APRIORI ESTIMATES FOR FRACTIONAL DIFFUSION

APRIORI ESTIMATES FOR FRACTIONAL DIFFUSION EQUATION K. BURAZIN AND D. MITROVIC

α/2

Abstract. We derive L2 ([0, T ); Hloc (Rd )), α ∈ [1, 2), apriori estimate for solutions to the fractional or anomalous diffusion equation using a generalization of the Leibnitz rule for the fractional Laplacean. The equation models a wide range of physical phenomena and, in particular, it is a linearized variant of the fractional porous media equation. The apriori estimates can be further used to rate convergence of corresponding numerical schemes, in the control and optimization theory and for various non-linear fractional PDEs. We use them here to prove existence of solution to a Cauchy problem for the fractional porous media equation as well as a result concerning optimal control.

1. Introduction and Notation In the paper, we consider the following partial differential equation: ∂t u(t, x) = ∆α/2 (|u(t, x)|m u(t, x)) + f (t, x), (t, x) ∈ [0, T ) × Rd , T > 0, (1.1) where f ∈ L2 ([0, T ) × Rd ) ∩ L1 ([0, T ) × Rd ) and m ≥ 0. The operator ∆α/2 , α ∈ (1, 2) is a generalization of the Laplace operator ∆ and we define it here via the Fourier transform F = b: F(∆α/2 u)(ξ) = −|ξ|α F(u)(ξ) := −|ξ|α u ˆ(ξ), ξ ∈ Rd ,

(1.2)

where we extended the function u by zero out of Ω. This kind of equation is widely investigated recently both for m = 0 [3, 4] as well as m > 0 [5, 6, 7, 9] (in which case it is called the fractional porous media equation) and many others which we unfortunately cannot mention due to space limits. The equation has very important role in the theory of fractional calculus which is intensively developing in various directions [12, 13, 14, 15, 11] etc. We particularly address readers on [2] for thorough information on the equation. We shall prove the following apriori estimate for (1.1) with m = 0 which is essentially the main result of the paper (existence of solution for m > 0 then follows using the standard vanishing viscosity arguments).

Key words and phrases. fractional Laplacean, fractional diffusion, apriori estimates. Permanent address of Darko Mitrovic is University of Montenegro, Montenegro. 1

2

K. BURAZIN AND D. MITROVIC

Theorem 1.1. Let u be a weak solution to (1.1) with m = 0. There exists a constant C > 0 such that for any compactly supported φ ∈ Cc2 (Rd ) it holds ∫

∫

Rd

T

|u(T, x)| φ (x)dx + 2

∫

2

0

Rd

|∆α/4 (φ(x)u(t, x))|2 dx

(1.3)

d L1 (Rd ) + ≤ C∥φu∥L2 ([0,T )×Rd ) ∥uχsupp(φ) ∥L1 ([0,T );L2 (Rd )) ∥∆φ∥ ∥f φ∥L2 ([0,T )×Rd ) ∥uφ∥L2 ([0,T )×Rd ) +

d ∑

α∥φu∥H ϵ (Rd ) ∥u∆1/2 φ∥L2 ([0,T )×Rd ) ,

j=1

where ϵ = α − 1 and χsupp(φ) is the characteristic function of the supp(φ). Apriori estimates are fundamental in investigating partial differential equations. There are many similar results on this issue (see [16] and references therein) and here we add another contribution to this issue. General problem in the theory of fractional differential equations is first several different ways of defining fractional derivatives (see e.g. a classical book [11]) which implies that the same equation does not mean the same depending on the chosen fractional framework. Next and more serious problem is lack of the Leibnitz rule which makes gaining of the apriori estimates quite non-trivial. One can find numerous attempts to generalize the Leibnitz rule [1, 8, 10] and each of them is certainly useful but we cannot apply it on our situation. We thus derive a new variant of Leibnitz type rule for the fractional Laplacean and apply it to obtain the apriori estimates. We stress that the Leibnitz type rule given here significantly simplifies derivation of the estimates and, we believe, opens new ways for proving different kinds of existence theorems, optimization results, and rating of the numerical schemes. Some possible applications are briefly illustrated in subsections 2.2 and 2.3.

