Existence, uniqueness and regularity of the solution of the time-fractional Fokker-Planck equation with general forcing

TL;DR

This paper establishes the existence, uniqueness, and regularity of solutions for a time-fractional Fokker-Planck equation with general forcing, extending previous work to more complex forcing functions and providing estimates useful for numerical analysis.

Contribution

The paper proves existence, uniqueness, and regularity of solutions for a time-fractional Fokker-Planck equation with general forcing functions, including classical solutions for certain initial data.

Findings

Existence and uniqueness of mild solutions under $L^2$ initial data.

Classical solutions obtained for $u_0 ext{ in } H^2 ext{ and } H_0^1$ when $1/2< ext{alpha}<1$.

Derivation of estimates for time derivatives of solutions.

Abstract

A time-fractional Fokker-Planck initial-boundary value problem is considered, with differential operator $u_t-\nabla\cdot(\partial_t^{1-\alpha}\kappa_\alpha\nabla u-\textbf{F}\partial_t^{1-\alpha}u)$, where $0<\alpha <1$. The forcing function $\textbf{F} = \textbf{F}(t,x)$, which is more difficult to analyse than the case $\textbf{F}=\textbf{F}(x)$ investigated previously by other authors. The spatial domain $\Omega \subset\mathbb{R}^d$, where $d\ge 1$, has a smooth boundary. Existence, uniqueness and regularity of a mild solution $u$ is proved under the hypothesis that the initial data $u_0$ lies in $L^2(\Omega)$. For $1/2<\alpha<1$ and $u_0\in H^2(\Omega)\cap H_0^1(\Omega)$, it is shown that $u$ becomes a classical solution of the problem. Estimates of time derivatives of the classical solution are derived---these are known to be needed in numerical analyses of this problem.

Equations

$$u_t(t,x)-\nabla \cdot ({\partial }_t^{1-\alpha }{\kappa }_{\alpha }\nabla u-\text{F}{\partial }_t^{1-\alpha }u)(t,x)$$

$$u(0,x)$$

$$u(t,x)$$

$$J^{\beta }u={\int }_0^t{\omega }_{\beta }(t-s)u(s)ds\text{ where }{\omega }_{\beta }(t):=\frac{t^{\beta -1}}{\Gamma (\beta )}\text{ for }\beta >0.$$

$$J^{1-\alpha }(u_t)+Au=h(t,u,\nabla u,g,\text{F}),$$

$$\Vert \text{F}{\Vert }_{\infty }:=\underset{1\le i\le d}{\max }\underset{(t,x)\in [0,T]\times \overline{\Omega }}{\max }|F_i(t,x)|\text{and}\Vert \text{F}{\Vert }_{1,\infty }:=\Vert \text{F}{\Vert }_{\infty }+\underset{(x,t)\in [0,T]\times \overline{\Omega }}{\max }|\nabla \cdot \text{F}(t,x)|.$$

$$\Vert v{\Vert }_{C([0,T];X)}:=\underset{0\le t\le T}{\max }\Vert v(t){\Vert }_X.$$

$$\|v\|_{L^{p}(0,T;X)}:=\begin{cases}\bigl{(}\int_{0}^{T}\|v(t)\|_{X}^{p}\,dt\bigr{)}^{1/p}<\infty&\text{when $1\leq p<\infty$,}\\ \operatorname{ess\,sup}_{0\leq t\leq T}\,\|v(t)\|_{X}<\infty&\text{when $p=\infty$.}\end{cases}$$

$$\Vert v{\Vert }_{W^{1,p}(0,T;X)}:=\Vert v{\Vert }_{L^p(0,T;X)}+\Vert v^{\prime }{\Vert }_{L^p(0,T;X)}\text{ is finite}.$$

