with ϕ(T ) = 0, φ(T ) = 0. 3.27 Theorem. Let Y := V2 ∩ H2(Ω)3, p := r + 1, Iδ := [δ, T ) for a small δ > 0. Assume (U) with (3.68)2, p < ∞, v0 ∈ V2, θ0 ∈ L2(Ω). (i) If p > 59 , then there exists a weak solution (v, θ) of (3.71), (3.72) with

(ii) If p ≥

11 , 5

∂tv ∈ Lp0 (I; Y ∗) ,

v ∈ L∞ (I; H) ∩ Lp(I; Vp) , θ ∈ L∞ (I; L2(Ω)) ∩ L2(I; V) .

(3.79) (3.80)

then the weak solution is unique and fulfils

v ∈ C(¯I; H) ∩ L∞ (I; V2) ∩ L2(I; D(A)) , v ∈ L∞ (Iδ; Vp) , 2

θ ∈ L∞ (Iδ; V) ∩ L2(Iδ; H (Ω)) ,

∂tv ∈ L2(Iδ; H) ,

∂tθ ∈ L2(Iδ; L2(Ω)) .

(3.81) (3.82) (3.83)

For θ0 ∈ V in addition,

θ ∈ C(¯I; L2(Ω)) ∩ L∞ (I; V) ∩ L2(I; H2(Ω)) ,

∂tθ ∈ L2(I; L2(Ω)) .

(3.84)

For v0 ∈ Vp moreover,

v ∈ L∞ (I; Vp) ,

∂tv ∈ L2(I; H) .

(3.85)

3.28 Remark. Boundary values. In the proof we derive second energy estimates, i.e. we test equation (3.78)1 by Av. Due to the periodic boundary conditions it holds Av = −∆v (for Dirichlet conditions on parts of the boundary this is not true, [84, Ch. A4]), which is decisively used in the treatment of the nonlinear terms. We will return to this subject at the end of this section.

3.4

Power-law rheology



3.29 Remark. Less regular initial value. If v0 ∈ H, with the cut-off procedure explained below one can still prove the following: (1) Theorem 3.27 (i) remains true. (2) In case p ≥ 11 we find ∂tv ∈ Lp0 (I; (Vp)∗ ) and thus, v ∈ C(¯I; H). The other relations 5 in (3.81) hold if I is replaced by Iδ and (3.82) is true as well. (3) In case p ≥ 52 uniqueness can be proved as done in [84, Thm. 5.4.29]. 3.30 Remark. Other results. (i) The classical papers are restricted to p ≥ 11 due to 5 the properties of the nonlinear term stated in Lemma 3.25 (ii). (ii) Local in time (strong) solutions (for arbitrary data) and (global in time strong) solutions for small data exist for p > 35 , [88]. For short time and large data the result has recently been improved to p > 57 , [29]. (iii) p < 6 in [84] is due to the fact that considering the functional h∂tv, ϕi only the continuity of the Galerkin projection operator in H2(Ω) has been used. Nevertheless, as 1 proved in [119], one can take H3(Ω) which is continuously embedded in W∞ (Ω). 6 (iv) In case p > 5 the rest state is stable, [88]. Plan of the Proof. Mostly we skip the element of integration dxdt; ΩT := Ω × (0, T ). As in the proof of Theorems 3.7 & 3.14, the existence of weak solutions is shown with the Faedo-Galerkin approximation, i.e. we set [n]

v (t, x) :=

n X

fn,j(t) ωj(x) ,

[n]

θ (t, x) :=

j=1

n X

gn,j(t) τj(x) ,

(3.38)

j=1

[n]

[n]

and (v0 , θ0 ) is the orthogonal projection P[n] in H × L2(Ω) of (v0, θ0) onto the space spanned by (ω1, τ1), · · · , (ωn, τn). The corresponding weak equations read

[n] [n] ^ ∂tv , ωj + hv[n] · ∇v[n], ωji + hT(D(v )), D(ωj)i = hθ[n]b, ωji , ∂tθ[n], τj + (v[n] · ∇θ[n], τj) + (∇θ[n], ∇τj) = (v3, τj) .

(3.86)

(1) We prove the following apriori estimates: (a) Testing30 (3.86)1 by v[n] and (3.86)2 by θ[n] we find (v[n])N

is uniformly bounded in

(θ[n])N


(b) Testing (3.86)1 by Av[n] we prove (v[n])N


(v[n])N


L∞ (I; H) ∩ Lp(I; Vp) ,

L∞ (I; L2(Ω)) ∩ L2(I; V) .

L∞ (I; V2) ∩ L2(I; D(A)) for Lq(I; Wq1+σ ∩ Vq) for

(c) We show that ∂tv[n] ∈ Lp0 (I; Y ∗). 30

(3.87)

The precise procedure is stated in the proof of Theorem 3.7.

(3.88)

p > 3,

q < p , σ ∈ (0, 1) .

3

 (d) Testing (3.86)1 by Av[n] we see for p ≥

11 5


and v0 ∈ Vp that

∂tv ∈ L2(I; H) and v ∈ L∞ (I; Vp) . 11 5

(e) Testing (3.86)2 by −∆θ and ∂tθ we derive for p ≥

and θ0 ∈ V that

θ[n] ∈ L∞ (I; V) ∩ L2(I; H2(Ω)) and ∂tθ[n] ∈ L2(I; L2(Ω)) .

(2) Limit in the convective terms:

(a) We apply Aubin/Lions for X0 = Vp, X = H, X1 = Y ∗ , i.e. from v[n] ∈ Lp(I; Vp) and ∂tv[n] ∈ Lp0 (I; Y ∗) we conclude v[n] ∈ Lp(I; H) and (at least for a subsequence) v[n] → v in

Lp(I; H) .

(3.89)

(b) We show that for ϕ ∈ C1(¯I; V) with ϕ(T ) = 0 we then control Z

[n]

ΩT

[n]

v ·∇v

ϕ=−

Z

[n]

ΩT

v ·∇ϕ v

[n]

→−

Z

v·∇ϕ v =

ΩT

Z

v·∇v ϕ . (3.90)

ΩT

(c) Obviously, (at least for a subsequence) θ[n] → θ in L2(I; V), and the analogue of (3.90) for the term with v[n] · ∇θ[n] is an immediate consequence.

(3) Limit for the time derivatives: First we observe Z

[n]

∂tv fn,j(t) ωj(x) = −

ΩT

Z

[n]

v ∂tfn,j(t)ωj(x) −

ΩT

Z

[n]

v0 fn,j(0)ωj(x) .

Ω

Due to (3.87) we see v[n] v weakly in Lq(I; H) ∩ Lp(I; Vp) for any 1 < q < ∞ (at least [n] for a subsequence). In addition v0 → v0 in H. Since functions of the form fn,j(t) ωj(x) are dense in the space chosen for ϕ we finally arrive at Z

ΩT

[n]

∂tv ϕ → −

Similar considerations hold for (4) Limit in the stress term:

R

ΩT

Z

v∂tϕ − ΩT

Z

v0ϕ(0) .

Ω [n]

∂tθ[n]gn,j(t)τj(x) and the limit θ0 → θ0.

(a) Step (1b) provides estimates for the second order derivatives stated in (3.93) and (3.96) below. As a result moreover, v[n] ∈ Lq(I; Wq1+σ(Ω)3 ∩ Vq) for some q < p and σ ∈ (0, 1). Together with ∂tv[n] ∈ Lp0 (I; Y ∗ ) (which has been shown in (1c)) the application of Aubin/Lions yields ∇v[n] → ∇v strongly in Lq(ΩT )9

and this implies convergence of ∇v[n] → ∇v a.e. in ΩT , [80, Lemma II.2.1].

3.4

Power-law rheology



[n] ^ is continuously differentiable, we may conclude that T(D(v ^ ^ (b) Since T )) → T(D(v)). Thus, for all measurable Q ⊂ ΩT we find with Lemma 3.22 that

Z Z Z p−1 p ^ [n] p−1 [n] ≤c (1 + kD(v[n])k)p |Q|1/p . T(D(v )) ≤ c (1 + kD(v )k) Q

Q

Q

The last term is bounded due to Korn’s inequality and since v[n] ∈ Lp(I; Vp). Now ↑ Vitali’s lemma ensures the limit n → ∞.

(c) (2c) yields in particular that ∇θ[n] R→ ∇θ strongly in RL2(ΩT )9 which implies its convergence a.e. in ΩT and the limit ΩT ∇θ[n] · ∇φ → ΩT ∇θ · ∇φ.

(5) Existence of weak solutions: After the limiting processes in (2)-(4) we have proved the existence of a weak solution with the properties (3.79) and (3.80). (6) The case p ≥

11 5

:

(a) Lemma 3.25 states that v · ∇v ∈ Lp0 (I; (Vp)∗ ). Since θ ∈ L2(I; V) and p > 2, also θ ∈ Lp0 (I; (Vp)∗ ) and thus, ∂tv ∈ Lp0 (I; (Vp)∗ ). Parabolic embedding then provides v ∈ C(¯I; H).

(b) From (3.99) in step (1b) we know that v ∈ L2(I; D(A)) ∩ L∞ (I; V2).

(c) The estimates (3.85) follow from the apriori estimates for ∂tv in (1d). The counterparts with Iδ for v0 ∈ V2 are shown with a cut-off procedure.

(7) Uniqueness: Let (v1, θ1) and (v2, θ2) be two solutions to the same initial value (v0, θ0) and v := v1 − v2, θ := θ1 − θ2. As done in the uniqueness proof at the end of Section 3.2 we subtract the corresponding weak formulations from one another and test by v and θ, respectively. After some obvious calculations this yields Z Z Z 1 d 2 ^ ^ kvk2 + T(D(v1)) − T(D(v2)) · D(v) = θv3 − v · ∇v1v , (3.91) 2 dt Ω Ω Ω 1 d kθk22 + k∇θk22 = −(v · ∇θ1, θ) + (θ, v3) . 2 dt The estimates for the temperature equation have been derived in Section 3.2. Moreover, Z Z 1/2 3/2 v · ∇v1v ≤ |v|2|∇v1| ≤ k∇v1k2kvk24 ≤ k∇v1k2kvk2 k∇vk2 . − Ω

Ω

Here we used interpolation of L4(Ω) between L2(Ω) and L6(Ω) and then the embedding of L6(Ω) into H1(Ω). Inserting this estimate and Korn’s inequality into (3.91) we find with (3.76) and Hölder’s inequality that 1 d kvk22 + c1k∇vk22 ≤ c1k∇vk22 + c2(k∇v1k42kvk22 + kθk2) . 2 dt

The right-hand side is bounded since in particular, v1 ∈ L∞ (I; V2). d (kvk22 + kθk22) ∈ L1(Ω) . Together with the estimates for the temperature this shows 21 dt Since v(0) = 0, θ(0) = 0 we conclude v = 0, θ = 0. This finishes the proof.

3




Apriori estimates. Since these estimates concern just the approximate solutions, there is no danger of confusion and we omit the superscript [n] from now on. (a) We observe that (3.68)2 with Lemma 3.22 and Korn’s inequality provide ^ hT(D(v)), D(v)i ≥ c1

Z

(kD(v)k − 1)p ≥ c2(k∇vkpp − |Ω|) .

Ω

Thus, the first energy estimates for v and θ read 1 d kvk22 + c2k∇vkpp ≤ c3(kvk22 + kθk22 + |Ω|) , 2 dt 1 d kθk22 + k∇θk22 ≤ c4(kvk22 + kθk22) . 2 dt We set E(t) := kvk22 + kθk22 + |Ω| and add the two relations obtaining (here we drop the gradient terms). Then Gronwall’s Lemma yields

1 d E(t) 2 dt

≤ c E(t)

E(t) ≤ ectE(0) < c(T, kθ0k2, kv0k2, |Ω|) , which shows the first part of (3.79). The second part then follows by dropping the time derivatives and integrating from 0 to T . (b) Now we test by Av, i.e we multiply equation (3.86)1 by λjfn,j(t), where λj is the jth eigenvalue of A and add up. It holds λr(ωr, v) = (Aωr, v) = (∇ωr, ∇v). Moreover, we may use that Av = −∆v. Thus, in the convective term we can calculate hv · ∇v, Avi = h∇(v · ∇v), ∇vi. Applying Lemma 3.1 we observe hv · ∇(∇v), ∇vi = 0 and find that |hv · ∇v, Avi| = |h(∇v : ∇) ⊗ v, ∇vi| ≤ k∇vk33 . ^ = ∇D(v) : U 00 (D(v)) and For the stress term we exert property (U) twice: firstly, div T ^ ^ secondly, since hT(D(v)), D(Av)i = −h div T(D(v)), D(∇v)i it follows ^ |hT(D(v)), D(Av)i| ≥ c

Z

(1 + kD(v)k)p−2k∇D(v)k2 =: c Ip(v) .

(3.92)

Ω

A direct consequence of Korn’s inequality and the definition of Ip is that for p ≥ 2 k∇2vk22 ≤ c Ip(v) .

(3.93)

Again, we set the nonlinear term to the right-hand side. Then the estimate reads Z 1 d 2 k∇vk2 + c1Ip(v) ≤ − θ∆v3 + c2k∇vk33 , 2 dt Ω Z θ∆v3 ≤ kθk2k∇2vk2 ≤ c1kθk22 + γ Ip , − Ω

(3.94)

3.4

Power-law rheology



and γ Ip goes to the left. Now we treat the second term at the right-hand side of (3.94). For p ≥ 3 we can simply use that k∇vk3L3 (ΩT ) ≤ k∇vk3Lp (ΩT ). Integrating (3.94) in time for some t ≤ T and taking into account the previous calculations we find that 1 k∇vk22 + c1 2

Zt

1 Ip(v) dt ≤ k∇v0k22 + c2kθk2L2 (I;L2 (Ω)) + c3k∇vk3Lp (ΩT ) . 2 0

Then R t all terms at the right are bounded2under our assumptions. In particular this means I (v) dt < c. Due to (3.93) also k∇ vk2 is bounded and thus, v ∈ L2(I; D(A)). 0 p

Now we consider 95 < p < 3 . Then k∇vk3 must be interpolated between L2 and some Ls with s > 3 such that applying later on Young’s inequality the exponents can be controlled. For this we use the following Lemma. 3.31 Lemma. [84, Lemma 5.3.24] Let v ∈ C2per(Ω)3. Then for any 1 < p < 2 k∇2vkp ≤ c(Ip(v))1/2(1 + k∇vkp)(2−p)/2 ,

(3.95)

while for 1 ≤ q ≤ 2 and p > 1 also k∇vk3p/(3−q) ≤ c(Ip(v))q/(2p)(1 + k∇vkp)(2−q)/2 . We choose q ≥ 3 − p since then s :=

3p 3−q

> 3 and use both

β2 1 kwk3 ≤ ckwkβ 2 kwks ,

for β1 =

2(p+q−3) , 3p+2q−6

β2 =

p , 3p+2q−6

β3 =

(3.96)

β4 3 kwk3 ≤ ckwkβ p kwks

p+q−3 , q

β4 =

3−p q

.

We put k∇vk33 = k∇vk3(1−α)+3α and insert both interpolations obtaining :=2Q

z }| 1 { 3(1 − α)β1

k∇vk33 ≤ k∇vk2

With this and (3.96) we conclude that

:=Q

:=Q }|3 { z z }|2{ 3αβ + 3(1 − α)β 4 2 k∇vk3αβ3k∇vk .

p

s

1 d k∇vk22 + Ip(v) ≤ 2 dt ≤ c (k∇vk22)Q1 (1 + k∇vkp)Q2 (Ip(v))Q3 q/(2p)(1 + k∇vkp)Q3 (2−q)/2 . Now we want to apply Young’s inequality such that we can absorb Ip at the left, i.e. we η q η = 1 and (Q2 + Q3 2−q ) η−1 = p. With these requirements look for η such that Q3 2p 2 we can calculate η and α ∈ (0, 1). After finishing these technical calculations we see that p > 53 is necessary and find for λ := 2(3−p) that 3p−5 d (1 + k∇vk22) + c1Ip(v) ≤ c2(1 + k∇vk22)λ(1 + k∇vkp)p . dt

(3.97)

3




We divide (3.97) by (1 + k∇vk22)λ (for λ 6= 1) and integrate with respect to t obtaining 1 (1 + k∇vk22)−λ + c1 1−λ

ZT 0

Ip(v)(1 + k∇vk22)−λdt ≤ c(v0) .

