An Lq(Lp)-theory for diffusion equations with space-time nonlocal operators☆
Introduction
Many types of diffusion equations have been used to describe diverse phenomena in various fields including mathematics, engineering, biology, hydrology, finance, and chemistry. The classical heat equation describes the heat propagation in homogeneous media. When , the equation describes the anomalous diffusion exhibiting subdiffusive behavior caused by particle sticking and trapping effects (e.g. [30], [31]). On the other hand, the spatial nonlocal operator describes long range jumps of particles, diffusions on fractal structures, and long time behavior of particles moving in space with quenched and disordered force field (e.g. [5], [13]).
The space-time fractional diffusion equation can be used to describe the combined phenomena, for instance, jump diffusions with a higher peak and heaver tails (see e.g. [10], [16]). The space-time fractional equation is also related to the scaling limit of continuous time random walk (see [11], [16], [28]).
In this article we study the space-time fractional equation Here, , is the Caputo fractional derivative of order α, and ϕ is a Bernstein function satisfying , that is, such that The operator is defined by For instance, if , , then becomes the fractional Laplacian. It turns out that is a type of integro-differential operator, and the class of is characterized by the infinitesimal generators of subordinate Brownian motions. See Section 2 for details.
Probabilistic representation of solution to equation (1.1) has been introduced e.g. in [7], [8], [10], [28]. Actually the transition density of subordinate Brownian motion delayed by an inverse subordinator becomes the fundamental solution, and this transition density and its appropriate time-fractional derivative appear in the solution representation. See Sections 4 and 5.
The main purpose of this article is to present a Sobolev-regularity theory of equation (1.1). We prove the uniqueness and existence in Sobolev spaces and obtain the maximal -regularity of solutions. In particular, we prove for any , where is a Besov space on related to ϕ. Moreover, we obtain the maximal regularity of higher order derivatives as well as negative order derivatives of solutions.
Our proof for (1.2) is mainly based on BMO-estimate if , and Littlewood-Paley theory is used to treat the case . Specifically speaking, we prove that if then Here denotes the sharp function of . The BMO estimate and the Marcinkiewicz interpolation theorem lead to (1.2) for , and the case is covered based on the vector-valued Calderón-Zygmund theorem. For the implement of these procedures, we rely on sharp upper bounds of arbitrary order space-time derivatives of the fundamental solution, which are obtained in Section 3. Due to the non-integrability of derivatives of the fundamental solution, our proofs of e.g. (1.3) are much more delicate than the proofs for PDEs with local operators. Condition (0.1) is a minimal assumption on ϕ such that our derivative estimates of the fundamental solution hold for all . This is essential in this article because we are aiming to prove estimates for solutions which are independent of the time intervals where the solutions are defined.
Here are some related -theories for the diffusion equations with either time fractional derivative or spatial integro-differential operators. An -theory for the time fractional equation was introduced in [6], [36] when . The result of [6], [36] is based on semigroup theory, and similar approach is used in [39] to treat the equation with uniformly continuous coefficients. Recently, the continuity condition of [39] is significantly relaxed in [12], [22]. For instance, if then [12] only requires that the coefficients are only measurable in t and have small mean oscillation in x. The approach in [12] is based on the level set arguments. Regarding the equations with spatial integro-differential operators, an -theory of the diffusion equation the type was introduced in [34]. Here χ is a certain indicator function and the jump kernel is of the type , where is homogeneous of order zero and sufficiently smooth in y. Recently, the condition on has been generalized and weaken e.g. in [20], [23], [32], [33], [40].
This article is organized as follows. In Section 2, we introduce some basic facts on the fractional calculus, integro-differential operator , and related function spaces. We also introduce our main result, Theorem 2.8, in Section 2. In Section 3 we obtain sharp upper bounds of space-time derivatives of the fundamental solution. In section 4 we study the zero initial data problem, and non-zero initial data problem is considered in Section 5. Finally we prove our main result in Section 6.
We finish the introduction with some notations. We use “:=” or “=:” to denote a definition. The symbol denotes the set of positive integers and . Also we use to denote the set of integers. As usual stands for the Euclidean space of points . We set For , multi-indices , and functions we set We also use the notation for arbitrary partial derivatives of order m with respect to x. For an open set in or , denotes the set of infinitely differentiable functions with compact support in . By we denote the class of Schwartz functions on . For , by we denote the set of complex-valued Lebesgue measurable functions u on satisfying Generally, for a given measure space , denotes the space of all F-valued -measurable functions u so that where denotes the completion of with respect to the measure μ. If there is no confusion for the given measure and σ-algebra, we usually omit the measure and the σ-algebra. We denote and . By and we denote the d-dimensional Fourier transform and the inverse Fourier transform respectively, i.e. For any , we write if there is a constant independent of such that . Finally if we write , this means that the constant N depends only on what are in the parentheses. The constant N can differ from line to line.
Section snippets
Main results
First we introduce some definitions and facts related to the fractional calculus. For and , the Riemann-Liouville fractional integral of the order α is defined as We also define . By Jensen's inequality, for , Using Fubini's theorem, one can easily check for any , Let , . If is -times differentiable and is absolutely continuous on
Estimates of the fundamental solution
In this section we obtain sharp bounds of arbitrary order derivatives of the fundamental solution to equation (2.17).
We first study the derivatives of the transition density of d-dimensional subordinate Brownian motion.
Lemma 3.1 Let Assumption 2.1 hold. Then there exists a constant such that
Proof Note first that Assumption 2.1 combined with the concavity of ϕ gives By the change of variables and (3.2),
Key estimates: BMO and -estimates
In this section we prove some a priori estimates for solutions to the equation with zero initial condition
We first present the representation formula.
Lemma 4.1 (i) Let and denote . Then (ii) Let and define u as in (4.2). Then u satisfies equation (4.1) for each .
Proof See [22, Lemma 3.5], which treats the case . The proof for the general case is same. The only difference is one needs to use formula
Homogeneous equation
In this section we study the homogeneous equation with non-zero initial condition
We first show that q is a fundamental solution to equation (5.1).
Lemma 5.1 Let , and define u as (i) As , converges to uniformly on and also in for any . (ii) and u satisfies for . Proof (i) By (3.24), for any , For any ,
Proof of Theorem 2.8
Due to Remark 2.4 and Lemma 2.7 (iii), we only need to prove case .
Step 1 (Uniqueness). Let be a solution to equation (2.17) with and . Then by Corollary 5.5, . Hence, by Lemma 2.7 (ii), there exists such that in . Due to Lemma 4.1, it also holds that where . Note as . Thus by
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The authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2020R1A2C1A01003354).