$$\Vert v{\Vert }_{L_{\alpha }^2(0,T;X)}:=\underset{0\le t\le T}{\max }{\left[J^{\alpha }(\Vert v{\Vert }_X^2)(t)\right]}^{1/2}=\underset{0\le t\le T}{\max }{\left[\frac{1}{\Gamma (\alpha )}{\int }_{s=0}^t(t-s{)}^{\alpha -1}\Vert v(s){\Vert }_X^{2}ds\right]}^{1/2}.$$

$$\Vert v{\Vert }_{L_{\alpha }^2}=\Vert v{\Vert }_{L_{\alpha }^2(0,T;L^2(\Omega ))}\text{and}\Vert v{\Vert }_{L^2}=\Vert v{\Vert }_{L^2(0,T;L^2(\Omega ))}.$$

$$E_{\alpha }(z):=\sum _{k=0}^{\infty }\frac{z^k}{\Gamma (k\alpha +1)},$$

$$G(t):=u_0+{\int }_0^{t}g(s)ds\text{ for }0\le t\le T.$$

$$u-\kappa_{\alpha}\Delta(J^{\alpha}u)+\nabla\cdot(\mathbf{F}J^{\alpha}u)-\nabla\cdot\biggl{(}\int_{0}^{t}\mathbf{F}^{\prime}(s)J^{\alpha}u(s)\,ds\biggr{)}=G(t)\ \text{ a.e. on }(0,T)\times\Omega.$$

$$u^{\prime }\in L^2(0,T;L^2(\Omega ))\text{and}{\partial }_t^{1-\alpha }u\in L^2(0,T;H^2(\Omega )),$$

$$0\le y(t)\le a(t)+b(t){\int }_0^t{\omega }_{\beta }(t-s)y(s)ds\text{for 0≤t≤T,}$$

$$y(t)\leq a(t)E_{\beta}\bigl{(}b(t)t^{\beta}\bigr{)}\quad\text{for $0\leq t\leq T$.}$$

$$J^{\alpha }⟨J^{\alpha }v,v⟩(t)\ge \frac{1}{2}\Vert J^{\alpha }v(t){\Vert }^2\text{and}{\int }_0^t⟨J^{\alpha }v,v⟩(s)ds\ge \frac{1}{2}{J}^{1-\alpha }(\Vert J^{\alpha }v{\Vert }^2)(t).$$

$${\int }_0^t⟨J^{\alpha }v,v⟩(s)ds\ge \cos (\alpha \pi /2){\int }_0^t\Vert J^{\alpha /2}v{\Vert }^2(s)ds.$$

$$\Vert J^{\beta }\varphi (t){\Vert }^2\le {\omega }_{\beta +1}(t)J^{\beta }(\Vert \varphi {\Vert }^2)(t)\text{for 0≤t≤T.}$$

$$\Vert J^{\beta }\varphi (t){\Vert }^2$$

$$={\omega }_{\beta +1}(t){\int }_0^t{\omega }_{\beta }(t-s){\int }_{\Omega }{\varphi }^2(x,s)dxds={\omega }_{\beta +1}(t)J^{\beta }(\Vert \varphi {\Vert }^2)(t).$$

$$\Vert J^{\beta }\varphi (t)\Vert \le \frac{t^{\beta }}{\Gamma (\beta +1)}\Vert \varphi {\Vert }_{L^{\infty }(0,t;L^2)},\text{for all }\varphi \in L^{\infty }(0,t;L^2).$$

$$\Vert J^{\beta }\varphi (t)\Vert \le \frac{t^{\beta -1/2}}{\Gamma (\beta )\sqrt{2\beta -1}}\Vert \varphi {\Vert }_{L^2(0,t;L^2)},\text{for all }\varphi \in L^2(0,t;L^2).$$

$$\Vert J^{\beta }\varphi (t)\Vert$$

$$\le \Vert \varphi {\Vert }_{L^{\infty }(0,t;L^2)}{\int }_0^t{\omega }_{\beta }(t-s)ds=\frac{t^{\beta }}{\Gamma (\beta +1)}\Vert \varphi {\Vert }_{L^{\infty }(0,t;L^2)}.$$