(3.98)

d ln(1 + k∇vk22). So, if now λ ≤ 1, If λ = 1 then the first term in (3.97) is replaced by 12 dt , we calculate the supremum over t ∈ I of (3.98) and find i.e. p ≥ 11 5

∇v ∈ L∞ (I; H)

ZT

⇒

0

Ip(v) dt ≤ c

(3.93)

⇒

v ∈ L2(I; D(A)) .

(3.99)

If λ > 1, i.e. 95 < p < 11 , then the first term in (3.98) is negative and goes to the 5 1 . Thus, right-hand side. There it can be bounded from above by λ−1 ZT 0

Ip(v)(1 + k∇vk22)−λ dt ≤ c .

(3.100)

Let now in particular p ∈ [2, 11 ) . From (3.100) we deduce with the help of (3.93) for 5 λβ 1−β

=

2(3−p) β 3p−5 1−β

ZT 0

k∇

vk2β 2

2

= 1 (i.e. β :=

3p−5 p+1

and thus β ∈ [ 13 , 21 )) that

ZT

β Ip(v)(1 + k∇vk22)−λ (1 + k∇vk22)λβ 0 1−β ZT β Z T β 2 −λ ≤ c. ≤c Ip(v)(1 + k∇vk2) (1 + k∇vk22)λ 1−β ≤c

0

0

(3.101)

But for s = 6−p the space H2(Ω) is continuously embedded into Wp1+s(Ω) and in addi2p tion we interpolate 1−σ/s

kwk1+σ,p ≤ c kwk1,p

σ/s

kwk1+s,p ,

0 < σ < s.

(3.102)

Together with Hölder’s inequality and (3.101) we can choose q ∈ (1, p) such that ZT 0

kvkq1+σ,p dt ≤ c

for

σ=s

2β(p − q) . q(p − 2β)

Thus, v ∈ Lq(I; Wq1+σ(Ω)3) . 9 5

Whereas, if ZT 0

< p < 2 then as a simple consequence of (3.100) and (3.95) we find

N :=

ZT 0

k∇2vk2p (1 + k∇vk22) −λ (1 + k∇vkp) p−2 ≤ c . | {z } | {z } =:n2

=:np

(3.103)

3.4

Power-law rheology

 RT

Now we derive a bound for ZT 0

0

k∇2vk2β 2 as follows (notation in (3.103)): (3.103)⇒≤c

z Z }| { Z T β T λβ (2−p)β 1−β 2−p β β λβ N n n n21−β np 1−β k∇2vk2β = N ≤ p 2 2 0 0 0 Z T (2−p)β Z T λβ (2−p)β np 1−β 1−β + c ≤c n21−β np 1−β 1−β . } {z } | 0 {z | 0 ZT

=:N1

=:N2

1 To treat N2 we interpolate H1(Ω) between Wp1(Ω) and W3p/(3−p) (Ω), apply the embed2 1 ding of Wp(Ω) into W 3p (Ω) and Hölder’s inequality with exponent η, arriving at 3−p

N2 ≤ c

ZT

η Z T

npp

0

0

1−η k∇2vk2β p

(5p−9)p β 9 for η = ( 2−p + 5p−6 λ) 1−β and thus, β = 2(−p 2 +8p−9) . Here β > 0 is ensured by p > 5 . p p2 β N1 is finite since (2 − p) 1−β ≤ p. Finally,

ZT 0

k∇

2

vk2β p

ZT

≤ c1 + c2

0

(1−η)(1−β) k∇2vk2β p

and Young’s inequality provides the desired bound. Now, with interpolation of Wp1+σ(Ω) between Wp1(Ω) and Wp2(Ω), as in case p ≥ 2 we can choose q ∈ (1, p) such that ZT 0

kvkq1+σ,p ≤ c

for σ =

2β(p − q) . q(p − 2β)

Cut-off procedure. If v0 ∈ / V2 we cannot directly integrate the result of (3.97) with respect to t. We multiply it by a smooth function χ(t), which is 0 in a small neighbourhood of t = 0 and equal to 1 in Iδ. We obtain 1 d 1 d χn21−λ + c1χIp(v)n2−λ ≤ c npp + χn21−λ . 1 − λ dt 1 − λ dt

c The last term in (3.104) is bounded by 1−λ η(1 + k∇vkpp) for λ < 1 (i.e. p ≥ c η for λ > 1. Thus, we can integrate (3.104) from δ to t ∈ [δ; T ] and find 1−λ

1 n1−λ + 1−λ 2

Zt δ

(3.104) 11 ) 5

and by

Ip(v)n−λ 2 ≤ c.

This coincides with (3.98) and we may continue as above and find that v[n] ∈ Lq(Iδ; Wq1+σ). This provides the limit in the stress term for Iδ × Ω. For the remaining part of ΩT we argue as follows:

3




Since ∇v and ∇v[n] are uniformly bounded in Lp(ΩT ) for all ε > 0 there exists δ > 0 such that for ϕ with k∇ϕkL∞ (ΩT ) ≤ 1 we find ZδZ 0

Z δ p−1 1 p (|Ω|δ) p ≤ ε , T^ (D(v )) − T^ (D(v)) · D(ϕ) ≤ 2ck∇ϕkL∞ (ΩT ) npp

[n]

Ω

0

and the limit in the stress term is finally justified.

(c) Let P be the projection onto the span of ω1, ..., ωn. Then kPϕk3,2 ≤ kϕk3,2 (see 1 e.g. [119]) and each function in H3(Ω) is bounded in W∞ (Ω) as well. Then we show ∗ ∂tv ∈ Lp0 (I; Y ) in the following way (note, that v stands for v[n]): Z Z ∂tv ϕ = ∂tv Pϕ Ω Ω Z Z Z ^ θ Pϕ =: I1 + I2 + I3 , ≤ v · ∇v Pϕ + T(D(v)) · D(Pϕ) + Ω Ω Ω ZT ZT while I1 ≤ c k∇vk2kvk3k∇Pϕk6 ≤ ckvkL∞ (I;V2 )kϕkL1 (I;H2 ) , 0 0 ZT Z ZT p−1 I2 ≤ c (1 + kD(v)k) |∇Pϕ| ≤ c (1 + k∇vkp)p−1k∇Pϕkp 0

ZT 0

ΩT

0

≤ ckϕkLp (I;Vp )(1 +

k∇vkLp−1 ), p (I;Lp )

I3 ≤ ckθkL2 (ΩT )kϕkL2 (I;H3 ) .

(d) For p ≥

11 5

we test (3.86)1 by ∂tv, isolate and absorb γ k∂tvk22 at the left and find d 1 k∂tvk22 + 2 dt

If p ≥ 3 then we can estimate ZT 0

I≤

ZT 0

kvk

2 2p p−2

Z

Ω

U(D(v)) ≤

c1kθk22

+ c2

Z

|∇v|2|v|2 . } |Ω {z :=I

k∇vk2p

≤ c1

ZT 0

k∇vk22k∇vk2p

≤ c2

ZT 0

k∇vk2p ≤ c

due to the first and second energy estimates above and the embedding of H1(Ω) into 2p L 2p (Ω) (since p > 3 we have indeed that p−2 ≤ 6). p−2 RT In case 11 ≤ p < 3 we benefit from I (v) < c and k∇vkp3p ≤ Ip(v) (this is (3.96) 5 0 p with q = 2) deriving ZT 0

I≤

Moreover, U provide

ZT 0

k∇vk23pkvk2

6 3p−2

Zt 0

k∂tvk22 +

k∇vkp ≤ c

Z

Ω

≤c

ZT 0

k∇vk22kvk23p

≤ ckvkL∞ (I;V2 )

ZT 0

k∇vk23p ≤ c .

U(D(v)) ≤ c . Now Korn’s inequality and the properties of

⇒

∂tv ∈ L2(I; H)

⇒

v ∈ L∞ (I; Vp) if v0 ∈ Vp.

3.4

Power-law rheology



(e) Higher order estimates for θ are valid if p ≥ 11 and for θ0 ∈ V. The corresponding 5 estimates are derived testing (3.86)2 first with −∆θ and then by ∂tθ as done in the previous sections. The nonlinear term is placed at the right-hand side and we isolate k∆θk22 and k∂tθk22, respectively, such that it can be absorbed at the left-hand side with Hölder’s and Young’s inequality: 1 d k∇θk22 + k∆θk22 ≤ γ k∆θk22 + c (kvk22 + kvk42k∇θk22) , 2 dt 1 d k∇θk22 + k∂tθk22 ≤ γ k∂tθk22 + c (kvk22 + kvk42k∇θk22 + k∆θk22) . 2 dt Since for p ≥ 11 we proved that v ∈ L∞ (I; V2), the right-hand side of the first line 5 is bounded and we find that θ ∈ L∞ (I; V) ∩ L2(I; H2(Ω)). While applying Gronwall’s lemma we use θ0 ∈ V. Then also the terms on the right-hand side of the second estimate is bounded and ∂tθ ∈ L2(I; L2(Ω)). Proof of (3.90). Z v[n] · ∇ωj v[n] − v · ∇ωj v = ΩT Z Z [n] [n] = (v − v) · ∇ωj v + ΩT

|V1| ≤ k∇ωjk∞

ZT 0

ΩT

[n]

kv

(v[n] − v) · ∇ωjv =: V1 + V2 ,

[n]

[n]

− vk2kv k2 ≤ k∇ωjk∞ kv kL∞ (I;H)

ZT 0

kv[n] − vk → 0 . | {z }2 →0

For v[n] v weakly in Lq(I; H) we observe that |V2| → 0 as well. Since the functions of the form fn,j(t) ωj(x) are dense in the space chosen for ϕ we finally arrive at Z Z [n] [n] v · ∇ϕ v → v · ∇ϕ v . ΩT

ΩT

Some extensions. In case p ≥ 2 the property (3.68)2 implies (3.68)1. In case p < 2 we still have a gap, which is filled by the following theorem. 3.32 Theorem. Assume (U) with (3.68)1, 95 < p < 2, v0 ∈ H, θ0 ∈ L2(Ω). Then there is a weak solution (v, θ) of (3.71), (3.72)with v ∈ L∞ (I; H) ∩ Lp(I; Vp) ,

θ ∈ L∞ (I; L2(Ω)) ∩ L2(I; V) .

Proof. The first apriori estimates simplify since |Ω| drops out. In the second ones we change the definition of Ip as follows: Z Ip := kD(v)kp−2kD(∇v)k2 dx Ω

and proceed as in the proof of Theorem 3.27. Since (3.95) and (3.96) hold for this Ip as well there are no essential changes.

3




The last facet we want to discuss is the possibility to change the periodic boundary condition (3.72) to Dirichlet conditions on the surfaces. Therefor firstly, we replace Vp by Vp := closure of

1 3 C∞ 0,per,σ(Ω) in Wp(Ω) .

Then one can derive the first energy estimate without serious changes. One finds for p > 56 , that the Galerkin approximation for the velocity is in L∞ (I; H) ∩ Lp(I; Vp) (the bound for p is due to the application of Lemma 3.25(i) to the convective term). This is not enough to justify the limit in the stress term in the classical sense. Nevertheless, it is possible to characterise the limit with the help of Young measure valued functions. 3.33 Lemma. Let v[n] be a sequence of measurable functions for which kv[n]kL∞ (Rm )s ≤ c .

Then there exists a subsequence v[nk ] (which converges *-weakly) and a family of probability measures {νy}y∈Rm on Rs, called Young measures. For a compact set K ⊂ Rs it holds supp(νy) ⊆ K for a.e. y ∈ Rs. This family {νy}y∈Rm represents the subsequence v[nk ] in the following sense: For any w ∈ C(Rs)p we observe [nk ] ∗

w◦v

m p

b in L∞ (R ) , w

b w(y) =

Z

w(λ) dνy(λ)

Rs

for a.e. y ∈ Rm .

Here a probability measure ν is a ↑ Radon measure which is nonnegative and ν(Rs) = 1. We denote the space of probability measures by Prob(Rs). Furthermore L∞,w(ΩT ; Prob(Rs)) contains all functions ν : ΩT → Prob(Rs) for which the map Z x 7→ F(x, λ) dνx(λ) Rs

is measurable for all F ∈ L1(ΩT ; C0(Rs)).

3.34 Definition. A triple (v, ν; θ) is a measure valued solution to (3.71), (3.34) if v ∈ L∞ (I; H) ∩ Lp(I; Vp) , ν ∈ L∞,w(ΩT ; Prob(R9)) , θ ∈ L∞ (I; L2(Ω)) ∩ L2(I; V) ,

and satisfies (3.78)2 together with Z Z − v ∂tϕ − (v ⊗ v) : ∇ϕ + D(ϕ) ΩT

R9

^ T

1 (λ 2

+ λ ) dνt,x(λ) dxdt = Z Z θbϕ dxdt + v0ϕ dx >

ΩT

for all ϕ ∈ C∞ 0 ((−∞, T ); C0,σ(Ω)) and ∇v(t, x) =

R

R9

Ω

λ dνt,x(λ) a.e. in ΩT .

3.4

Power-law rheology



3.35 Theorem. Let v0 ∈ H, θ0 ∈ L2(Ω) and p > valued solution to (3.71), (3.34).

6 5

. Then there exists a measure

Ideas of the Proof. Choose s > 25 and let Ws := closure of C0,σ(Ω) with respect to the W2s(Ω)3-norm. If v ∈ W2s(Ω)3 then ∇v ∈ W2s−1(Ω)9, which is embedded in L∞ (Ω)9 for such s > 25 . Instead of the eigenvalue problem for the Stokes system one takes (ωr, ϕ)W2s = λr(ωr, ϕ)L2 (Ω)3

for

ϕ ∈ W2s(Ω)

and selects a basis of eigenvectors of this problem which are orthonormal in H and orthogonal in Ws. In addition to the apriori estimates considered at the beginning of this paragraph on proves k∂tv[n]kLp 0 (I;Ws∗ ) ≤ c choosing ϕ ∈ Lp(I; Ws) with kϕkLp (I;Ws ) ≤ 1 and estimating all terms in the velocity equation for such ϕ separately. Then Z [n] ∂tv[n]ϕ dxdt k∂tv kLp 0 (I;Ws∗ ) = sup ϕ∈Lp (I;Ws ),kϕk≤1

ΩT

yields the uniform estimate for ∂tv[n]. The limit in the stress term is justified with the following result, [84, Corollary 4.2.10], setting u[n] := ∇v[n], q := p − 1, s := 9, and Q := ΩT . 3.36 Lemma. Let Q ⊂ Rd be a bounded open set and u[j] uniformly bounded in Lp(Q)s. Then there exists a subsequence (which we still denote by u[j]) and a measure valued ^ : Rs → R obeying the growth condition function ν such that for all T ^ |T(ξ)| ≤ c (1 + |ξ|)q

∀ ξ ∈ Rd

^ [j]) T ^ weakly in Lr(Q), T(y) ^ for some q > 0, it holds T(u = p that 1 < r < q .

R

Rs

^ dνy a.e. provided T

This finishes the proof.

Finally, let us draw the attention to the fact that the existence of weak solutions to problem (3.71), (3.34) in case p ≥ 11 can be proved as done by Ladyˇzenskaya and Lions [77, 82] 5 for the velocity equation. Thus, there is still a gap between 95 obtained for the periodic case and 11 . 5 Adapting the method of [86] one can show, that there exists a weak solution for p ≥ 2 (the technique yields also an upper limit p < 3 which is not important), which is strong and unique if p > 94 . We avoid to summarise the details of the proof in [86] since it is very delicate to replace the second energy estimates. Indeed, a twofold approximation is necessary. Firstly the convective term is ↑ mollified and secondly the potential is replaced by a quadratic approximation. Then both limits have to be carried out.