2. The main result The fractional derivatives very often cause significant technical problems since we do not have the classical Leibnitz rule. Here, we need to adapt the approach from [8] them in order to get appropriate estimates. We need the following lemma: Lemma 2.1. Let Ω ⊆ Rd be a domain in Rd . For any φ ∈ Cc2 (Ω) and u ∈ H α (Ω), α = 1 + ϵ ∈ (1, 2), and a constant C: ∥∆α/2 (uφ) − φ∆α/2 u +

d ∑

α∆ϵ/2 Rj (u∆1/2 φ)∥L2 (Ω)

j=1

(2.1)

d L1 (Rd ) ≤ C∥uχsupp(φ) ∥L2 (Ω) ∥∆φ∥ where Rj is the Riesz potential i.e. the Fourier multiplier operator with the symbol iξj |ξ| . Proof. In order to simplify the notation, we shall omit the characteristic function appearing on the right-hand side of (2.1). It appears there since we start from uφ = uχsupp(φ) φ. Now, we proceed with the proof. We have

APRIORI ESTIMATES FOR FRACTIONAL DIFFUSION

3

∫ F(∆α/2 (uφ)) = −|ξ|α F(uφ) = −|ξ|α F(u) ⋆ F(φ) = −

|ξ|α u ˆ(ξ − η)φ(η)dη ˆ η

∫

(2.2)

∫

=−

(|ξ|α − |ξ − η|α ) u ˆ(ξ − η)φ(η)dη ˆ − η

η

∫ =−

|ξ − η|α u ˆ(ξ − η)φ(η)dη ˆ

(|ξ| − |ξ − η| ) u ˆ(ξ − η)φ(η)dη ˆ + F(φ∆α/2 u). α

α

η

By the Taylor formula, we have for a constant C d ∑ ξj α α ϵ ˜ α−2 |η|2 , − |ξ − η| + α|ξ| ⟨ , ηj ⟩ ≤ C|ξ| |ξ| |ξ| j=1

(2.3)

where ξ˜ belongs to the ball centered at ξ with the radius |ξ − η|. From (2.2) and (2.3), we conclude d ∑ αF(∆ϵ/2 Rj (u∆1/2 φ)) F(∆α/2 (uφ)) − F(φ∆α/2 (u)) +

(2.4)

j=1

∫

˜ ϵ−1 |η|2 u |ξ| ˆ(ξ − η)φ(η)dη. ˆ

≤C η

Now, since ξ˜ is in the ball centered at ξ with the radius |ξ − η|, we have ˜ ϵ−1 ≤ |ξ|ϵ−1 + |ξ − η|ϵ−1 . |ξ|

(2.5)

We thus have ∫ ˜ ϵ−1 |η|2 u |ξ| ˆ(ξ − η)φ(η)dη ˆ (2.6) η ∫ ∫ ≤ |ξ|ϵ−1 |η|2 |ˆ u(ξ − η)| |φ(η)|dη ˆ + |ξ − η|ϵ−1 |η|2 |ˆ u(ξ − η)| |φ(η)|dη ˆ η

η

Let us consider L2 norm of the right-hand side in (2.6). We have for the first summand (we recall Tϵ−1 is the Riesz transform) ∫ ( ) −1 d ∥L2 (Rd ) (2.7) ∥ |ξ|ϵ−1 |η|2 |ˆ u(ξ − η)| |φ(η)|dη∥ ˆ (|ˆ u| ⋆ |∆φ|) L2 (Rd ) = ∥Tϵ−1 F η

d L2 (Rd ) ≤ ∥∆φ∥ d L1 (Rd ) ∥u∥L2 (Ω) , ≤ ∥|ˆ u| ⋆ |∆φ|∥ where we used the Young inequality for convolution and L2 -continuity of the Riesz transform. As for the second summand on the right-hand side of (2.6), we have ∫ ∥ |ξ − η|ϵ−1 |η|2 |ˆ u(ξ − η)| |φ(η)|dη∥ ˆ (2.8) L2 (Rd ) η

d L2 (Rd ) ≤ ∥u∥L2 (Ω) ∥∆φ∥ d L1 (Rd ) . = ∥Tϵ−1 (F −1 (|ˆ u| ⋆ |∆φ)|∥ □ Now, we can easily derive the apriori estimate from Theorem 1.1.