$$\Vert J^{\beta }\varphi (t)\Vert \le {\int }_0^t{\omega }_{\beta }(t-s)\Vert \varphi (s)\Vert ds\le {\left({\int }_0^t{\omega }_{\beta }^2(t-s)ds\right)}^{1/2}\Vert \varphi {\Vert }_{L^2(0,t;L^2)}=\frac{t^{\beta -1/2}}{\sqrt{2\beta -1}\Gamma (\beta )}\Vert \varphi {\Vert }_{L^2(0,t;L^2)}$$

$${\int }_0^t⟨{\partial }_s^{1-\alpha }v,v⟩ds\ge {\rho }_{\alpha }{t}^{\alpha -1}{\int }_0^t\Vert v(s){\Vert }^{2}ds\text{where}{\rho }_{\alpha }={\pi }^{1-\alpha }\frac{(1-\alpha {)}^{1-\alpha }}{(2-\alpha {)}^{2-\alpha }}\sin (\frac{1}{2}\pi \alpha ).$$

$$\bigl{\|}\nabla\cdot\bigl{(}\textbf{F}(t)\phi(t)\bigr{)}\bigr{\|}\leq\|\textbf{F}\|_{1,\infty}\|\phi(t)\|_{H^{1}(\Omega)}\quad\text{for $0\leq t\leq T$.}$$

$$\displaystyle\bigl{\|}\nabla\cdot\bigl{(}\textbf{F}(t)\phi(t)\bigr{)}\bigr{\|}^{2}$$

$$\displaystyle\leq\Bigl{(}\|\nabla\cdot\textbf{F}(t)\|_{L^{\infty}(\Omega)}+\|\textbf{F}(t)\|_{L^{\infty}(\Omega,\mathbb{R}^{d})}\Bigr{)}^{2}\|\phi(t)\|_{H^{1}(\Omega)}^{2}$$

$$\displaystyle\leq\Bigl{(}\|\textbf{F}\|_{1,\infty}\|\phi(t)\|_{H^{1}(\Omega)}\Bigr{)}^{2},$$

$$E_{p,r}:=\bigl{\{}v\in L^{p}((0,T);B_{0}):\partial_{t}v\in L^{r}((0,T);B_{2})\bigr{\}}$$

Full text

Keywords: Fokker-Planck equation, Riemann–Liouville derivative, time-fractional, regularity of solution.

MSC 2010 Classification: 35R11.

1 Introduction

In this paper, we study the existence, uniqueness and regularity of solutions to the following inhomogeneous, time-fractional Fokker–Planck initial-boundary value problem:

[TABLE]

where $\kappa_{\alpha}>0$ is constant and $\Omega$ is an open bounded domain with $C^{2}$ boundary in $\mathbb{R}^{d}$ for some $d\geq 1$. In (1a), one has $0<\alpha<1$ and $\partial_{t}^{1-\alpha}$ is the standard Riemann–Liouville fractional derivative operator defined by $\partial_{t}^{1-\alpha}u=(J^{\alpha}u)_{t}$, where $J^{\beta}$ denotes the Riemann–Liouville fractional integral operator of order $\beta$, viz.,

[TABLE]

Regularity hypotheses on $\textbf{F},g$ and $u_{0}$ will be imposed later.

The problem (1) was considered in [5, 6, 14]. We describe it as “general forcing” since $\textbf{F}=\textbf{F}(t,x)$; this is a more difficult problem than the special case where $\textbf{F}=\textbf{F}(x)$, which can be reduced to a problem already studied by several authors (see, e.g., [3, 7, 8, 12, 15]). More precisely, when the force F may depend on $t$ as well as $x$, equation (1) cannot be rewritten in the form of the fractional evolution equation

[TABLE]

in which the first term is a Caputo fractional derivative, the operator $A=-\kappa_{\alpha}\Delta$, and the function $h$ does not depend explicitly on $\partial_{t}^{1-\alpha}u$.