4



The Boussinesq limit

ε

# % (

4 The Boussinesq limit In Section 2 we derived several systems approximately describing the flow of generalised incompressible fluids. In our approximation scheme we started with general Balance laws, inserted the material properties, then chose a nondimensionalisation and arrived at the respective nondimensional system which governs the flow of the material with the implemented properties (compare (2.20), (2.33), and (2.50)). After that we have expanded the physical quantities v, θ and p into power series with respect to the small parameter ε. This parameter measures the compressibility of the configuration. The new approximate quantities called v, θ and p in systems (2.26), (2.40), (2.49), (2.52), (2.57), and (2.59) combine the first terms of the respective expansion such that a closed approximate system of equations for these new quantities emerges. Nonetheless, this is a formal procedure and the ”right sort of ↑ convergence” of the expansion into powers of ε is silently assumed. More precisely, if in one of the systems (2.20), (2.33), (2.50) we let ε → 0, i.e. the material becomes incompressible, do we arrive at the usual approximation for the incompressible case, i.e. for ε = 0? We have to provide a mathematical frame apt to make the limiting process rigorous. All systems have in common that we prescribe div v = cε3θ˙ with c ∈ {1, Re −1} as constitutive relation. Thus, we are not treating the limit of a compressible to incompressible flow (see e.g. [55, 90] and references therein). Among the problems (2.20), (2.33), and (2.50) we choose the system which we found on our way to the Oberbeck-Boussinesq approximation, namely ε3

ρ = e− Re θ ,

div v =

ε3 ˙ θ, Re

ε3 1 ε3 (2.20) ρv˙ − ∆v + ∇( div v) + ∇p = ρb , Re Re 2 3 2 ˜ − ε5Υ(θ + Θ)p θ˙ − ε2 Re Υ(θ + Θ)p˙ − ∆θ = 2Υε2 kDk2 − |trD| . Pr Re C 3

Since system (2.20) is highly nonlinear and the velocity is no longer solenoidal we meet a lot of deep mathematical problems. At the moment we succeed to solve only some of them and thus, carry out the first step of the justification process only. Namely, from system (2.20) we extract the model problem (4.1) below in the following way: (1) We re-define ε to use it as compressibility measure, that is to say Re −1ε3 7→ ε3. This is the ε chosen when we derived the systems with dissipation.

 (2) We simplify the notation in the velocity equation setting Re −1 = 2µ. ˜ = cv is a constant. (3) In the heat equation we put Pr Re = 1 and C (4) We neglect all nonlinear terms in the divergence and the heat equation. (5) The expression for ρ is replaced by the continuity equation. (6) We can treat any conservative force and write ∇f for b.

Obviously, the first three steps above are only adjustments which do not oversimplify the system while the forth item collects the serious cuts in the highly nonlinear structure of the system. We have to pay for more simplicity by losing the easy formula for the density. The dependence of all quantities on ε is most important and we state and underline it by the superscript (ε). Precisely, we treat the problem div v(ε) = ε3∂tθ(ε) , 1 ρ(ε) 1 ρ(ε)v˙ (ε) − 2µ div D(v(ε)) − ∇ div v(ε) + 3 ∇p(ε) = 3 ∇f , 3 ε ε (ε) (ε) cv∂tθ − ∆θ = 0 ,

(4.1)

∂tρ(ε) + div (ρ(ε)v(ε)) = 0 . We consider problem (4.1) in the bounded ↑ domain Ω with boundary Γ and in the time interval I := (0, T ) under the boundary conditions v(ε) = a(ε) ,

∇θ(ε) · n = b(ε) on

ΓT := (0, T ) × Γ ,

(4.2)

where n is the unit outer normal to the boundary, and the initial conditions (ε)

v(ε)(0, ·) = v0 ,

(ε)

ρ(ε)(0, ·) = ρ0 ,

(ε)

θ(ε)(0, ·) = θ0 .

(4.3)

Since (4.1) - (4.3) is a new system first we must show that for fixed ε there exist solutions. The solvability properties are formulated in Theorem 4.4. Here without loss of generality we set µ = cv = 1. In our proof of Theorem 4.4 we benefit from the fact, that the equation for θ(ε) is decoupled. Hence, we can apply classical results about ↑ parabolic equations collected in Proposition 4.7. The investigations concerning velocity and density are more complicated. We carefully adapt a technique used in [6]. In contrast to the problem in [6] we have to deal with the case div v(ε) 6= 0. Therefore on the one hand we must construct ρ(ε) from equation (4.1)4 with non-solenoidal v(ε) and on the other hand we need results on the unsteady Stokes system with non-zero divergence condition which were only recently proved, [131]. After that in Theorem 4.5 we discuss the limit ε → 0. For most of the terms it is immediately clear from Theorem 4.4 and Proposition 4.7. Nevertheless, the treatment of the term ρ(ε)ε−3 needs special attention. We add − ε13 ∇f on both sides of (4.1)2. Here we explicitly benefit from the fact that the forces have a potential since this term can be hidden in the pressure. Then we show that ε13 (ρ(ε) − 1) has a limit and that v(ε) converges to a solution of the usual Boussinesq system (4.13) as ε → 0.

4




$ # % (

4.1 Prerequisites and Results Notation. In the following we use the abbreviation Z Z 2 A(u, v) := 2 D(u) · D(v) dx − div u · div v dx . 3 Ω Ω We use the letter A to refer to the Stokes operator A, since the bilinear form A(u, v) plays the role of the Stokes term if the velocity is not solenoidal. Moreover, if v ∈ V then we compute A(u, v) = 2(D(u), D(v)) = (Au, v) = (∇u, ∇v). We distinguish the following domains: In contrast to the cells considered in the previous chapter here Ω ⊂ R3 is a bounded domain with sufficiently smooth boundary Γ . We employ the symbols ΩT := (0, T ) × Ω ,

ΓT := (0, T ) × Γ ,

ΓT := [0, T ] × Γ ,

Γ0 := {t = 0} × Γ ,

I := (0, T ) , ¯I := [0, T ] .

Concerning function spaces and norms: One finds precise definitions in [6, Chapter 1, Section 2.1], [80], [130], and [131]. Additional to the previous chapters we make the following agreements for our notation. In accordance with the boundary conditions in (4.2) in this chapter - as usual - H and V 3 1 3 are the closures of C∞ 0,σ(D) in L2(D) and H (D) , respectively. Hdiv consists of vector fields in L2(Ω)3 with divergence in L2(Ω). This space is equipped with the norm kvkHdiv := (kvk22 + k div vk22)1/2 . Depending on the context on the one hand we use C1([0, T ]; Wql (Ω)), Lp(0, T ; Wql (Ω)), and Wpm(0, T ; Wql (Ω)) for m, l ∈ N, 1 ≤ p, q ≤ ∞, together with the corresponding trace spaces. For convenience we often skip the reference to Ω in the norms. For example, k . kL2 (0,T;W21 (Ω)) =: k . kL2 (I;W21 ) .

On the other hand anisotropic Sobolev and Hölder spaces for l, s ≥ 0, i.e. Wq2l,l(ΩT ) and C2s,s(ΩT ) are natural (definitions (2.9)/(2.10) in Chapter 1 [6]; see also [80], [130], [131]). They have different smoothness assumptions with respect to the t- and x-variables in accordance with the number of respective derivatives. For example, for l ∈ N kvkWq2l,l (ΩT ) :=

2l X X

j=0 2r+|α|=j

k∂rt∂α x kq,ΩT .

(4.4)

The definition of Wq2l,l(ΩT ) for noninteger l is analogous to the Wql (Ω)-spaces taking into account that the order of derivatives counts as in (4.4).

4.1

Prerequisites and Results



Namely, for l ∈ R, 1 < q < ∞ we already sketched before that for some domain D α q 1/q X Z Z |∂α x φ(x) − ∂y φ(y)| dydx kφkWql (D) := kφkW[l] (D) + . (4.5) q |x − y|n+q(l−[l]) D D |α|=[l]

Here, [l] is the integer31 part of l and n the dimension of D, i.e. n = 3 for D = Ω while n = 2 for D = ∂Ω. Adapted to the anisotropic spaces this means X kφkWq2l,l (DT ) := k∂rt∂α x φkLq (DT )+ 0≤2r+|α|

2r+|α|=[2l]

X

(l−r−s/2)

q,t,DT xφ >

,

0q,t,DT

1/q Z T Z Z |ψ(t, x) − ψ(t, y)|q dydxdt := , |x − y|(n−1)+qν 0 D D Z Z T Z T |ψ(t, x) − ψ(τ, x)|q 1/q := dτdtdx . |t − τ|1+qν D 0 0

Now we characterise the traces on t = 0 and on ΓT in this scale. 4.1 Lemma. [80, Lemma II.3.4] Let φ ∈ Wq2l,l(ΩT ) then

(1)

for

2r + s < 2l −

2 q 1 q

it holds

∂rt∂sxφ|t=0 ∈ W 2l−2r−s−2/q(Ω) while

it holds ∂rt∂sxφ|ΓT ∈ W 2l−2r−s−1/q,l−r−s/2−1/(2q)(ΓT ) (2) for 2r + s < 2l − together with the corresponding estimates. Mathematical Frame. From the combination of (4.1)1 and (4.1)3 together with the Gauss theorem there follows the compatibility condition: Z Z (ε) 3 a · n ds = ε b(ε) ds. Γ

Γ

Since (4.1)3 shall be satisfied a.e. in ΩT , this condition has to hold for almost all t ∈ I. We will prescribe a(ε)·nR= 0, i.e. an impermeable wall. Thus, our compatibility condition in space takes the form Γ b(ε) ds = 0. The velocity v(ε) is constructed as the sum of two parts u(ε) and w(ε), namely v(ε) = u(ε) + w(ε) with div u(ε) = 0 in Ω ,

u(ε) = 0 on ΓT .

Our general procedure to find solutions to (4.1) consists of four steps: First we solve the (decoupled) initial boundary value problem for the heat equation ∂tθ(ε) − ∆θ(ε) = 0 in ΩT , 31

∇θ(ε) · n = b(ε) on ΓT ,

Note that if q 6= 2 this scale of spaces is not continuous for l ∈ Z.

(ε)

θ(ε)(0, ·) = θ0 .

(4.6)

4




(ε)

The second step is to find w0 from (ε)

(ε)

(ε)

(ε)

div w0 = ε3∆θ0 , −∆w0 + ∇π0 = 0 in Ω ,

(ε)

w0 = a(ε)(0, ·) on Γ0 . (4.7)

Thirdly, we determine w(ε) as the solution of the unsteady Stokes problem div w(ε) = ε3∂tθ(ε) ,

∂tw(ε) − ∆w(ε) + ∇q(ε) = 0

in ΩT ,

w(ε) = a(ε)

on ΓT ,

(ε)

w(ε)(0, ·) = w0

(4.8)

on Γ0 ,

(ε)

with the initial value w0 from Step 2. To construct u(ε) in the forth step we can use standard techniques with solenoidal test functions. Namely, it is determined via the condition that v(ε) is a weak solution of (4.1)2, i.e. for all ϕ ∈ C1(¯I; V) with ϕ(T ) = 0 ρ(ε)(∂tu(ε) + u(ε) · ∇u(ε)), ϕ + A(u(ε), ϕ) = 1 = 3 (ρ(ε)∇f, ϕ) − ρ(ε)∂tw(ε), ϕ − A(w(ε), ϕ)− ε − ρ(ε)w(ε) · ∇u(ε), ϕ − ρ(ε)(u(ε) + w(ε)) · ∇w(ε), ϕ ,

(4.9)

(ε)

(u(ε)(0, ·), ϕ(0, ·)) = (u0 , ϕ) .

Note, that since ϕ ∈ V the terms A( . , ϕ) simplify to (∇·, ∇ϕ). In (4.9) also the density is not known, i.e. ρ(ε) and u(ε) must be calculated together. 4.2 Definition. (v(ε), ρ(ε), θ(ε)) is called a weak solution of the initial boundary value problem (4.1) - (4.3) if for all T > 0 it satisfies v(ε) = u(ε) + w(ε), u(ε) ∈ L∞ (0, T ; H) ∩ L2(0, T ; V) ,

w(ε) ∈ W22,1(ΩT ) ,

ρ(ε) ∈ L∞ (ΩT ) ,

(4.10)

θ(ε) ∈ W24,2(ΩT ) ,

and for all ϕ ∈ C1(¯I; V) with ϕ(T ) = 0 and φ ∈ C1(¯I; W21(Ω)) with φ(T ) = 0 −

ZT

(ε) (ε)

(ρ v

0

, ∂tϕ + v

(ε)

· ∇ϕ) dt + =

−

ZT 0

ZT 0

A(v(ε), ϕ) dt =

(ε) (ε) (ρ0 v0 , ϕ(0)) (ε)

(ρ(ε), ∂tφ + v(ε) · ∇φ) dt = (ρ0 , φ(0))

1 + 3 ε

(4.11) ZT 0

(ρ(ε) ∇f, ϕ) dt , (4.12)

hold, while (4.6) is satisfied a.e. in ΩT . 5/2,5/4

(ΓT ), 4.3 Remark. Note, θ(ε) ∈ W24,2(ΩT ) matches n · ∇θ(ε)|ΓT = b(ε) ∈ W2 3/2,3/4 (ε) (ε) (ε) (ε) 2,1 3 (ΓT ) due θ0 ∈ W2 (Ω), while w ∈ W2 (ΩT ) fits with w |ΓT = a ∈ W2 to Lemma 4.1.

4.1

Prerequisites and Results



4.4 Theorem. We assume for the data of system (4.1) - (4.3), that 3/2,3/4

(a.1)

a(ε) ∈ W2

(a.2)

θ0 ∈ W23(Ω) ,

a(ε) · n = 0 ,

(ΓT ) ,

(ε)

5/2,5/4

b(ε) ∈ W2

(ΓT )

gether with the compatibility condition (ε)

0 < m ≤ ρ0 ≤ M < ∞ ,

(a.3)

(ε)

v0 ∈ Hdiv , ∇f ∈ L2(I; L6/5(Ω)) ; R where b(ε) ds = 0 holds toΓT (ε)

∇θ0 · n = b(ε)(0, ·) m

for some constants

on Γ0 ;

M.

and

Then there exists a weak solution (v(ε), ρ(ε), θ(ε)) of (4.1) - (4.3). 4.5 Theorem. In addition to (a.1) - (a.3) assume that ∇f ∈ L2(I; L3(Ω)) and as ε → 0 3/2,3/4

(A.1)

a(ε) → a ∈ W2

(ΓT ) ,

kb(ε)kW5/2,5/4 (Γ ) ≤

c kbkW1/2,1/4(Γ ) , ε3 T 2

(ε)

1/2,1/4

b(ε) → b ∈ W2

θ0 → θ0 ∈ W21(Ω) ,

(A.2) (A.3)

T

2

(ε)

ρ0 → 1 ∈ L∞ (Ω) ,

(A.4)

(ε)

a·n = 0,

∃ r0

s.t.

v0 → v0 ∈ Hdiv ;

(ΓT ) ;

(ε)

kθ0 k3,2 ≤

(ε) 1 (ρ0 ε3

c kθ0k1,2 ; ε3

− 1) → r0 ∈ L2(Ω) .

Correspondingly, let (v(ε), ρ(ε), θ(ε)) be the solution obtained through Theorem 4.4 for any fixed ε > 0. Then there exists a sequence ε 0 and v, θ, r such that as ε 0 → 0 0

v(ε ) → v 0

θ(ε ) θ 0

∗

ρ(ε ) 1

strongly in L2(ΩT ) , *-weakly in L∞ (0, T ; L2(Ω)) ,

weakly in L2(0, T ; W21(Ω)) ,

weakly in L2(ΩT ) ,

1 (ε0) (ρ − 1) r ε 03

*-weakly in L∞ (ΩT ) ,

weakly in L2(ΩT ) ,

and (v, θ) is a solution of the Boussinesq system

 div v = 0  ∂tv + v · ∇v − ∆v + ∇p1 = r ∇f  ∂tθ − ∆θ = 0 v = a, v(0, ·) = v0 ,

in ΩT , (4.13)

∇θ · n = b θ(0, ·) = θ0 ,

on ΓT ,

i.e. v ∈ L∞ (0, T ; H) ∩ L2(0, T ; V), θ ∈ W22,1(ΩT ) and there holds the equation ZT ZT ZT − (v, ∂tϕ + v · ∇ϕ) dt + (∇v, ∇ϕ) dt = (v0, ϕ(0)) + (r∇f, ϕ) dt 0

0

(4.14)

0

for all ϕ ∈ C1(¯I; V) with ϕ(T ) = 0 while (4.13)3 is satisfied a.e. in ΩT , and r solves ZT ZT − r, ∂tφ + v · ∇φ dt = (r0, φ(0)) + (∂tθ, φ)dt (4.15) 0

for all φ ∈ C1(¯I; W21(Ω)) with φ(T ) = 0.