4


2.1. Proof of Theorem 1.1. Let us take an arbitrary φ ∈ Cc2 (Rd ) and multiply (1.1) by φ2 u. By integrating the obtained equation over [0, T ) × Rd , we get after adding and subtracting appropriate terms: ∫ T∫ φu∂t (φu)dxdt 0 Rd   ∫ T∫ d ∑ = φu φ∆α/2 u − ∆α/2 (φu) − α∆ϵ/2 Rj (u∆1/2 φ) dxdt Rd

0

∫

T

∫

∫

∫

T

φu ∆α/2 (φu) +

f φ2 udtdx +

+ Rd

0

j=1



Rd

0

d ∑

 α∆ϵ/2 Rj (u∆1/2 φ dxdt.

j=1

By using integration by parts, we get from the above: ∫

∫

T

Rd

0

∫

T

∫

∂t (φu)2 dxdt + 2 

= Rd

0

∫

T

0

Rd

T

∫ Rd

0

|∆α/2 (φu)|2 dxdt

φu φ∆α/2 u − ∆α/2 (φu) −

d ∑

 α∆ϵ/2 Rj (u∆1/2 φ) dxdt

j=1

∫

+

∫

∫

∫

T

f φ2 udtdx −

∫

Rd

0

|∆α/4 (uφ)|2 dxdt +

T

∫ φu

0

Rd

d ∑

Rj (u∆1/2 φ)dxdt.

j=1

Now, we can use Lemma 2.1 and L2 -continuity of the Riesz potential to get: ∫ Rd

φ(x)u(T, x) dx + 2

∫

T

∫ Rd

0

|∆α/4 (φu)|2 dxdt

≤ ∥φu∥L2 ([0,T )×Rd ) ∥φ∆α/2 u − ∆α/2 (φu) −

d ∑

α∆ϵ/2 Rj (u∆1/2 φ)∥L2 ([0,T )×Rd )

j=1

+ ∥f φ∥L2 ([0,T )×Rd ) ∥φu∥L2 ([0,T )×Rd ) + d L1 (Rd ) ≤ C∥φu∥L2 ([0,T )×Rd ) ∥∆φ∥

∫

d ∑

α∥φu∥L2 ([0,T );H ϵ (Rd )) ∥u∆1/2 φ∥L2 ([0,T )×Rd )

j=1 T

∥u(t, ·)∥L2 (Rd ) dt 0

+ ∥f φ∥L2 ([0,T )×Rd ) ∥φu∥L2 ([0,T )×Rd ) +

d ∑

α∥φu∥L2 ([0,T );H ϵ (Rd )) ∥u∆1/2 φ∥L2 ([0,T )×Rd ) .

j=1

This completes the proof. 2 α ϵ Using the Kondrachov embedding theorem (Hloc (Rd ) ⊂ Hloc (Rd ), ϵ < α), we conclude from Theorem 1.1: Corollary 2.2. It holds ∫ ∫ |u(T, x)|2 φ2 (x)dx + Rd

0

T

∫ Rd

|∆α/4 (φ(x)u(t, x))|2 dx ≤ C(φ)∥u∥L2 ([0,T )×supp(φ)) ,

where C(φ) depends on φ as well as its derivatives.


5

2.2. Application to the fractional porous media equation. We shall now show how to use the latter result in order to simplify existence results from [5]. More precisely, we consider the problem ∂t u = ∆α/2 (|u|m u) + f (t, x), (t, x) ∈ R+ × Rd ∞

u|t=0 = u0 (x) ∈ L (R ), d

(2.9) (2.10)

d where α ∈ [1, 2) and f ∈ Lm+2 loc ([0, T ) × R ). A similar problems have been considered in [5, 17] but on bounded domains. We can also confine ourselves on bounded domains since we know (by Lemma 2.1) how to commute a cutoff function with the fractional Laplacean. We shall prove the following theorem. α/2

Theorem 2.3. There exists a function u ∈ L2 ([0, T ); Hloc (Rd )) solving (2.9), (2.10). Proof. We start with the vanishing viscosity approximation to (2.9), (2.10): ∂t uϵ = ∆α/2 (|uϵ |m uϵ ) + f (t, x) + ϵ∆uϵ , (t, x) ∈ R+ × Rd ∞

uϵ |t=0 = u0 (x) ∈ L (R ), d

(2.11) (2.12)

1 which, for every fixed ϵ > 0 admits a unique L2 ([0, T ); Hloc (Rd )) solution (e.g. [18]). ′ m By noticing that for η (u) = |u| u the function η is convex (it is actually equal 1 to η(u) = m+2 |u|m+2 ), we get by multiplying (2.11) by φη ′ (uϵ ) for a non-negative compactly supported φ ∈ Cc2 (Rd ):

( ) ∂t (φη(uϵ )) = φη ′ (uϵ )∆α/2 (|uϵ |m uϵ ) + f (t, x)φη ′ (uϵ ) + ϵφ ∆η(uϵ ) − η ′′ (uϵ )|∇uϵ |2 . Now, we integrate the latter equation over [0, T ) × Rd and use convexity of the function η to get ∫

∫

Rd

Rd

∫ η(u0 (x))φ(x)dx +

0

∫ Rd

0

∫

≤

T

η(uϵ (T, x))φ(x)dx − T

∫

Rd

φη ′ (uϵ )∆α/2 (|uϵ |m uϵ )dxdt f (t, x)φη ′ (uϵ )dxdt + ϵ

∫ 0

T

(2.13) ∫ Rd

φ(x)∆η(uϵ (t, x))dxdt.