The regularity of the solution to the Cauchy problem for (2) was studied in [3]; there a fundamental solution of that problem was constructed and investigated for a more general evolution equation where the operator $A$ in (2) is a uniformly elliptic operator with variable coefficients that acts on the spatial variables. The Cauchy problem was also considered in [12] where $h=h(t,u,g,\textbf{F})$ lies in a space of weighted Hölder continuous functions, and in [15] for the case where $A$ is almost sectorial. Existence and uniqueness of a solution to the initial-boundary value problem where (1a) is replaced by (2) is shown in [7, 8].

To the best of our knowledge, the well-posedness and regularity properties of solutions to (1) are open questions at present, apart from a recent preprint [11] which treats a wider class of problems that includes (1) as a special case. The analysis in [11] proceeds along broadly similar lines to here—relying on Galerkin approximation, a fractional Gronwall inequality and compactness arguments—but employs a different sequence of a priori estimates and does not make use of the weighted $L^{2}$-norm of Definition 2.2 or the Aubin–Lions–Simon lemma (Lemma 3.8). An interesting consequence of the approach taken here is that the constants in our estimates remain bounded as $\alpha\to 1$, which one expects since in this limit (1) becomes the classical Fokker–Planck equation. However, the estimates in sections 6 and 7 are valid only for $1/2<\alpha<1$, with constants that blow up as $\alpha\to 1/2$ (cf. the comment following Assumption 6.1). By contrast, the results in [11] hold for the full range of values $0<\alpha<1$, but with constants that blow up as $\alpha\to 1$. Also, the analysis is significantly longer than the one presented here.

The main contributions of our work are:

• •

A proof in Theorem 5.3 of existence and uniqueness of the mild solution of (1) for the case $\alpha\in(0,1)$ and $u_{0}\in L^{2}(\Omega)$;

• •

By imposing a further condition on $u_{0}$ and restricting $\alpha$ to lie in $(1/2,1)$, the mild solution becomes the classical solution of (1) described in Theorem 6.7;

• •

Estimates of time derivatives of the classical solution in Theorem 7.3.

The paper is organized as follows. Section 2 introduces our basic notation and the definitions of mild and classical solutions of (1). Various technical properties of fractional integral operators that will be used in our analysis are provided in Section 3. In Section 4, we introduce the Galerkin approximation of the solution of (1) and prove existence and uniqueness of approximate solutions. Properties of the mild and classical solutions are derived in Sections 5 and 6, respectively. Finally, in Section 7, we provide estimates of the time derivatives of the classical solution in $L^{2}(\Omega)$ and $H^{2}(\Omega)$, needed for the error analysis of numerical methods for solving (1); see, e.g., [5, 6, 14].

2 Notation and definitions

Throughout the paper, we often suppress the spatial variables and write $v$ or $v(t)$ instead of $v(t,\cdot)$ for various functions $v$. We also use the notation $v^{\prime}$ for the time derivative. Let $\|\cdot\|$ denote the $L^{2}(\Omega)$ norm defined by $\|v\|^{2}=\langle v,v\rangle$, where $\langle\cdot,\cdot\rangle$ is the $L^{2}(\Omega)$ inner product. Let $\|\cdot\|_{H^{r}(\Omega)}$ and $|\cdot|_{H^{r}(\Omega)}$ be the standard Sobolev norm and seminorm on the Hilbert space of functions whose $r$th-order derivatives lie in $L^{2}(\Omega)$. We borrow some standard notation from parabolic partial differential equations, e.g., $C([0,T];L^{2}(\Omega))$.

Assume throughout the paper that the forcing function $\textbf{F}=(F_{1},\dots,F_{d})^{T}\in W^{1,\infty}((0,T)\times\Omega)$ and that its divergence $\nabla\cdot\textbf{F}$ is continuous on $[0,T]\times\overline{\Omega}$. Then F is continuous on $[0,T]\times\overline{\Omega}$ and we set

[TABLE]

Stronger assumptions on the regularity of F will be made in some sections.