0

4




4.6 Remark. (i) It turns out that the weak formulation above is equivalent to another one (for details we refer to [6, pp. 102-103]) which yields that the initial condition are attained in the sense (ε) lim+ ρ(ε)(t)v(ε)(t), ϕ(t) = ρ0v0 , ϕ(0) , t→0 (ε) lim+ ρ(ε)(t), φ(t) = ρ0 , φ(0) . t→0

(ε)

(ii) We prove that θ ∈ W24,2(ΩT ). With the definition of this space one immediately concludes (very roughly) that ∂tθ(ε) ∈ L2(I, W22(Ω)) as well as ∂tθ(ε) ∈ W21(I, L2(Ω)). This implies ∂tθ(ε) ∈ C(¯I, W21(Ω)) by the parabolic embedding provided in Theorem A.4. (iii) In our proof of Theorem 4.4 the forces need not have a potential. Useful known results. 4.7 Proposition. [130, Theorem 17] Consider the boundary value problem for the heat equation ∇θ · n = b

∂tθ − ∆θ = 0 in ΩT ,

on ΓT ,

θ(0, ·) = θ0 .

If for some l ∈ N the compatibility conditions in the sense of [80, Chapter IV.5] for x ∈ Γ0 1 1 ,l− 12 − 2q 2l−1− q

3 up to the order32 [l − 2q − 12 ] are fulfilled, Γ ∈ C2l+δ and b ∈ Wq 2l− q2

θ0 ∈ Wq

(ΓT ) while

(Ω) then there exists a unique solution θ ∈ Wq2l,l(ΩT ) and

kθkWq2l,l (ΩT ) ≤ c kθ0kW2l−2/q (Ω) + kbkW2l−1−1/q,l−1/2−1/(2q) (ΓT ) . q

q

4.8 Proposition. Consider the steady Stokes problem −∆w + ∇π = 0 ,

div w = g

in Ω ;

w = w0

on Γ .

R R 1+s− 1 Let 1 < q < ∞, s ∈ [0, 1], g ∈ Wqs (Ω), w0 ∈ Wq q (Γ ), and Γ w · n ds = g dx. 0 Ω R 1+s s Then there is a unique solution (w, π) ∈ Wq (Ω) × Wq(Ω) with Ω π dx = 0 and kwkWq1+s (Ω) + kπkWqs (Ω) ≤ c (kw0kW1+s−1/q (Γ) + kgkWqs (Ω)) . q

Proof: According to [40, Theorem IV.6.1] there is a unique weak solution (w, π) ∈ Wq1(Ω) × Lq(Ω) and in addition a unique strong one (w, π) ∈ Wq2(Ω) × Wq1(Ω) for the 1−1/q 2−1/q corresponding data (g, w0) ∈ Lq(Ω) × Wq (Γ ) and (g, w0) ∈ Wq1(Ω) × Wq (Γ ), respectively (fulfilling the compatibility condition related to the Gauss theorem). If we interpolate w ∈ Wq1+s(Ω) between Wq1(Ω) and Wq2(Ω) (and similarly for the pressure) we see, that a solution (w, π) ∈ Wq1+s(Ω) × Wqs (Ω) exists and is unique. The estimate stated above is derived as the apriori estimates for the cases s = 0 or s = 1, respectively.

32

Here [ . ] is the integer part of a real number.

4.2

Proof of Theorem 4.4



4.9 Proposition. [131, Theorem 1.2] Consider the time dependent Stokes problem div w = g , w(0, ·) = w0 ,

∂tw − ∆w + ∇q = 0 w=a 2− 1 ,1−

in ΩT , on ΓT .

1

2− q2

Let q ≥ 2, g ∈ Lq(I, Wq1(Ω)), a ∈ Wq q 2q (ΓT ) with a · n = 0, w0 ∈ Wq Moreover, the data shall fulfil the following compatibility conditions Z

g dx = 0

Ω

for all t ∈ I ,

Besides, assume that ∂tg = div h

div w0 = g(0, ·) ,

(Ω).

w0 = a(0, ·) on Γ0 .

for some h ∈ Lq(ΩT ) with

Then there exists a solution (w, ∇q) satisfying the estimate

Z

Γ

h · n ds ∈ Lq(I) .

kwkWq2,1 (ΩT ) + k∇qkLq (ΩT ) ≤

≤ c kw0kW2−2/q (Ω) + kakW2−1/q,1−1/(2q)(ΓT ) + kgkLq (I;Wq1 (Ω)) + khkLq (ΩT ) . q

q

# # % (

4.2 Proof of Theorem 4.4 Scheme. 1 Proposition 4.7 states classical results for ↑ parabolic equations which under our assumption (a.2) guarantee the existence of a unique solution of problem (4.6) θ(ε) ∈ W24,2(ΩT ) and the estimates (ε) kθ(ε)kW22,1 (ΩT ) ≤ c kθ0 kW21 (Ω) + kb(ε)kW1/2,1/4 (ΓT ) , 2 (ε) (ε) (ε) kθ kW24,2 (ΩT ) ≤ c kθ0 kW23 (Ω) + kb kW5/2,5/4 (ΓT ) .

(4.16) (4.17)

2

Note, that in this case q = 2 and l = 1, l = 2, respectively. R For that no compatibility condition occurs. Furthermore, the solution θ(ε) satisfies Ω ∂tθ(ε) dx = 0 due to the assumptions for b(ε) in (a.2). 2 Later on, we will construct u(ε) with a Galerkin approximation. Hence, we need data of a certain smoothness. For that (for fixed ε) for n ∈ N we introduce a sequence (ε,n)

θ

∞

∈ C (ΩT ) with

Z

Ω

∂tθ(ε,n) dx = 0 and

θ(ε,n) → θ(ε) in W24,2(ΩT ) .

4




Indeed, let g(ε,n) ∈ C∞ (ΩT ) converge strongly to θ(ε) in W24,2(ΩT ). We set (ε,n)

θ

:= g

(ε,n)

−

Z

∂tg(ε,n) dxdt

ΩT

θ(ε) in

→

W24,2(ΩT )

R and they have the property Ω ∂tθ(ε,n) dx = 0. Obviously, there exists a constant c, which does not depend on n such that kθ(ε,n)kW24,2 (ΩT ) ≤ c + kθ(ε)kW24,2 (ΩT ) .

(4.18)

3 Now, we approximate a(ε) by a sequence a(ε,n) ∈ C∞ (ΓT ) with a(ε,n) · n = 0 . Indeed, let us extend n to a smooth function defined in a neighbourhood of ΓT and let ˜ (ε,n) − (a ˜ (ε,n) · n)n . ˜ (ε,n) ∈ C∞ (ΓT ) converge to a(ε) in W23/2,3/4(ΓT ). Set a(ε,n) := a a 3/2,3/4 Obviously a(ε,n) have the desired properties, converge to a(ε) in W2 (ΓT ) and it holds ka(ε,n)kW3/2,3/4 (ΓT ) ≤ c + ka(ε)kW3/2,3/4 (ΓT ) , 2

(4.19)

2

with a constant c independent of n. (ε)

(ε,n)

4 We approach the solutions w0 of (4.7) by solutions w0 (ε,n)

div w0

= ε3∂tθ(ε,n)(0, ·) ,

(ε,n)

−∆w0

of

(ε,n)

=0

(ε,n)

= a(ε,n)(0, ·) on Γ0 .

+ ∇π0 w0

in Ω ,

(4.20)

Since a(ε,n) and ∂tθ(ε,n)(0, ·) are smooth, Proposition 4.8 yields, for 2 ≤ q < ∞, a (ε,n) 2−2/q unique w0 ∈ Wq (Ω) which obeys kW2−2/q (Ω) ≤ c ka(ε,n)(0)kW2−3/q(Γ) + ε3k∂tθ(ε,n)(0)kW1−2/q (Ω) .

(ε,n)

kw0

q

q

q

(4.21)

(ε)

5 The next step is to find w(ε,n) as a solution to (4.8) with θ(ε), a(ε) and w0 replaced (ε,n) by θ(ε,n), a(ε,n), and w0 , respectively, i.e. div w(ε,n) = ε3∂tθ(ε,n) , (ε,n)

w(ε,n)(0, ·) = w0

∂tw(ε,n) − ∆w(ε,n) + ∇q(ε,n) = 0

w(ε,n) = a(ε,n)

,

in ΩT , on ΓT .

(4.22)

Here we want to apply Proposition 4.9 and have to check the compatibility conditions and the assumptions for g = div w(ε,n) = ε3∂tθ(ε,n). However, since θ(ε,n) is smooth Z

Ω

3

g dx = ε

Z

(ε,n)

∂tθ Ω

dx = 0 ,

Z

Ω

∂tg dx = ∂t

Z

Ω

g dx = 0

4.2




and thus (cf. [40, Theorem III.3.2]), there is a smooth h(ε,n) that solves div h(ε,n) = ∂tg ,

h(ε,n) = 0 on ΓT ,

and satisfies the estimate khkLq (I;Wq1 (Ω)) ≤ c ε3k∂2tθ(ε,n)kLq (I;Lq (Ω)) ≤ c ε3kθ(ε,n)kWq4,2 (ΩT ) .

(4.23)

Now Proposition 4.9 yields w(ε,n) ∈ Wq2,1(ΩT ) and, if 2 ≤ q < ∞, the bound (ε,n)

kw(ε,n)kWq2,1 (ΩT ) ≤ c kw0

3

kW2−2/q (Ω) + ka(ε,n)kW2−1/q,1−1/(2q)(Γ )+ q

T

q

(ε,n)

+ ε k∂tθ

k∂2tθ(ε,n)kLq (I;Lq (Ω))

3

kLq (I;Wq1 (Ω)) + ε

(4.24) .

4.10 Remark. (i) Note, that in general θ(ε,n) does not solve problem (4.6). However, in the limit n → ∞ it holds ∂tθ(ε) = ∆θ(ε) and thus the problems (4.7) and the limiting problem of (4.20) when n → R ∞ coincide. Moreover, the condition ∂ θ(ε,n)Rdx = 0 transfers to our usual compatibility condiΩ t R (ε) R tion 0 = Γ b ds = Ω ∆θ(ε) dx = Ω ∂tθ(ε) dx .

(ii) Later on while constructing the density ρ(ε,n) also v(ε,n) must be smooth enough to justify the limit as n → ∞ - precisely, v(ε,n) ∈ C(¯I; C1(Ω)) is necessary. Hence, we must ensure that w(ε,n) ∈ C(¯I; C1(Ω)). 2−2/q We note that Wq2,1(ΩT ) ⊂ C(¯I; Wq (Ω)), since by the definition Φ ∈ Wq2,1(ΩT ) 2−2/q implies Φ ∈ Wq1(I; Wq1(Ω)), ∂tΦ ∈ L2(I; Wq2(Ω)) and Wq (Ω) (q ≥ 2!) lies 1 2 between Wq(Ω) and Wq(Ω) (compare this to Remark 4.6 (ii) and use Theorem A.4). 2−2/q For q > 5 we find that Wq (Ω) ⊂ C1+δ(Ω) (since then the embedding property mq > n = 3 is fulfilled for m = 2 − 2/q) and w(ε,n) ∈ C(¯I; C1+δ(Ω)). We emphasise that the estimates (4.21) and (4.24) with q 6= 2 are only needed to ensure that w(ε,n) is smooth enough to justify all following computations.

(iii) Obviously, due to (4.18) and (4.19) the inequalities (4.21) and (4.24) induce estimates (ε,n) with q = 2 for w0 and w(ε,n), respectively, by θ(ε) and a(ε), which are independent of n, and finally by θ0, a and b, which are independent of ε. Indeed, from (4.17), (4.18), and the assumption (A.3) we derive ε3kθ(ε,n)kW24,2 (ΩT ) ≤ c 1 + ε3kθ(ε)kW24,2 (ΩT ) (4.25) ≤ c (1 + kθ0kW21 (Ω) + kbkW1/2,1/4(ΓT )) . 2

Applying parabolic embeddings and (4.18), (4.19) we arrive at (ε,n)

kw0

kW21 (Ω) ≤ c ka(ε,n)kW3/2,3/4(ΓT ) + ε3k∂tθ(ε,n)kW22,1 (ΩT ) 2

≤ c 1 + ka kW3/2,3/4 (ΓT ) + ε3 1 + kθ(ε)kW24,2 (ΩT ) (ε)

2

(4.26)

≤ c 1 + kakW3/2,3/4(ΓT ) + kθ0kW23,2 (Ω) + kbkW1/2,1/4(ΓT ) , 2

2

4




where we used in the last step (4.25) and assumption (A.1). Analogously we obtain kw(ε,n)kW22,1 (ΩT ) ≤ c 1 + ka(ε)kW3/2,3/4 (ΓT ) + ε3 1 + kθ(ε)kW24,2 (ΩT )

≤ 1 + kakW3/2,3/4(ΓT ) + kθ0kW23,2 (Ω) + kbkW1/2,1/4(ΓT ) . 2

2

(4.27)

2

6 Construction of u(ε) and ρ(ε). Up to this point we could carefully fit known results to our needs. To construct velocity and density we adapt ideas of the proof of Theorem 1.1 in Chapter 3 of [6]. Nevertheless, since in our case div v(ε) 6= 0 there are two main differences. The first (minor) difference to their procedure is that we must split v(ε) into one part - u(ε) - fulfilling the conditions of the Theorem in [6] and the ”remainder” - w(ε) as done above. As a consequence our weak formulation for u(ε) contains additional terms with w(ε) and ∂tw(ε) and we must take care for the estimates of the corresponding nonlinear terms. The second (major) difference is that the representation of the density is more delicate. Note, that we can re-write the transport equation (4.1)4 as ρ˙ (ε)/ρ(ε) = − div v. In the standard case div v(ε) = 0 this means ρ˙ (ε)/ρ(ε) = (ln ρ(ε))· = 0 and integration yields (ε) ρ(ε) = ρ0 . In case of the full system (2.20) we find (ln ρ(ε))· = −ε3θ˙ (ε)

⇒

(ε)

3 (θ(ε) +θ(ε) ) 0

ρ(ε) = ρ0 e−ε

.

However, from our simplified equation (4.1)4 we obtain (ln ρ(ε))· = −ε3∂tθ(ε) , i.e. there are different types of derivatives on both sides. The general plan to construct u(ε) and ρ(ε) is the following: (1) Consider sequences of approximate (smooth) solutions (u(ε,n), ρ(ε,n)): . u(ε,n) fulfils the nth order Galerkin scheme derived according to (4.9), . v(ε,n) := u(ε,n) + w(ε,n) and ρ(ε,n) solves ∂tρ(ε,n) + div (ρ(ε,n)v(ε,n)) = 0 ,

(ε,n)

ρ(ε,n)(0, ·) = ρ0

.

(4.28)

(2) Show compactness of the sequences. (3) Justify the limit n → ∞ in all terms.

Galerkin approximation. First we construct a smooth initial value for ρ(ε,n) with the (ε,n) ˜ ⊃ Ω be bounded and set property m ≤ ρ0 ≤ M in Ω. Let Ω

(ε,n)

 (ε)  ρ0 in Ω , (ε) ˜ ρ˜0 := m in Ω\Ω ,  3 ˜ 0 in R \Ω . (ε)

(4.29)

Then ρ0 shall be a ↑ mollification of ρ˜0 by a smooth kernel Ψn (see [129, Sect. II.1.7]) (ε) (ε) (ε,n) (ε,n) such that ρ0 := ρ0 (Ψn, ρ˜0 ) → ρ0 as n → ∞.