Next, notice that according to the Young inequality ∫ 0

T

∫ Rd

≤ ∥f φ

f (t, x)φ(x)|uϵ |m |uϵ |dtdx

m+2 m+1

(∫

T

(2.14)

∥Lm+2 ([0,T )×Rd ) 0

m+1 ) m+2

∫

∫ ( m+2 ≤ C ∥f φ m+1 ∥m+2 + m+2 d L ([0,T )×R )

Rd

0

T

φ(x)|uϵ |m+2 dxdt ∫ Rd

) φ(x)η(uϵ )dxdt .

Integrating (2.13) with respect to T we get after taking (2.14) into account:

6


(

)∫ T ∫ ∫ T∫ (m + 2)T η(uϵ )φdtdx − T φη ′ (uϵ )∆α/2 (|uϵ |m uϵ )dxdt (2.15) C 0 0 Rd Rd ∫ m+2 ≤T η(u0 (x))φ(x)dx + T ∥f φ m+1 ∥m+2 Lm+2 ([0,T )×Rd ) 1−

Rd

∫

T

∫

+ Tϵ Rd

0

φ(x)∆η(uϵ (t, x))dxdt.

Now, choosing C large enough and repeating the procedure from the proof of Theorem 1.1 and Corollary 2.2, we get ∫

∫ T∫ η(uϵ (t, x))φ(x)dxdt + |∆α/4 (|uϵ |m uϵ )|2 dxdt 0 Rd 0 Rd ) ( 2 2 ≤ C(φ) ∥u0 ∥2L2 (supp(φ)) + ∥uϵ ∥L 2 (supp(φ)) + ∥f ∥L2 ([0,T )×supp(φ)) ∫ T∫ +ϵ φ(x)∆η(uϵ (t, x))dxdt, T

∫

0

Rd

where C(φ) depends on φ as well as of its derivatives up to the second order. α/2 From here, we conclude that (|uϵ |m uϵ ) is bounded in L2 ([0, T ); Hloc (Rd )) from where, together with linearity with respect to t-derivative and the Aubin-Lions lemma, we conclude that (|uϵ |m uϵ ) is strongly precompact L2loc ([0, T )×Rd ). Clearly, this implies L2loc ([0, T ) × Rd )-convergence along a subsequence of (uϵ ) and the limiting function is the weak solution to (2.9), (2.10). □ 2.3. Optimal control. We shall now briefly illustrate how above result can be easily fitted in some standard frameworks of optimal control. We could choose working on bounded domains, however we shall present an optimal control problem where governing equation si given on the whole space, while control, as well as performance, can be influenced and measured only on some bounded subsets of the whole space. For simplicity, we shall restrict ourselves to the linear fractional diffusion equation, i. e. m = 0. To be precise, let K1 and K2 be compact subsets of Rd , and f ∈ L2 ((0, T ) × K2 ), z ∈ L2 ((0, T ) × K1 ). Their extensions by zero to the whole (0, T ) × Rd we shall denote the same. For given constant k > 0, and initial u0 ∈ L∞ (Rd ) we are interested in minimizing the cost functional J(f ) := ∥u − z∥2L2 ((0,T )×K1 ) + k∥f ∥2L2 ((0,T )×K2 ) , where u is the solution of ∂t u = ∆α/2 (u) + f (t, x), (t, x) ∈ (0, T ) × Rd u|t=0 = u0 .

(2.16) (2.17)

Clearly, f represent control which can be influenced only on part K2 of the domain, while z is desired state and we evaluate our state only on part K1 of the domain. Existence of unique minimizer f of functional J can be obtained by standard optimization techniques, similarly as in [14]. To be precise, the following theorem is valid.