We use $C$ to denote a constant that depends on the data $\Omega,\kappa_{\alpha},\textbf{F}$ and $T$ of the problem (1) but is independent of any dimension of finite-dimensional spaces to be used in our Galerkin approximations. Here the unsubscripted constants $C$ are generic and can take different values in different places throughout the paper.

We now recall the definitions of some Banach spaces from [4, p.301]:

Definition 2.1**.**

Let $X$ be a real Banach space with norm $\|\cdot\|_{X}$. The space $C([0,T];X)$ comprises all continuous functions $v:[0,T]\to X$ with

[TABLE]

Let $p\in[1,\infty]$. The space $L^{p}(0,T;X)$ comprises all measurable functions $v:[0,T]\to X$ for which

[TABLE]

The space $W^{1,p}(0,T;X))$ comprises all measurable functions $v:[0,T]\to X$ for which

[TABLE]

Recall that $0<\alpha<1$.

Definition 2.2**.**

Given a Banach space $X$ with a norm (or seminorm) $\|\cdot\|_{X}$, define $L^{2}_{\alpha}(0,T;X)$ to be the space of functions $v:[0,T]\to X$ for which the following norm (or seminorm) is finite:

[TABLE]

For any Banach space $X$, clearly $L^{2}_{\alpha}(0,T;X)\subset L^{2}(0,T;X)$ for $0<\alpha<1$ and $\|\cdot\|_{L^{2}_{1}(0,T;X)}=\|\cdot\|_{L^{2}(0,T;X)}$ if we formally put $\alpha=1$ in Definition 2.2. For brevity, when $X=L^{2}(\Omega)$ we write

[TABLE]

The Mittag-Leffler function $E_{\alpha}(z)$ that is used in the fractional Gronwall inequality of Lemma 3.1 is defined by

[TABLE]

for $z\in\mathbb{R}$. Its properties can be found in, e.g., [2].

We now introduce the definitions of mild solutions and classical solutions to problem (1). Set

[TABLE]

Definition 2.3** (Mild solutions).**

A mild solution of problem (1) is a function $u\in L^{2}(0,T;L^{2}(\Omega))$ such that J^{\alpha}u\in L^{2}\bigl{(}0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\bigr{)} and $u$ satisfies

[TABLE]

Definition 2.4** (Classical solutions).**

A classical solution of problem (1) is a function $u$ belonging to the space $C([0,T];L^{2}(\Omega))\cap L^{\infty}(0,T;H^{1}_{0}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega))$ such that

[TABLE]

with $u$ satisfying (1a) a.e. on $(0,T)\times\Omega$, and (1b) a.e. on $\Omega$.

3 Technical preliminaries

This section provides some properties of fractional integrals that will be needed in our analysis.

Lemma 3.1**.**

[16, Corollary 2]** Let $\beta>0$. Assume that $a$ and $b$ are non-negative and non-decreasing functions on the interval $[0,T]$, with $a\in L^{1}(0,T)$ and $b\in C[0,T]$. If $y\in L^{1}(0,T)$ satisfies

[TABLE]

then

[TABLE]

The following lemmas will be used several times in in our analysis.

Lemma 3.2**.**

[5, Lemma 2.2]** If $\alpha\in(1/2,1)$ and $v(x,\cdot)\in L^{2}(0,T)$ for each $x\in\Omega$, then for $t\in[0,T]$,

[TABLE]

Lemma 3.3**.**

[13, Lemma 3.1 (ii)]** If $\alpha\in(0,1)$ and $v(x,\cdot)\in L^{2}(0,T)$ for each $x\in\Omega$, then for $t\in[0,T]$,

[TABLE]

Lemma 3.4**.**

[5, Lemma 2.1]** Let $\beta\in(0,1)$. If $\phi(\cdot,t)\in L^{2}(\Omega)$ for $t\in[0,T]$, then

[TABLE]

Proof.