4.2




Now we are ready to construct approximate solutions (u(ε,n), ρ(ε,n)), n ∈ N. Namely, ρ(ε,n) is the integral of (4.28) along the characteristics X(ε,n). The characteristics (or trajectory) is the solution of the Cauchy problem (see [6, Ch. 2] below formula (1.5)) d (ε,n) X (t, x) = v(ε,n)(t, X(ε,n)(t, x)) , X(ε,n)(0, ·) = x . dt

(4.30)

Let (ωj)j∈N be the orthonormal basis of H consisting of the eigenfunctions of the Stokes (ε,n) (ε) (ε,n) operator (see Section 3.2) and u0 is the orthogonal projection in H of v0 − w0 onto the space spanned by ω1, · · · , ωn, i.e. (ε,n) u0

:=

n X

(ε) (v0

−

j=1

(ε,n) w0 )j, ωj ωj

=:

n X

fjωj .

(4.31)

j=1

For each fixed n ∈ N and j = 1, 2, · · · , n we solve the system 1 ρ(ε,n)u˙ (ε,n), ωj + A(u(ε,n), ωj) = 3 (ρ(ε,n)∇f, ωj)− ε (ε,n) (ε,n) (ε,n) − ρ u · ∇w , ωj − ρ(ε,n)w(ε,n) · ∇u(ε,n), ωj − ˙ (ε,n), ωj) − A(w(ε,n), ωj) , − (ρ(ε,n)w

(4.32)

(ε,n)

(u(ε,n)(0, ·), ωj(0, ·)) = (u0

, ωj)

for the unknown coefficients fn,j ∈ C1(¯I) in u(ε,n)(t, x) =

n X

fn,j(t) ωj(x) .

(4.33)

j=1

Furthermore we introduce Zt (ε) ϑ (t) := ε3k∂tθ(ε,n)k∞ dτ ,

(ε) (T)

m(ε) := m e−ϑ

,

(ε) (T)

M(ε) := M eϑ

.

0

Note, that due to (4.25) there are constant 0 < m0, M0 < ∞ independent of ε such that m0 ≤ m(ε) ≤ M(ε) ≤ M0 .

4.11 Proposition. (A priori estimates) Let (u(ε,n), ρ(ε,n)) be an approximate solution on [0, T ]. Then for (t, x) ∈ ΩT 0 < m(ε) ≤ ρ(ε,n)(t, x) ≤ M(ε) , Zt (ε,n) 2 ku (t)k2 + k∇u(ε,n)(s)k22 ds ≤ c^ . 0

Here c^ > 0 does not depend on n.

(4.34)

4




To prove this proposition in addition to the Galerkin method we use a representation for ρ(ε,n), and the fact that X(ε,n) does not leave Ω (note, a(ε) · n = 0). 4.12 Proposition. (Existence of approximate solutions) The approximate solutions (u(ε,n), ρ(ε,n)) for n ∈ N exist on [0, T ] for all T > 0. This is shown with the ↑ Schauder fixed point principle on the basis of the apriori estimates in Proposition 4.11. They ensure that we can choose subsequences from (u(ε,n), ρ(ε,n)) such that as n 0 → ∞ ∗

0

u(ε,n ) u(ε) ρ

(ε,n0 )

*-weakly in L∞ (0, T ; H) and

∗

ρ(ε)

weakly in L2(0, T ; V) ,

*-weakly in L∞ (ΩT ) .

This is not enough to carry out the limit in (4.35) below. Proving compactness of (u(ε,n))n∈N in L2(ΩT ) we see that ρ(ε,n)u(ε,n) converges weakly in L2(ΩT ). 4.13 Proposition. (Compactness) For any 0 < δ < T there is c > 0 independent of n such that T−δ Z

ku(ε,n)(s + δ) − u(ε,n)(s)k22 ds ≤ c δ1/2 .

0

From Proposition 4.13, the apriori estimates and due to embedding relations we may conclude compactness of the sequence (u(ε,n)) in L2(ΩT ) as well as in Lq(I, Lp(Ω)) for 3 > 34 . Finally, with this information in (4.32) we want 2 ≤ p ≤ ∞, 2 ≤ q < 6 if p1 + 2q to let n → ∞ so as to arrive at (4.11). Namely, we multiply system (4.32) by functions hj(t) ∈ C1(¯I) with hj(T ) = 0, then sum with respect to j, integrate from 0 to T , and make an integration by parts. In this way we obtain the approximate form of our equations (4.11) for test functions of the type n P ϕ= hj(t)ωj(x). The difficult term is j=1

lim

ZT

n→∞ 0

(ρ

(ε,n) (ε,n)

u

(ε,n)

,u

? · ∇ϕ) dt =

ZT 0

(ρ(ε)u(ε), u(ε) · ∇ϕ) dt .

(4.35)

Remember that ρ(ε,n)u(ε,n) converges weakly in L2(ΩT ). We write (ρ(ε,n)u(ε,n),u(ε,n) · ∇ϕ) − (ρ(ε)u(ε), u(ε) · ∇ϕ) = = (ρ(ε,n) − ρ(ε))u(ε), u(ε) · ∇ϕ + ρ(ε,n)(u(ε,n) − u(ε)), u(ε) · ∇ϕ + + ρ(ε,n)u(ε,n), (u(ε,n) − u(ε)) · ∇ϕ .

The first term on the right-hand side tends to zero since ρ(ε,n)ρ(ε) weakly, the second and third terms go to zero due to the uniform with respect to n boundedness of the norms kρ(ε,n)k∞,ΩT , ku(ε,n)kL∞ (0,T;H) and the strong ↑ convergence of u(ε,n) in L2(ΩT ). Smooth functions of the type ϕ are dense and thus, Theorem 4.4 is proved.

4.2




Proof of Proposition 4.11. Here we simply write u instead of u(ε,n), ρ in place of ρ(ε,n), θ for θ(ε,n) and so on. Our assumptions ensure that a · n = 0 on ΓT . Thus, one can assure33 oneself of the fact that X (i.e. the solution of (4.30)) stays inside Ω and we do not need boundary conditions for ρ. Now we show that ρ has the representation 3

ρ(t, x) = ρ0(X−1(t, x)) e−ε

Rt

0

∂t θ(s,X(s,X−1 (t,x)))ds

.

(4.36)

Indeed, if we differentiate (4.36) with respect to t and x all derivatives are classical ones. Due to the regularity of w (for q > 5 we already stated that Wq2,1(ΩT ) ⊂ C(I, C1+δ(Ω))) and u we notice that v ∈ C(I, C1+δ(Ω)) . Hence, div v ∈ C(I, Cδ(Ω)). In addition X ∈ C1(¯I, C1(Ω)) and it is unique. If we also use that div v = ε3∂tθ (4.37) and notice that ρ0 is smooth we find that ρ from (4.36) fulfils the transport equation (4.28) and moreover, ρ ∈ C1(ΩT ). This is a uniqueness class and thus, the one solution of (4.28) has the representation (4.36). Note, that k∂tθk∞ ≤ c k∂tθk2,2 and m < ρ0 < M. Thus, we obtain the apriori estimate for ρ immediately from (4.36), independent of n due to (4.18). We next deduce the apriori estimate for u. We multiply (4.32) by fn,j, j = 1, · · · , n, and add up. Taking into account (4.28) and (4.33) we calculate 1 d 1/2 2 kρ uk2 + A(u, u) = − A(w, u) − (ρu · ∇w, u)− 2 dt 1 − (ρ∂tw, u) − (ρw · ∇w, u) + 3 (ρ∇f, u) . ε

(4.38)

We will show that the right-hand side of (4.38) is bounded by 1 k∇u(t)k22 + c F(t)kρ1/2u(t)k22 + G(t) 2

and the functions F(t) and G(t) have the right integrability to apply the Gronwall lemma, namely F(t), G(t) ∈ L1(I). Indeed, for any small γ > 0 we may estimate A(w, u) ≤ γ k∇uk22 + c kwk21,2 , |(ρu · ∇w, u)| ≤ kρ1/2k∞ kρ1/2uk2 ku · ∇wk2 √ ≤ M(ε)kρ1/2uk2 kuk6 k∇wk3

≤ γk∇uk22 + c(M(ε), γ)kρ1/2uk22k∇wk23 .

33 If X would cross the boundary at some time t then there would be a velocity component in normal direction to Γ which contradicts a · n = 0. Since all fields are smooth, already in a small neighbourhood of Γ this direction is made impossible for X.

4




Analogously we find |(ρw · ∇w, u)| ≤ kρ1/2k∞ kw · ∇wk2kρ1/2uk2 √ ≤ M(ε)k∇wk3kwk1,2kρ1/2uk2 , |(ρ∂tw, u)| ≤ M(ε)kρ1/2uk22 + k∂twk22 . As long as ε > 0 is fixed the ∇f-term can be estimated in the same manner under the minimal assumption that ∇f ∈ L2(I; L 6 (Ω)). If γ > 0 is suitably small, we find the 5 bound for the right-hand side of (4.38) with F(t) = c(M(ε), γ)(k∇wk23 + k∇wk3kwk1,2 + 1) , 1 G(t) = c(M(ε), γ)(kwk21,2 + k∂twk22 + 3 k∇fk26 ) . 5 ε Of course, k∇wk3 ≤ c kwk2,2 and in fact, all w(ε,n)-norms are bounded from above independently of n due to (4.27). Then the apriori bound for u immediately follows from the resulting differential inequality by means of the Gronwall lemma. This completes the proof of Proposition 4.11. Proof of Proposition 4.12. Again we fix ε and n, and skip the superscripts (ε,n). We will reformulate our problem in terms of a linear map M and a compact convex set K and show that M is a continuous self-map in K. Then the ↑ Schauder fixed point theorem ensures the existence of an approximate solution. (ε,n) Namely, let f1, · · · , fn be the coefficients of the initial value u0 in (4.31) and c^ the constant in the apriori estimate for u(ε,n) in Proposition 4.11. Then we set K := {Φ(t) ∈ C(¯I)n : kΦkC(¯I)n ≤ c^ ,

Φj(0) = fj ,

j = 1, · · · , n} .

Let Φ, Ψ ∈ K and s ∈ [0, 1]. Then it holds (1 − s)Φj(0) + sΨj(0) = fj and k(1 − s)Φ + sΨkC(¯I)n ≤ (1 − s)kΦkC(¯I)n + skΨkC(¯I)n = c^. Thus, K is convex. (1) We choose an arbitrary f∗ ∈ K, set u∗ (t, x) :=

n X

f∗j (t)ωj(x) ,

v∗ := u∗ + w ,

(4.39)

j=1

and find ρ∗ as solution to the transport equation ∂tρ∗ + div (ρ∗ v∗ ) = 0 , ρ∗(0, ·) = ρ0 . Since u∗ ∈ C(¯I; C1(Ω)) and thus, v∗ ∈ C(¯I; C1(Ω)) (due to the aforementioned properties of w), there is a unique solution ρ∗ ∈ C1(ΩT ).

^ from the linearised system (4.32), that is to say from (2) Now we determine u 1 ^ + v∗ · ∇^ v), ωj + A(^ v, ωj) = 3 (ρ∗ ∇f, ωj) , ρ∗ (∂tv ε

(4.40)

^ := u ^ + w and u ^ := f^jωj, i.e. we find the coefficients f^j as solutions to the so inserting v d^ generated ordinary differential equations. The coefficients of dt f are (ρ∗ ωj, ωl) =: kjl. Thus, the system is solvable if for all t ∈ [0, T ] we find det (kjl(t)) 6= 0.

4.2




Let t0 ∈ [0, T ] such that det (kjl(t0)) = 0. This Pn means that there exist numbers λ1, · · · , λn, which are not all equal to zero such that j=1 λjkjl(t0) = 0 for l = 1, · · · , n. If we multiply the jth relation by λj and sum over j then we produce (ρ∗(t0)Φ, Φ) = 0

for

Φ=

n X

λjωj .

j=1

But this means, that Φ = 0 which is impossible since ωj form a basis. (3) Steps (1) & (2) define a map M : K → C(¯I)n ,

M(f∗ ) = ^f .

Due to our definition M is linear. Moreover it is continuous, since the solutions of the system of ordinary differential equations depend continuously on the given data. If we multiply the jth equation in (4.40) by f^j and sum up then as in the proof of the apriori estimates in Proposition 4.11 one shows, that k^fkC[0,T]n ≤ c^, i.e. M(K) ⊂ K.

(4) Now we show that M : K → K is compact. d^ We restrict our consideration to the estimate for dt fj. d ^ th We multiply the j equation in (4.40) by dt fj and sum with respect to j to obtain

1 ^ ) − A(w, ∂tu ^ )− ^ + u∗ · ∇^ ^ + A(^ ^ ) = 3 (ρ∗ ∇f, ∂tu ρ∗ (∂tu u), ∂tu u, ∂tu ε (4.41) ˙ ∂tu ^ ) − ρ∗ u∗ · ∇w, ∂tu ^ − ρ∗ w · ∇^ ^ . −(ρ∗ w, u, ∂tu

This leads to the estimate

d 1 ^ k22 ≤ Mk∂tu ^ k2 c k∇fk2 + k∇wk2 + k∂twk2+ k∇^ uk22 + mk∂tu dt M ∗ + ku k∞ + kwk∞ k∇^ uk2 + k∇wk2 .

Obviously, by definition, u∗ ∈ C(¯I; C(Ω)3). We apply Young’s inequality to isolate ^ k22 with some small factor. This term is hidden at the left-hand side. Then integrating k∂tu from 0 to T with respect to t we find ZT ^ k22 dt ≤ c(n) ⇒ k^fkW21 (0,T) ≤ c(n) ⇒ k^fkC[0,T] ≤ c(n) . k∂tu 0

c depends on n through ku∗ k∞ . Finally, the ↑ Schauder fixed point theorem states that the map M has a fixed point in the set K. This is our solution.

Proof of Proposition 4.13. We fix n, skip (ε,n), and start with the two relations d (ρu, ϕ) + A(u, ϕ) = (ρv · ∇ϕ, u) + (ρu · ∇ϕ, w) − A(w, ϕ)− dt 1 − (ρ∂tw, ϕ) − ρw · ∇w, ϕ + 3 (ρ∇f, ϕ) , ε d (ρ, φ) = (ρv, ∇φ) dt

4




for all ϕ ∈ span(ω1, · · · , ωn) ⊂ V and all φ ∈ W21(Ω). We integrate these identities on (t, t + δ) and then set ϕ := u(t + δ) − u(t) while φ := u(t) · u(t + δ) − u(t) . After that we use the obvious relation ρ1u1 − ρ2u2 = ρ1(u1 − u2) + (ρ1 − ρ2)u2

to combine both equations and arrive at the following: Z t+δ 2 ρ(s)v(s), ∇ u(t) · u(t + δ) − u(t) ds− kρ (t + δ) u(t + δ) − u(t) k2 = − t Z t+δ Z t+δ A w(s), u(t + δ) − u(t) ds+ A u(s), u(t + δ) − u(t) ds − − t t Z t+δ + ρ(s)v(s) · ∇ u(t + δ) − u(t) , u(s) ds+ t Z t+δ + ρ(s)u(s) · ∇ u(t + δ) − u(t) , w(s) ds− t Z t+δ ρ(s)∂tw(s), u(t + δ) − u(t) ds− − t Z t+δ − ρ(s)w(s) · ∇w(s), u(t + δ) − u(t) ds− t Z t+δ ρ(s) − ∇f(s), u(t + δ) − u(t) ds . ε3 t 1 2

From the left-hand side we obtain the desired compactness estimate through m(ε) ≤ ρ if we find appropriate upper bounds for all terms at the right-hand side. Let Vδ(t) := |u(t + δ)| + |u(t)| ,

Wδ(t) := |∇u(t + δ)| + |∇u(t)| .

We calculate writing M and m for M(ε) and m(ε), respectively, ku(t + δ) − u(t)k22 ≤ t+δ Z Z 1 M 2 Wδ(t) |u(s)| + (|∇u(s)| + |∇w(s)|) + (|u(s)| + |w(s)|)|u(t)| + ≤ m M t Ω + Vδ(t) (|u(s)| + |w(s)|)|∇u(t)| + c |∇f(s)| + |∂tw(s)| + |w(s)||∇w(t)| dxds .