7

Theorem 2.4. The functional J : L2 ((0, T ) × K2 ) −→ R is weakly (sequentially) lower semi-continuous and strictly convex, which implies that there exists unique f ∈ L2 ((0, T ) × K2 ) such that J(f ) = inf L2 ((0,T )×K2 ) J. Proof. The proof of weakly (sequentially) lower semi-continuity of J follows from the fact that operator f 7→ u(f ) that maps f to the corresponding solution u of (2.16)-(2.17) is weakly (sequentially) continuous L2loc ((0, T ) × Rd ) −→ L2loc ((0, T ) × Rd ). Indeed, if we combine this with weakly (sequentially) lower semi-continuity of mappings u 7→ ∥u−z∥2L2 ((0,T )×K1 ) from L2 ((0, T )×K1 ) to R, and f 7→ ∥f ∥2L2 ((0,T )×K2 ) from L2 ((0, T ) × K2 ) to R we easily get that J is weakly (sequentially) lower semicontinuous. Similarly, using strict convexity of above two mappings and the property u(λf + (1 − λ)g) = λu(f ) + (1 − λ)u(g), for λ ∈ [0, 1], which easily follows from linearity of equation (2.16), we conclude that J is strictly convex. □ 3. Acknowledgement This work was partially supported by project P30233 of the Austrian Science Fund FWF. References [1] T.J.Osler, Leibniz Rule for Fractional Derivatives Generalized and an Application to Infinite Series, SIAM J. Appl. Math., 18 (1970), 658-674. [2] L. R. Evangelista, E. K. Lenzi, Fractional Diffusion Equations and Anomalous Diffusion, Cambridge University Press, 2018. [3] K. Logvinova and M.-C. Nel, A fractional equation for anomalous diffusion in a randomly heterogeneous porous medium, Chaos 14, 982 (2004). [4] P.A. Santoro, J.L. de Paula, E.K.Lenzi, L.R.Evangelista, Anomalous diffusion governed by a fractional diffusion equation and the electrical response of an electrolytic cell, J Chem Phys. 135 114704 (2011). [5] A. de Pablo, F. Quirs, A. R. Juan, L. Vazquez, A fractional porous medium equation, Advances in Mathematics 226 (2011), 1378–1409. [6] F. del Teso, Finite difference method for a fractional porous medium equation, Calcolo 51 (2014), 615-638. [7] C.Imbert, Finite speed of propagation for a non-local porous medium equation, Colloquium Mathematicae 143 (2016), 149–157. [8] D.Mitrovic, On a Leibnitz type formula for fractional derivatives, Filomat 27 (2013), 1141– 1146. [9] D. Stan, F. del Teso, J.L. Va zquez, Finite and infinite speed of propagation for porous medium equations with nonlocal pressure, J. Differential Equations 260, 2 (2016), 1154-1199. [10] A. Alsaedi, B. Ahmad, M. Kirane, A survey of useful inequalities in fractional calculus, J. of Pseudo-differential calculus and application 20 (2017), 574–594. [11] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Academic Press, San Diego - New York - London, 1999. [12] M. Weiss, M. Elsner, F. Kartberg, T. Nilsson, Anomalous Subdiffusion Is a Measure for Cytoplasmic Crowding in Living Cells, Biophysical Journal (2004) 3518–3524. [13] H.l. Li, L. Zhang, C. Hu, Y.L. Jiang, Z. Teng, Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, Journal of Applied Mathematics and Computing 54 (2017), 435–449. [14] G. Mophou, Optimal control of fractional diffusion equation, Comp. Math. Appl. 61 (2011), 68–78. [15] F.A. Rihan, Numerical modeling of fractional-order biological systems, Abstract and Applied Analysis (Vol. 2013).

8


[16] M. Bonforte, J. L. Vzquez, A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains, Archive for Rational Mechanics and Analysis 218 (2015), 317–362. [17] G. Grillo, M. Muratori, F. Punzo, Fractional porous media equations: existence and uniqueness of weak solutions with measure data, Calculus of Variations and Partial Differential Equations 54 (2015), 3303–3335. [18] S. Cifani, E. R. Jakobsen, Entropy solution theory for fractional degenerate convectiondiffusion equations Annales de l’Institut Henri Poincare. Analyse non lin´ eare. 28 (2011), 413–441. Kreˇ simir Burazin, Department of Mathematics, University of Osijek, Trg Lj. Gaja 6, 31000 Osijek, Croatia E-mail address: [email protected] Darko Mitrovic, Faculty of Mathematics, University of Vienna, Oscar Morgenstern Platz 1, Austria E-mail address: [email protected]