As the proof is short, we give it here for completeness. The Cauchy–Schwarz inequality yields

[TABLE]

∎

Lemma 3.5**.**

For any $t>0$ and $\beta>0$,

[TABLE]

If $\beta>1/2$, then

[TABLE]

Proof.

Minkowski’s integral inequality gives

[TABLE]

To prove the second inequality, apply Hölder’s inequality to (4) to obtain

[TABLE]

for any $\beta>1/2$, which completes the proof of this lemma. ∎

Lemma 3.6**.**

[10, Theorem A.1]** For $t>0$,

[TABLE]

The following estimate involving the force F is used several times in our analysis.

Lemma 3.7**.**

If $\phi:[0,T]\to H^{1}(\Omega)$, then

[TABLE]

Proof.

The vector field identity $\nabla\cdot(\textbf{F}\phi)=(\nabla\cdot\textbf{F})\phi+\textbf{F}\cdot(\nabla\phi)$ implies that

[TABLE]

which gives the desired estimate. ∎

We now recall a fundamental compactness result that will be used several times in the proofs of our main results.

Lemma 3.8** (Aubin–Lions–Simon).**

Let $B_{0}\subset B_{1}\subset B_{2}$ be three Banach spaces. Assume that the embedding of $B_{1}$ in $B_{2}$ is continuous and that the embedding of $B_{0}$ in $B_{1}$ is compact. Let $p$ and $r$ satisfy $1\leq p,r\leq+\infty$. For $T>0$, define the Banach space

[TABLE]

with norm

[TABLE]

Then,

• •

the embedding $E_{p,r}\subset L^{p}((0,T),B_{1})$ is compact when $p<+\infty$, and

• •

the embedding $E_{p,r}\subset C([0,T],B_{1})$ is compact when $p=+\infty$ and $r>1$.

Proof.

See, e.g., [1, Theorem II.5.16]. ∎

4 Galerkin approximation of the solution

In this section we prove existence and uniqueness of a finite-dimensional Galerkin approximation of the solution of (1). This is a standard classical tool for deriving existence and regularity results for parabolic initial-boundary value problems; see, e.g., [4, Section 7.1.2].

Let $\{w_{k}\}_{k=1}^{\infty}$ be a complete set of eigenfunctions for the operator $-\Delta$ in $H_{0}^{1}(\Omega)$, with $\{w_{k}\}$ an orthonormal basis of $L^{2}(\Omega)$ and an orthogonal basis of $H_{0}^{1}(\Omega)$; see [4, Section 6.5.1]. For each positive integer $m$, set $W_{m}=\operatorname{span}\{w_{1},w_{2},\ldots,w_{m}\}$ and consider $u_{m}:[0,T]\to W_{m}$ given by

[TABLE]

Let $\Pi_{m}$ be the orthogonal projector from $L^{2}(\Omega)$ onto $W_{m}$ defined by: for each $v\in L^{2}(\Omega)$, one has

[TABLE]

The projections of the source term and initial data are denoted by

[TABLE]

We aim to choose the functions $d_{m}^{k}$ so that for $k=1$, $2$, …, $m$ and $t\in(0,T]$ one has

[TABLE]

Existence and uniqueness of a solution to (5) are guaranteed by the following lemma.

Lemma 4.1**.**

[6, Theorem 3.1]** Let $F\in W^{1,\infty}(0,T;L^{\infty}(\Omega))$ and $g\in L^{1}(0,T;L^{2}(\Omega))$. Then for each positive integer $m$, the system of equations (5) has a solution $\{d_{m}^{k}\}_{k=1}^{m}$ with $u_{m}:[0,T]\to H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$ absolutely continuous. This solution is unique among the space of absolutely continuous functions mapping $[0,T]$ to $H_{0}^{1}(\Omega)$.