Now we estimate the terms. We start with I :=

T−δ Z t+δ Z Z 0

t Ω

|u(s)|2|∇u(t + δ)| dxdsdt ≤ c

T−δ Z 0

k∇u(t + δ)k2

t+δ Z

ku(s)k21,2 dsdt .

t

Here first we applied the Hölder inequality for the exponents 41 + 14 + 12 = 1 and then the embedding W21(Ω) ⊂ L4(Ω). Now we may change the order of integration, at t > T and

4.2




t < 0 setting the vector function u(t) = 0. Then I≤c

ZT 0

ku(s)k21,2

Zs

s−δ

k∇u(t + δ)k2 dtds .

Now in the integral with respect to t we use the Hölder inequality to obtain a factor δ1/2. Then we extend the integration interval from [s − δ, s] to [0, T ]. With our apriori estimate we see that ZT 3/2 1/2 ≤ c1c^3/2δ1/2 . ku(t)k21,2 dt I ≤ c1δ 0

The term |u(s)|2|∇u(t)| is treated the same way. We continue with T−δ Z t+δ Z Z 0

|∇u(s)||∇u(t + δ)| dxdsdt ≤ c(δT )

t Ω

1/2

ZT

k∇u(t)k22 dt .

0

We note that the fact that div u = 0 and u = 0 on the boundary is not used in the calculations. Thus, w can be treated in same way as u, e.g. we calculate as above to find Z T−δ Z t+δ Z 0

t

|u(s)||w(s)||∇u(t + δ)| dxdsdt ≤ Z T−δ Z t+δ ≤c k∇u(s)k2kw(s)k1,2k∇u(t + δ)k2dsdt 0 t Zs ZT 1 · k∇u(t + δ)k2dtds . ≤ c k∇u(s)k2kw(s)k1,2 Ω

s−δ

0

In the second integral we apply Hölder’s inequality, extend the integration to [0, T ] and obtain the bound cM(ε)δ1/2kuk2L2 (I;V)kwkL2 (I;W21 ). For the ∂tw-terms we arrive at T−δ Z Z Z t+δ 0

|∂tw(s)||u(t + δ)| dxds dt ≤

t Ω

T−δ Z t+δ Z 0

≤ cδ

k∂tw(s)k2 ds ku(t + δ)k2 dt

t 1/2 1/2

T

k∂twkL2 (ΩT )kukL2 (I;V) .

Treating all terms we arrive at a bound of the type T−δ Z

ku(t + δ) − u(t)k22 dt ≤

0

≤c

2 M(ε) 1/2 δ kuk kuk + kwk 1 ) + k∂twkL (Ω ) + k∇fkL (I;L ) L (I;V) L (I;V) L (I;W 2 2 2 T 2 6 2 2 m(ε) 5

with 0 < c = c(T, Ω). This completes the proof of Proposition 4.13.

4




#

% (

4.3 Proof of Theorem 4.5 Again we indicate the dependence on ε by the superscript (ε). Our procedure to prove Theorem 4.5 in principle is the same as used in Theorem 4.4 but there are two new aspects. On the one hand our estimates must be independent of ε which follows for most of the quantities from the proof of Theorem 4.4 taking into account the assumptions on the limits in Theorem 4.5, i.e. we may immediately deduce the existence of the limit while ε → 0 for most of the terms in (4.1). On the other hand we must be more careful concerning the right-hand side of (4.1)2 than in Theorem 4.4. For ε → 0 we expect that ρ(ε) → 1. But this is not enough information to control the term ε13 ρ(ε)∇f as ε → 0. (ε)

3 (θ(ε) −θ(ε) ) 0

For system (2.33) we calculated ρ(ε) = ρ0 e−ε (ε)

and Taylor expansion at 0 yields

(ε)

ρ(ε) = ρ0 (1 − ε3(θ(ε) − θ0 )) + O(ε6) . (ε)

(ε)

(ε)

From this one may conclude that ε13 (ρ(ε)−1) = ε13 (ρ0 −1)−ρ0 (θ(ε)−θ0 )+O(ε3). The (ε) right-hand side of this relation converges if ε13 (ρ0 − 1) converges (cf. the assumptions of Theorem 4.5). For that in our case we expect that also the limit r(ε) := exists as ε → 0.

1 (ε) ρ − 1 →r ε3

(4.42)

4.14 Remark. In the spirit of our approximation procedure one might assume that (ε)

(ε)

(ε)

ρ0 = ρ0(1 − ε3θ0 ) + O(ε6) = 1 − ε3θ0 + O(ε6) (ε) (note, ρ0 = 1). This corresponds to ε13 ρ0 − 1 → r0 = −ρ0θ0 = −θ0.

To investigate the limit (4.42) first we note that the term ε13 (ρ(ε,n)∇f, ωj) on the righthand side of (4.32) can be replaced by (r(ε,n)∇f, ωj) with r(ε,n) :=

1 (ε,n) (ρ − 1) ε3

since div ω(j) = 0 and ω(j)|ΓT = 0. Consequently, instead of (4.38) we obtain 1 d (ε,n)1/2 (ε,n) 2 u k + A(u(ε,n), u(ε,n)) = −A(w(ε,n), u(ε,n))− kρ 2 dt −(ρ(ε,n)u(ε,n) · ∇w(ε,n), u(ε,n)) − (ρ(ε,n)∂tw(ε,n), u(ε,n))−

−(ρ(ε,n)w(ε,n) · ∇w(ε,n), u(ε,n)) + (r(ε,n)∇f, u(ε,n)) .

(4.43)

4.3




Now - in addition to the proof of Theorem 4.4 - we need an apriori bound for r(ε,n) independent of n and ε. 4.15 Proposition. Under the assumptions of Theorem 4.5 it holds sup kr(ε,n)(t)k2 ≤ c , t∈I

with a constant independent of n and ε. Proof. From the equation for ρ(ε,n) we find the differential equation for r(ε,n), namely ∂tr(ε,n) + v(ε,n) · ∇r(ε,n) = −∂tθ(ε,n)ρ(ε,n) ,

(ε,n)

r0

1 (ε,n) (ρ − 1) . ε3 0

:=

(4.44)

Estimate (4.16) provides a bound for θ(ε,n) in W22,1(ΩT ), which is independent of n and ε. Moreover, (4.25) yields a bound for ε3θ(ε,n) in W24,2(ΩT ) which is independent of n and ε. This means in particular that ∂tθ(ε,n) is bounded independently of n and ε in L2(ΩT ) and ε3∂tθ(ε,n) in L2(I; L∞ (Ω)), respectively. The scalar product in L2(Ω) of (4.44) with r(ε,n) furnishes Z Z 1 d (ε,n) 2 1 (ε,n) (ε,n) 2 kr k2 = − v · ∇(|r | ) dx − ∂tθ(ε,n) ρ(ε,n) r(ε,n) dx 2 dt 2 Ω Ω Z 1 (ε,n) 2 |r | div v(ε,n) dx + kρ(ε,n)k∞ k∂tθ(ε,n)k2 kr(ε,n)k2 ≤ Ω2 1 3 ≤ ε k∂tθ(ε,n)k∞ kr(ε,n)k22 + c(M0)k∂tθ(ε,n)k22 + kr(ε,n)k22 , 2 where we also used (4.37). The Gronwall lemma then provides (ε,n) 2 k2)

kr(ε,n)k22 ≤ c (1 + kr0 (ε,n)

Since ρ0

∀ t ∈ I. (ε,n)

(ε)

is a ↑ mollification of ρ0 we observe that kr0 (ε,n)

kr0

(ε)

k2 ≤ c (1 + kr0 k2) .

(ε)

− r0 k2 → 0 and hence,

From this we find our estimate for r(ε,n) and can let n → ∞. (ε) (ε) We assumed that ε13 (ρ0 − 1) → r0 in L2(Ω). For that kr0 k2 and kr(ε,n)k2 as well are bounded uniformly in ε and our estimate is proved.

Via Proposition 4.15 we may then proceed as done in the proof of Theorem 4.4. As indicated in (4.43) we have the only difference that deriving the apriori bounds for u(ε,n) we come across the new term (r(ε,n)∇f, u(ε,n)), which we treat as follows (r(ε,n)∇f, u(ε,n)) ≤ γk∇u(ε,n)k22 + c(γ) kr(ε,n) ∇fk26/5 . The application of Hölder’s inequality with the exponents 35 for r(ε,n) and 25 at ∇f, respectively, reveals that we need more integrability in space for the force term than in the proof of Theorem 4.4 and find kr(ε,n) ∇fk26/5 ≤ kr(ε,n)k22k∇fk23 .

4




As done below (4.38) also for (4.43) we can obtain an estimate with functions F(t) and G(t) that do not depend on ε. Finally we arrive at the second estimate of Proposition 4.11 with a constant that does not depend on n and ε and we may summarize that we have bounds independent of n and ε such that v(ε,n) → v ,

div v(ε,n) → 0 , ρ

and

(ε,n) ∗

∂tθ(ε,n) → ∂tθ

in L2(ΩT ) ,

while

ρ *-weakly in L∞ (ΩT ) .

4.16 Proposition. The limit ρ = 1 a.e. in ΩT as n → ∞ and then ε → 0. ∗

Proof. We already showed that ρ(ε,n) ρ *-weakly in L∞ (ΩT ) as n → ∞ and then ε → 0. Now we want to show that ρ = 1 a.e. in ΩT . (ε,n) (ε,n) Let σ(ε,n) := ρ(ε,n) − 1 and σ0 := ρ0 − 1. For our purpose it is enough to show that kσ(ε,n)k2 → 0 as n → ∞ and then ε → 0 since then σ(ε,n) → 1 a.e. in ΩT and thus, ρ = 1 a.e. in ΩT . Obviously, σ(ε,n) fulfils the differential equation ∂tσ(ε,n) + v(ε,n) · ∇σ(ε,n) + ε3∂tθ(ε,n)ρ(ε,n) = 0

and the L2(Ω) scalar product with σ(ε,n) yields Z Z 1 1 d (ε,n) 2 (ε,n) (ε,n) 2 3 ∂tθ(ε,n) ρ(ε,n) σ(ε,n) dx . kσ k2 = − v · ∇(|σ | ) dx − ε 2 dt 2 Ω Ω In the first term at the right-hand side we make partial integration and then replace through the equation div v(ε,n) = ε3∂tθ(ε,n). This leads to the estimate 1 d (ε,n) 2 1 3 kσ k2 ≤ ε k∂tθ(ε,n)k∞ kσ(ε,n)k22 + ε3kρ(ε,n)k∞ k∂tθ(ε,n)k2 kσ(ε,n)k2 2 dt 2 1 1 ≤ (ε3k∂tθ(ε,n)k2,2 + 1 kσ(ε,n)k22 + ε6c(M0)k∂tθ(ε,n)k22 . 2 2

Now (4.16), (4.17), (4.34), and Gronwall’s lemma provide as n → ∞, ε → 0 (ε,n) 2 k2)

kσ(ε,n)k22 ≤ c (ε6 + kσ0

(ε,n)

(ε,n)

→0

since ρ0 → 1 by our assumptions and thus, σ0 → 0 as n → ∞, ε → 0 (all limits in the sense of strong ↑ convergence in L2(ΩT )). This together with (4.34) and Proposition 4.11 yields the assertion of Proposition 4.16. Due to our apriori bound for r(ε,n) we know moreover, that also r(ε,n) r

weakly in L2(ΩT ) .

Thus, taking the limit in all terms in (4.44) r is identified as the weak solution to ∂tr + v · ∇r = −∂tθ , This finishes the proof of Theorem 4.5

r(0, ·) = r0 .

Appendix



Appendix: I. Linearised operator at the motionless state In this section we prove the estimate (3.54) for the semigroup e−tL by showing Theorem 3.13. We use the analytic perturbation theory for two parameters (see [72, p. 66ff, 116f, 119]). Usually the case of several parameters is delicate. However, after some reduction, we can transform our problem to an eigenvalue problem for which the critical eigenvalue is known to be simple. We consider the eigenvalue problem linearised at the motionless state u = 0: −σu + Lu = 0 , (3.53) where L is defined in (3.52). For u := (v, θ) ∈ H × L2(Ω) we agreed upon the norm kuk := (kvk22 + kθk22)1/2 . Here and in the following we denote the scalar product of H × L2(Ω) by (·, ·) which for uj := (vj, θj) ∈ H × L2(Ω) (j = 1, 2) is defined by: p (u1, u2) := (v1, v2)L2 (Ω) + Pr (θ1, θ2)L2 (Ω) and kuk := (u, u) . Theorem 3.13 There exist ζc > 0 and Rac(ζ) ≥ Rac such that if 0 ≤ ζ ≤ ζc and Ra < Rac(ζ), then σ1( Ra , ζ) > 0. Moreover, if 0 ≤ ζ ≤ ζc and Ra > Rac(ζ), then σ1( Ra , ζ) < 0. Here the number Rac(ζ) satisfies Rac(0) = Rac and

Rac(ζ) > Rac for 0 < ζ ≤ ζc .

Proof. Performing the transformation introduced in (3.16) we find the following problem, which is equivalent to (3.53), −σu + L(λ, ζ)u = 0 , L(λ, ζ)u :=

(A.1) Av − λP(θb) 1 λ ^ − x3) − 1 v3 − ∆θ + ζ(Θ Pr Pr

!

.

It is known that for ζ = 0, . all eigenvalues (σn(λ) := σn(λ, 0))n≥1 are real, . the smallest eigenvalue has even multiplicity, say 2m (m ∈ N), and

σ0(λ) := σ1(λ) = · · · = σ2m(λ) < σ2m+1(λ) ≤ · · · ≤ σn(λ) ≤ · · · → +∞ , (L(λ, 0)u, u) ≥ σ0(λ)kuk2 for all u ∈ D(L) := D(A) × D(B) .

(A.2)

√ . Furthermore, σ0(λ) > 0 (resp. σ0(λ) < 0) if and only if λ < λ0 := Rac (resp. λ > λ0) while σ0(λ) = 0 if and only if λ = λ0. . There exists γ0 = γ0(l1, l2, Pr ) > 0 such that if j ≥ 2m + 1 and λ ≤ λ0 then σj(λ) ≥ γ0. . If 1 ≤ j ≤ 2m each σj(λ) is continuous in λ. In particular, for any ε > 0 there exists δ(ε) > 0 such that if λ < λ0 − ε, then σ0(λ) ≥ δ(ε).

Appendix

 We now consider the case 0 < ζ 1. We write (A.1) as −σu + L0u + (λ − λ0)M1u + ζM2u + M3(λ, ζ)u = 0, 0 −P(θb) where L0 = L(λ0, 0), M1u = , M2u = λ0 ^ 1 v3 − Pr (Θ − x3)v3 Pr and

M3(λ, ζ)u =

0 λ−λ0 ^ − x3)v3 ζ(Θ Pr

.

Let first λ < λ0 − ε for some ε > 0. A.1 Proposition. For any ε > 0 there exists δ(ε) > 0 and ζ1(ε) > 0 with σ(L(λ, ζ)) ⊂ {σ ; Re σ ≥ δ(ε)/2}

if λ < λ0 − ε and 0 ≤ ζ ≤ ζ1(ε).

^ − x3)v3k2 ≤ c kuk, we see from (A.2) that Proof. Since k(Θ Re (L(λ, ζ)u, u) = (L(λ, 0)u, u) + Re (ζM2u, u) + Re (M3(λ, ζ)u, u) ≥ (σ0 − cλ0ζ)kuk2.

Now recall that for any ε > 0 there exists δ = δ(ε) > 0 such that if λ < λ0 − ε, then δ(ε) and λ < λ0 − ε, then σ0 ≥ δ(ε). Thus, if ζ ≤ 2cλ 0 Re (L(λ, ζ)u, u) ≥ δ(ε) 21 kuk2, which implies that σ(L(λ, ζ)) ⊂ {σ ; Re σ ≥ This shows Proposition A.1

1 2

δ(ε)} for λ < λ0 − ε and 0 ≤ ζ ≤

δ(ε) . 2cλ0

We next investigate L(λ, ζ) for |λ − λ0| ≤ ε 1 and 0 < ζ 1. A.2 Proposition. (i) There exist ε2 > 0 and ζ2 > 0 such that σ(L(λ, ζ)) ⊂ {σ ; |σ| ≤ 14 γ0} ∪ {σ ; Re σ ≥ 43 γ0}

(A.3)

if |λ − λ0| ≤ ε2 and 0 ≤ ζ ≤ ζ2. (ii) There exist 0 < ε3 ≤ ε2 and 0 < ζ3 ≤ ζ2 such that the eigenvalues of L(λ, ζ) in {σ ; |σ| ≤ 41 γ0} have the form σ = σ(1,0)(λ − λ0) + σ(0,1)ζ + O(|λ − λ0|2 + ζ2)

(A.4)

with constants σ(1,0) < 0 and σ(0,1) > 0, if |λ − λ0| ≤ ε3 and 0 ≤ ζ ≤ ζ3. Moreover, there exists λc = λc(ζ) > 0 satisfying λc(0) = λ0 and

λc(ζ) > λ0 for 0 < ζ ≤ ζ3

and it holds σ1(λ, ζ) > 0 if λ < λc(ζ) and σ1(λ, ζ) < 0 if λ > λc(ζ), provided that |λ − λ0| < ε3 and 0 ≤ ζ ≤ ζ3.

Appendix



Proof. First we observe kMjuk2 ≤ ckuk (j = 1, 2, 3).

(A.5)

Since L0 is self-adjoint there exists an orthonormal basis of eigenfunctions which are used to express u and L0u for u ∈ D(L0). From this we compute k(−µ + L0)uk2 ≥ (min | − µ + σk|)2kuk2.

(A.6)

k≥1

^ Pr ) > 0, Combining (A.5) and (A.6) we obtain for some constant a = a(λ0, Θ, k (λ − λ0)M1 + ζM2 + M3(λ, ζ) (−µ + L0)−1uk 1 a(|λ − λ0| + ζ) kuk ≤ kuk, ≤ min | − µ + σk| 2

(A.7)

k≥1

provided that µ ∈ S := {σ ; |σ| > 41 γ0} ∩ {σ ; Re σ < 43 γ0}, |λ − λ0| ≤ ε2 and 0 ≤ ζ ≤ ζ2 for some small ε2 > 0 and ζ2 > 0. Using the identity −µ + L(λ, ζ) = (I + ((λ − λ0)M1 + ζM2 + M3(λ, ζ))(−µ + L0)−1)(−µ + L0) we see that S is included in the resolvent set of L(λ, ζ) and (A.3) follows. To prove (A.4) we note that the problem (A.1) is equivalent to   −σv − ∆v − λθb + ∇p = 0, 1 λ ^ − x3) − 1)v3 = 0, (A.8) −σθ − Pr ∆θ + Pr (ζ(Θ  div v = 0

with boundary conditions under consideration. To solve (A.8) we expand v, θ and ∇p into Fourier series in x1 and x2, and so we assume v, θ and ∇p to have the form k

e

k

2πi( l 1 x1 + l 2 x2 ) 1

2

h(x3) ,

where (k1, k2) ∈ Z2. We first consider the case (k1, k2) = (0, 0), namely, vj = vj(x3) (j = 1, 2, 3), θ = θ(x3). Equation div v = 0 leads to dxd3 v3 = 0. This, together with v = 0 on {x3 = 0, 1}, yields v3 := 0. We then obtain −σkvjk2L2 (0,1) + k dxd3 vjk2L2 (0,1) = 0 (j = 1, 2),

−σkθk2L2 (0,1) +

1 k d θk2L2 (0,1) Pr dx3

= 0.

This implies that σ ≥ aπ2 = a inf

k dxd3 hk2L2 (0,1) khk2L2 (0,1)

; h ∈ H10(0, 1), h 6= 0 ,

where a = min{1, Pr −1}. Therefore, we see that σ ∈ {σ ; Re σ ≥ 43 γ0}.

Appendix



We next consider (k1, k2) 6= (0, 0). This is the case for which there can occur σ ∈ {σ; |σ| ≤ 14 γ0}. Taking curl curl of (A.8)1, we obtain

σ∆v3 + ∆2v3 + λ∆2θ = 0, λ 1 ^ − x3) − 1)v3 = 0 ∆θ + Pr (ζ(Θ −σθ − Pr

(A.9)

with boundary conditions v3 = ∂3v3 = θ = 0 at x3 = 0, 1 and the periodic boundary conditions in x1 and x2. Here ∆2 = ∂11 + ∂22. We now substitute k

v3 = e

k

k

2πi( l 1 x1 + l 2 x2 ) 1

2

f(x3) ,

θ=e

k

2πi( l 1 x1 + l 2 x2 ) 1

2

g(x3)

for (k1, k2) 6= (0, 0) into (A.9). Then we find the eigenvalue problem :  2 2  −σDωf + Dωf − λω g = 0 1 λ ^ − x3) − 1)f = 0 −σg + Pr Dωg + Pr (ζ(Θ  d f = dx3 f = g = 0

(0 < x3 < 1), (x3 = 0, 1),

2

(A.10)

2

d d 2 2 2 2 1 2 2 2 where ω2 := ( 2πk ) + ( 2πk ) > 0, Dω := (− dx 2 + ω ) and Dω := ( dx2 − ω ) . l1 l2 3 3 It is easily verified that the eigenvalues and eigenfunctions of (A.8) for (k1, k2) 6= (0, 0) can be obtained from those of (A.10) with suitable ω2 > 0 and vice versa, since ω2 > 0. We write (A.10) as

−σMf + L(λ, ζ)f = 0, f = {f, g}. Here

M :=

Dω 0 0 1

, L(λ, ζ) :=

D2ω λ ^ − x3) − 1) (ζ(Θ Pr

(A.11) −λω2 1 D Pr ω

and the operators Dω and D2ω are defined as above for g ∈ H2(0, 1) ∩ H10(0, 1) and f ∈ {f ∈ H4(0, 1) ; f = dxd3 f = 0 at x3 = 0, 1}, respectively. The eigenvalues σj(λ0) of L0 are given by the eigenvalues of the eigenvalue problem (A.11) with λ = λ0 and ζ = 0, and moreover, the eigenvalues of L(λ, ζ) in {σ ; |σ| ≤ 41 γ0} are given by those of L(λ, ζ) in {σ ; |σ| ≤ 41 γ0}. Especially, σ(L(0, 0))∩{σ ; |σ| ≤ 41 γ0} = {σ0(λ0) = 0}. The following lemma summarises facts contained in [112, p. 38] which are based on results obtained already earlier and quoted therein. A.3 Lemma. (i) The eigenvalue σ0(λ0) = 0 of L(0,0) := L(0, 0) is simple. (ii) One can choose an eigenfunction f0 = {f0, g0} of L(0,0) associated with σ0(λ0) = 0 in such a way that f0(x3) > 0 and g0(x3) > 0 for 0 < x3 < 1. Since σ0(λ0) is simple by Lemma A.3 (i), there exists only one eigenvalue σ = σ(λ, ζ) of L(λ, ζ) in {σ ; |σ| ≤ γ40 } when |λ − λ0| and ζ are sufficiently small. Furthermore, due to the simplicity of σ0(λ0), one can see that σ(λ, ζ) is analytic in λ and ζ near λ = λ0 and ζ = 0 and it is expanded as σ(λ, ζ) =

∞ P

σ(j,k)(λ − λ0)jζk with σ(0,0) = σ(λ0) = 0.

j,k≥0

(A.12)

Appendix



Let f(λ, ζ) be the eigenfunction associated with σ(λ, ζ) satisfying f(0, 0) = f0. Then f(λ, ζ) =

∞ P

(λ − λ0)jζkf(j,k)

(A.13)

j,k≥0

with f(0,0) = f0. Substituting (A.12) and (A.13) into (A.11) we obtain L(0,0)f0 = 0, (1,0)

(1,0)

+ L(1,0)f0 = 0,

(A.14)

(0,1)

(0,1)

+ L(0,1)f0 = 0

(A.15)

−σ

Mf0 + L(0,0)f

−σ and so on. Here L(λ, ζ) = L(1,0)f =

Mf0 + L(0,0)f

P

0≤j,k≤1(λ

− λ0)jζkL(j,k) with L(0,0) = L(0, 0),

1 λ0 ^ −ω g, − f , L(0,1)f = 0, (Θ − x3)f Pr Pr 2

(j,k) and L(1,1) = λ−1 we define h·, ·i by 0 L(0,1). To compute σ

1 hf1, f2i := 2 ω

Z1

f1(x3)f2(x3)dx3 + Pr

Z1

g1(x3)g2(x3)dx3

0

0

for fj = {fj, gj} ∈ L2(0, 1)2 (j = 1, 2). Here f denotes the complex conjugate of f. Note that hL(0,0)f1, f2i = hf1, L(0,0)f2i and hMf, fi > 0 for f 6= 0. Taking h·, ·i of (A.14) and (A.15) with f0 respectively, we obtain σ(1,0) =

hL(1,0)f0, f0i hMf0, f0i

and

σ(0,1) =

hL(0,1)f0, f0i , hMf0, f0i

respectively. The coefficient σ(1,0) must satisfy σ(1,0) < 0, since σ0(λ) > 0 if and only if λ < λ0, and σ0(λ) < 0 if and only if λ > λ0. Since f0, g0 > 0 by Lemma A.3 (ii) and ^ > 1 ≥ x3 for 0 ≤ x3 ≤ 1, we see that since Θ hL(0,1)f0, f0i =

Z1

^ − x3)f0(x3)g0(x3)dx3 > 0. λ0(Θ

0

Thus, σ(0,1) > 0, and we have obtained (A.4). Now we define λ0(ζ) by σ(λ0(ζ), ζ) = 0 and arrive at σ(0,1) λ0(ζ) = λ0 − (1,0) ζ + O(ζ2). σ Since λ0(ζ) also depends on ω2, we denote λ0(ζ) by λ0(ζ; ω2). Then the critical number Rac(ζ) is given by Rac(ζ) = λc(ζ)2 with λc(ζ) =

inf

(k1 ,k2 )∈Z2 \(0,0)

λ0(ζ; (

2πk2 2 2πk1 2 ) +( ) ). l1 l2

Appendix

 The calculation above can be justified as follows: For f ∈ D := H10(0, 1) × L2(0, 1) the problem (A.11) is equivalent to

−σf + L(0,0)f + M−1 (λ − λ0)L(1,0) + ζL(0,1) + (λ − λ0)ζL(1,1) f = 0, Dω −λ0ω2D−1 ω L(0,0) = λ0 1 − Pr D Pr ω 1/2

2 1 and Dω is the self-adjoint extension of D−1 ω Dω in D(Dω ) = H0(0, 1) with domain {f ∈ H3(0, 1) ; f = dxd3 f = 0 at x3 = 0, 1}. As for Dω we refer to [139, Th.V.2]. One can then obtain for certain a, b > 0,

|||L(j,k)f||| ≤ a |||L(0,0)f||| + b |||f|||. Here ||| · ||| denotes the norm in D : |||f|||2 = kM1/2fk2 =

1 kD1/2fk2 + Pr kgk2L2 (0,1). ω2 ω L2 (0,1)

Since σ0(λ0) = 0 is a simple eigenvalue of L(0,0), one can see that σ(λ, ζ) is analytic in λ and ζ near λ = λ0 and ζ = 0 in the same way as in [72]. Then σ(1,0) and σ(0,1) are immediately obtained as above. Theorem 3.13 now follows from Propositions A.1 and A.2 by taking ε = ε3 in Proposition A.1 and ζc = min{ζ1(ε3), ζ3}. This completes the proof of Theorem 3.13

II. Additional Parabolic Embedding Property Let X0 ⊂ X1 be two Banach spaces, p0, p1 ∈ [1, ∞), and I := [0, T ] for some T > 0. We denote d XI(X0, X1) := {φ ∈ L1,loc(I; X0) , dt φ ∈ L1,loc(I; X1)} , VI := {φ ∈ XI(X0, X1) : kφkVI < ∞} ,

d φkLp1 (I;X1 )} . kφkVI := max {ktα0 φkLp0 (I;X0 ), ktα1 dt

A.4 Theorem. Let γ := kφkL∞ (I;A

1 γ, γ

)

p1 p1 +p1 p0 −p0

. The space VI is embedded into C(I; Aγ, 1 ) and γ

k d φkγLp ≤ c max {kφkLp0 (I;X0 ), kφkL1−γ p (I;X0 ) dt 0

Here Aγ, 1 is the interpolation space indicated in the estimate above. γ

1

(I;X1 )} .

Part C References & Tables

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Glossary



Glossary Adiabatic: Thermodynamic process in which there is no heat transfer between a system and its surrounding environment. Banach space: A complete normed vector spaces. This means that a Banach space is a vector space V over the real or complex numbers with a norm k.k such that every ↑ Cauchy sequence in V has a limit in V. BUC(I,X): Bounded, uniformly continuous functions defined in I with values in X. Bulk viscosity: Viscosity associated with changes in volume. Buoyancy: Without heating buoyancy is due to pressure differences across a fluid packet which when balanced by the weight of the packet, lead to static equilibrium. With heating the fluid packet considered has less density relative to the surrounding which makes it rise owing to the increase in buoyancy force which disrupts the static equilibrium. d v(t) = f(t, v), v(t0) = v0 ∈ Rn of Carathéodory conditions: Consider the system dt ordinary differential equations on some time interval Iψ := (t0 − ψ, t0 + ψ). Assume f : Iψ × K → Rn where K := {v ∈ Rn : |v − v0| < ε} for some ε > 0. Then this function f is said to satisfy the Carathéodory conditions if

. t 7→ fi(t, v) is measurable for all i = 1, ..., n and v ∈ K,

. v 7→ fi(t, v) is continuous for almost all t ∈ Iψ,

. there exists an integrable function g : Iψ → R such that

|fi(t, v)| ≤ g(t) for all (t, v) ∈ Iψ × K ,

i = 1, ..., n .

If f satisfies the Carathéodory conditions, then there exists ψ 0 ∈ (0, ψ) and a continuous d function v : Iψ0 → Rn such that dt v exists for almost all t ∈ Iψ0 and v solves the system of ordinary differential equations above.

Cauchy sequence: A sequence in a metric space (M, d) is called a Cauchy sequence if for every positive real number ε, there is an integer N such that for all integers m and n greater than N the distance d(xm, xn) < ε. Convergence: As n → ∞ a sequence {xn} converges . strongly, i.e. {xn} → x in X . weakly, i.e. {xn} x in X ∗

⇔

⇔

kxn − xkX → 0 ,

ϕ(xn) → ϕ(x) ∀ ϕ ∈ X∗ ,

. *-weakly, i.e. {xn} x in X ⇔ xn(ξ) → x(ξ) ∀ ξ ∈ Y , where Y is a Banach space with the property Y ∗ = X (i.e. predual space to X) .

If X has a separable predual Y (i.e. there is a countable y ⊂ Y with y = Y) and the ∗ sequence {xn} is bounded in X then there exists a subsequence {xnk } x in X. Lq-spaces are ↑ reflexive if 1 < q < ∞. In this case (Lq)∗ can be identified with Lq0 and weak and *-weak convergence coincide. Creep: Slow deformation of material, usually measured under constant stress.

Glossary



Cretaceous Period: 146 to 65 million years ago, first fossils of many insect groups, modern mammal and bird groups, and the first flowering plants. The breakup of the world-continent Pangaea, which began to disperse during the Jurassic, continued. This led to increased regional differences in floras and faunas between the northern and southern continents. The end of the Cretaceous brought the end of many previously successful and diverse groups of organisms, such as non-avian dinosaurs and ammonites. This laid open the stage for those groups which had previously taken secondary roles to come to the forefront. The Cretaceous was thus the time in which life as it now exists on earth came together. Deformation gradient: The deformation gradient is the derivative of the configuration of the body. It gives the mapping between points in the original geometry to current geometry and has the basic information for the different strain measures that can be formed. It can be computed when the coordinate system and the motion are defined. Demon, Maxwell’s: Imagine a box filled with a gas at some temperature with an average speed of the molecules. Suppose that a partition is placed across the middle of the box separating the two sides into left and right. Maxwell imagined a molecule sized trap door in the partition with his minuscule creature poised at the door who is observing the molecules. When a faster than average molecule approaches the door he makes certain that it ends up on the left side (by opening the tiny door if it’s coming from the right) and when a slower than average molecule approaches the door he makes sure that it ends up on the right side. So after these operations he ends up with a box in which all the faster than average gas molecules are in the left side and all the slower than average ones are in the right side. Then one can use this separation of temperature to run a heat engine by allowing the heat to flow from the hot side to the cold side. Dilatant: ↑ Shear Thickening.

Dual space: Let X be a Banach space equipped with the norm k.kX. Its dual space X∗ consists of all functionals, i.e. the linear and continuous operators from X to R. Domain: An open connected subset of Rn. Elliptic: A second-order partial differential equation, of the form A · (∂2iku(x))3i,k=1 + b · ∇u(x) = c(x) is called elliptic if the (constant) matrix A is positive definite. It is called parabolic, if one eigenvalue of A is zero and the others have the same sign. The basic example of an elliptic partial differential equation is Laplace’s equation: ∆u(x) = 0. The notions elliptic and parabolic are adapted to partial differential equations with nonconstant coefficients and to systems as well. For example, Lu(x) := div (A(x)∇u(x)) for some matrix A(x). L is elliptic in the domain D if for any ξ ∈ R3 and x ∈ D it holds 0 < c1|ξ|2 ≤ ξ> A(x) ξ ≤ c2|ξ|2. Entropy: Entropy, an idea born from classical thermodynamics, is a quantitative entity, and not a qualitative one. That means that entropy is fundamentally defined via an equation. In classical thermodynamics, the entropy of a system is the ratio of internal heat energy to temperature. It was introduced in 1854 by Rudolph Clausius, who built on the work of Carnot. His ideas were later extended and clarified by Helmholtz and others.

Glossary Eulerian Coordinates: Coordinates used in fluid dynamics which are fixed in space. Euclidean transformation: Transformation that preserves lengths and angle measures, i.e. it consists of translation, rotation, and reflection. Fennoscandian ice sheet: Situated in NW Europe during last glacial period. Based on raised and tilted shorelines associated with Baltic lakes and seas one shows that the eustatic sea level during the last glacial maximum was between −85 and −90 m. Then the sea level rose to about −65 m by 15,000 BP, fell until after 13,000 BP when there was a rapid rise until 11,000 BP. Thereafter, it fluctuated slightly and began to rise rapidly after 10,000 BP, reaching the position of present sea level around 5,000 BP. Free-air anomaly: Difference between observed gravity and theoretical gravity that has been figured for latitude and corrected for elevation of the station above or below the ↑ geoid, by application of the normal rate of change of gravity for change of elevation.

Geodynamics: Examination of causes and consequences of plate motions over time scales for postseismic deformation to continental drift. Geoid: That equipotential surface of the earth gravity field that most closely approximates the mean sea surface. At every point the geoid surface is perpendicular to the local plumb line. It is therefore a natural reference for heights - measured along the plumb line. At the same time, the geoid is the graphical representation of the earth gravity field. The geoid surface is described by geoid heights that refer to a suitable earth reference ellipsoid. Geoid heights are relative small. The minimum of some −106 meter is located at the Indian Ocean. ¨ Gruneisen constant Γ : Ratio of the thermal pressure to thermal energy per unit volume, Γ := cVV (∂θp)V = αβcVθ V . Hilbert space: An inner product space which is complete with respect to the norm defined by the inner product (and is hence a ↑ Banach space).

Inertial frame of reference: It has a constant velocity. That is, it is moving at a constant speed in a straight line, or it is standing still (when something is standing still, it has the constant velocity zero meters per second), which is the same as saying that the frame is not accelerating. Isochoric: Volume preserving. It holds det F = 1 and thus, div v = 0 [25, § 4]. Lagrangian Coordinates: Coordinates used in fluid dynamics in which the coordinates are fixed to a given parcel of fluid, but move in space. Laurentide Ice Sheet: Situated in the north of America with three independent glaciation centers, one located over Nouveau Quebec/Labrador, one over Keewatin and one over the Fox basin. Separate ice caps were also centred over the northern Appalachians and where collectively nearly as large as the Fennoscandian ice sheet. Lipschitz domain: The boundary ∂G of a domain G ⊂ Rn belongs to the class Ck for k ∈ N or shorter: ∂G ∈ Ck, if for each point x0 on ∂G there exists a ball B0 with center x0 such that ∂G ∩ B0 can be represented in the form xi = h(x1, . . . , xi−1, xi+1, . . . , xn) for some i, where h is k times continuously differentiable. Analogously, ∂G ∈ Ck,µ means that h ∈ Ck,µ. G is said to be Lipschitz if h ∈ C0,1.



Glossary



Mollification: Is basically the same as ”smoothing” or ”convolving with an averaging kernel”. We turn a non-smooth function f into a C∞ function by convolving it with a smooth kernel. By dilating the kernel and making it closer and closer to a Dirac delta and convolving it with f one gets back f in the limit. Newton’s Law: The resistance which arises from the lack of slipperiness of the parts of the liquid, other things being equal, is proportional to the velocity with which the parts of the liquid are separated from one another. Objective quantity: Frame indifferent. Olivine: (Mg,Fe)2SiO4 is a nesosilicate of orthorhombic symmetry. Best studied is forsterite (Fo), the magnesium end-member, to which natural olivine is close (90% magnesium). Under the pressure of the earth mantle olivine undergoes several phase transitions to denser structures. The main transition is to a spinel structure. Parabolic: See ↑ elliptic.

Prandtl number: Relative importance of heat versus momentum diffusion. Pseudoplastic: ↑ Shear thinning.

Radon measures: Dual space of C(Ω) for a bounded domain Ω. If Ω = Rd, then C0(Rd) := {u ∈ C(Rd) : lim u(x) = 0} and |x|→∞

d M(Rd) := {µ : C0(Rd) → R, linear, |µ(f)| ≤ ckfk∞ ∀ f ∈ C∞ 0 (R )}. d It is a ↑ Banach space with the norm kµkM(Rd ) := sup |µ(f)| (f ∈ C∞ 0 (R )). kfk∞ ≤1

Reflexive Banach space: Let X be a ↑ Banach space. There is a natural map J from X to (X∗ )∗ defined by J(x)(f) = f(x) for all x ∈ X and f ∈ X∗ . This map is always one-to-one. If its range is equal to (X∗ )∗ , then X is called reflexive. Reflexive spaces have many important geometric properties. A space is reflexive if and only if its dual is reflexive, which is the case if and only if its unit ball is compact in the weak topology. Reynolds number: Ratio of inertia forces to viscous forces. Rheology: Rheology is the science of flow and deformation of matter and describes the interrelation between force, deformation and time. The term comes from Greek rheos meaning “to flow”. (The Greek philosopher Heraclitus said once upon a time “Παντα ρι - everything flows”. Translated into rheological terms by Marcus Reiner this means everything will flow if you just wait long enough). Schauder Fixed Point Theorem: Every continuous self-map of a compact convex subset of a ↑ Banach space has a fixed point. Shear Rate: The velocity gradient measured across the diameter of a fluid-flow channel, be it a pipe, annulus or other shape. Shear rate is the rate of change of velocity at which one layer of fluid passes over an adjacent layer. Unit: s−1.

Shear Stress: Force per unit area required to sustain a constant rate of fluid movement. Shear thickening (dilatant): Whenever the ↑ shear stress or ↑ shear rate is altered, the fluid will gradually move towards its new equilibrium state and at lower ↑ shear rates the ↑ shear thickening fluid is less viscous than the Newtonian Fluid, and at higher ↑ shear rates it is

Glossary



more viscous. Fluids that become more viscous as a function of time are called rheopectic. This term is often applied when the term dilatant is meant. shear-thickening liquids: clay slurries, candy compounds, corn starch in water, and sandwater mixtures. Shear thinning (pseudoplastic): At lower ↑ shear rates the shear thinning fluid is more viscous than the Newtonian Fluid, and at higher ↑ shear rates it is less viscous. In contrast, but often confused with the above are fluids that become less viscous as a function of time. These types of fluids are called thixotropic, although this term is often confused with shear thinning. shear-thinning liquids: tomato puree, molten chocolate, drilling mud, oil paint, printing inks, blood, shampoo Simple material: The stress tensor is determined, to within a hydrostatic pressure, by the history of the deformation gradient. Simple shear flow: The sketch to the left depicts the velocity profile in a simple shear flow. The fluid is moving to the right and the magnitude of the fluid velocity increases linearly with y. The velocity can be written v1 = γy ,

v2 = v3 = 0 ,

where γ is the constant shear rate. Support of a function: supp u := {x : u(x) 6= 0} . Specific heat: The amount of heat per unit mass required to raise the temperature by one degree Celsius (also specific heat capacity). The heat capacity of a substance can differ depending on what extensive variables are held constant, with the quantity being held constant usually being denoted with a subscript. For example, the specific heat at constant pressure is commonly denoted cp, while the specific heat at constant volume is commonly denoted cV . True polar wander: True polar wander is caused by an imbalance in the mass distribution of the planet itself, which is forced by the laws of physics to equalise itself in comparatively rapid time scales. During this redistribution, the entire solid part of the planet moves together, avoiding the internal shearing effects that impose a speed limit on conventional plate motions. While this happens, the entire earth maintains the original spin axis in relation to the plane of the solar system. Thus, true polar wander can result in land masses moving at rates hundreds of times faster than tectonic motion caused by convection in earth’s mantle. Viscoelasticity: Having both viscous and elastic properties. Viscosity: Property of material to resist deformations increasingly with increasing rate of deformation. Also the measure of the resistance of a fluid to flow, i.e. of the internal friction between contiguous layers of fluid in motion relative to one another. Viscous flows are dissipative, i.e. they generate heat. Measurement of viscosity is carried out with a viscometer, which can also measure the changes in viscosity due to time and temperature, or due to shear-thinning or shear-thickening. These all relate to increases or decreases in viscosity when there is an increase in the ↑ shear rate.

Acknowledgement



Isaac Newton (1680) was the first to formulate a mathematical description of the resistance of a fluid to deform or flow when a stress is applied. The resistance was described as the viscosity, and mathematically defined as the shear stress divided by the rate of shear strain. From that time until Couette developed the first rotational viscometer (1890), viscosity was measured using stress driven (gravity) flow. Many techniques which use this principle, such as flow cups, u-tubes and capillaries are still very popular. Unit: Pa s. Vitali’s Lemma [84, 1.2.11]. Let fn : D → R be integrable ∀ n ∈ N. Assume that

. limn→∞ fn(y) exists and is finite for almost all y ∈ D, R . ∀ ε > 0 ∃ δ > 0 : supn∈N H |fn(y)| dy < ε ∀ H ⊂ D with |H| < δ. Z Z Then lim fn(y) dy = lim fn(y) dy . n→∞ D

D n→∞

Yield stress: The pressure which a substance is capable of supporting without fracturing.

Acknowledgement Some persons are especially close to the contents of my book namely, I will not miss to thank all co-authors of papers related to Boussinesq approximation for the fruitful collaboration: C. Ferrario (Ferrara, I), Y. Kagei (Fukuoka, JP), J. Málek (Praha, CZ), A. Passerini (Ferrara, I), and M. R˚uzˇ iˇcka (Freiburg i.Br.). The first step in this direction and at the same time the first step in my scientific carreer was the invitation to join a project of J. Málek and M. R˚uzˇ iˇcka - both working at SFB 256 in Bonn at that time - on a power-law version of the Boussinesq approximation. I did not only learn a lot about the appropriate tools but we had a lot of discussions about the reliability of approximations. This was made more precise through lectures of K.R. Rajagopal and a stay at the Department of Mechanical Engineering at the University of Pittsburgh (PE). Years later, Y. Kagei was giving a lecture at Paderborn university and we had some discussion finding out our common interest in generalisations of the Boussinesq approximation. I was about to change to Bonn and with M. R˚uzˇ iˇcka we started to think more seriously about a model Y. Kagei had treated. The outcome of that process are several common papers and my fascination for the topic leading to this book. And, obviously, Ohne Begeisterung schlummern die besten Kräfte unseres Gemütes. (i.e. Without enthusiasm the best powers of our mind lie dormant.) (Herder)

Tables



Notation b c0 cp e g k q π, p r U v D L L T ^ T

N m3 J kgK J kgK kJ kg m s2 W mK W m2 N m2 m s m s −1

s m s−1 N m2 N m2 −1

α η θ θt, θb ϑ Θ

K

κ µ

m2 s kg ms m2 s kg m3 kg m3

ν ρ ρr σ Φ

kJ K kg

K K K −

density of external body forces mean specific heat specific heat specific internal energy gravity constant thermal conductivity heat flux vector pressure radiant heating mean velocity velocity field stretching/strain: 21 (∇v + (∇v)> ) typical length velocity gradient: ∇v Cauchy stress tensor extra stress tensor volumetric expansion coeff. specific entropy temperature θ at the top/bottom θb − θt 1 (θb + θt) 2ϑ thermal diffusity viscosity kinematic viscosity, ν := µρ density reference density density of entropy production entropy flux

Conversions N bar J

= kgs2m = 105 mN2 = 105 Pa = Nm = 2, 778 10−4Wh

∂k ∂t div v ∇ρ θ˙

:= ∂x∂k , k ∈ {1, 2, 3} ∂ := ∂t := ∂1v1 + ∂2v2 + ∂3v3 := (∂1ρ , ∂2ρ , ∂3ρ )>

b˙ ∂ts ∆

:= ∂tb + (b · ∇)v := ∂t∂s := ∂11 + ∂22 + ∂33

v·w

:=

T·L

:=

Di Gr Pr Ra Re U

:= c10 gαLρr := ν12 gαϑL3 := νκ 1 := νκ gαϑL3 = Gr Pr := ν1 UL √ := gαϑL √ ⇒ Re = Gr ,

:= ∂tθ + v · ∇θ

3 P

k=1 3 P

vkwk TklLkl

k,l=1

Rules for derivatives div (ρv) = ∇ρ · v + ρ div v

Tables

 Physical properties of certain materials

material Forsterite∗ Iron Olivine+ Silicone Spinel+

Mg2SiO4 Fe (Fe,Mg)2SiO4 Si (Fe,Mg)2SiO4 Fe2SiO4

ρ

κ

α

[ cmg 3 ]

cm2 [ 10 2s ]

[ 1015 K ]

3.21 7.87 3.30 2.33 3.56 4.40

0.94/1.10 1.35

3.3

2.31

1.9

Data from [94] and [109]; ∗ κ at 1100 K, 30/50 kbar, + κ at 1100 K, 0.001 kbar. Fluid properties of water and air at θ = 15o C and p = 1 atm [46] ρ

cp

ν

[ cmg 3 ]

[ kgkJK ]

2 [ cms ]

water: 1 4.2 × 103 1.1 × 10−2 air: 1.2 × 10−3 103 0.145

α Pr −1 [K ] 1.5 × 10−4 8.1 3.5 × 10−3 0.72

Model parameters for the upper mantle: from L ϑ θt Di ρ0 g c0 κ ν α

[km] [K] [K] g [ cm 3] m [ s2 ] kJ [ kgK ] cm2 [ 102 s ] 2 [ ms ] [ 1015 K ]

[19, p.480] [26] [56] [94] [104] 700 1500 700 700 300 2500 700 1420 500 0.12 0.3 0.12 0.12 3.7 3.3 3.5 3.7 3.3 10 10 10 10 10 1.2 1 1 1.2 1 1.5 0.8 1.5 1.0 17 17 2 × 10 2 × 10 2 4 2 3

Examples for nondimensionalisation

[10], [19] [44], [108]∗ [44]∗ [46]+ [92] [112]∗ ∗

x v p θ t 3 2 L κν/(gαL ) L /κ 2 L κ/L =: U ρ0U ϑ L/U L √ ϑ L2/ν L αϑgL =: U ρ0U2 ν/U U νU/L ν/U2 L/2 2ν/L 4ρ0ν2/L2 νϑ/2/κ

velocity scaling according to molecular diffusivities/ + according to “free fall”