Proof.

Our argument is based mainly on the proof of [6, Theorem 3.1], but we fill a gap in that argument by verifying that $u_{m}$ is absolutely continuous. Define the linear operator $B_{m}(t):W_{m}\to W_{m}$ by

[TABLE]

and rewrite (5a) as

[TABLE]

Formally integrating this equation in time we obtain the Volterra integral equation [6, p.1768]:

[TABLE]

where

[TABLE]

It is shown in [6] that (6) has a unique solution u_{m}\in C\bigl{(}[0,T];H_{0}^{1}(\Omega)\bigr{)}.

Now, g\in L^{1}\bigl{(}0,T;L^{2}(\Omega)\bigr{)} implies that g_{m}\in L^{1}\bigl{(}0,T;L^{2}(\Omega)\bigr{)}, and it follows that $G_{m}:[0,T]\to L^{2}(\Omega)$ is absolutely continuous. Furthermore, Theorem 2.5 of [2] implies (using the continuity of $u_{m}$) that $t\mapsto\int_{s=0}^{t}K_{m}(t,s)u_{m}(s)\,ds$ is absolutely continuous. Hence, (6) shows that $u_{m}:[0,T]\to L^{2}(\Omega)$ is absolutely continuous.

We are now able to differentiate (6) (to differentiate the integral term, imitate the calculation in the proof of [2, Lemma 2.12]), obtaining

[TABLE]

The absolute continuity of $u_{m}(t)$ implies that $\partial_{t}^{1-\alpha}u_{m}(t)$ exists for almost all $t\in[0,T]$ by [2, Lemma 2.12]. Hence from the above equation, $u_{m}$ satisfies (5a). From (6), one sees immediately that $u_{m}$ satisfies (5b), so we have demonstrated the existence of a solution to (5).

To see that this solution of (5) is unique among the space of absolutely continuous functions, one can use the proof of [6, Theorem 3.1] since the absolute continuity of the solution is now known a priori. ∎

5 Existence and uniqueness of the mild solution

In this section, we assume that $\alpha\in(0,1)$, $\textbf{F}\in W^{1,\infty}((0,T)\times\Omega)$ and that the initial data $u_{0}\in L^{2}(\Omega)$.

5.1 A priori estimates

In order to prove a priori estimates, we consider the integrated form of equation (5a):

[TABLE]

where $G_{m}(t)=\Pi_{m}G(t)$ as in (6).

Let $C_{\mathrm{P}}$ denote the Poincaré constant for $\Omega$, viz., $\|v\|^{2}\leq C_{\mathrm{P}}\|\nabla v\|^{2}$ for $v\in H^{1}_{0}(\Omega)$.

Lemma 5.1**.**

Let $m$ be a positive integer. Let $u_{m}(t)$ be the absolutely continuous solution of (5a) that is guaranteed by Lemma 4.1. Then for any $t\in[0,T]$ one has

[TABLE]

where

[TABLE]

Proof.

Taking the inner product of both sides of (7) with $J^{\alpha}u_{m}(t)\in W_{m}$ then integrating by parts with respect to $x$, we obtain

[TABLE]

Integrating by parts with respect to the time variable, and using Minkowski’s integral inequality and Hölder’s inequality, we have

[TABLE]

It follows from (5.1) and (5.1) that

[TABLE]

Integrating in time and invoking Lemma 3.3, we deduce that

[TABLE]

But Lemma 3.4 gives us

[TABLE]

Thus, setting $\psi_{m}(t):=J^{1}(\|J^{\alpha/2}u_{m}\|^{2})(t)$, we deduce from (5.1) that

[TABLE]

Applying Lemma 3.1, one obtains

[TABLE]

This inequality and (13) together yield

[TABLE]

Now (8) follows immediately on recalling (5.1)–(14) and the Poincaré inequality.

In a similar fashion, we take the inner product of both sides of (7) with $u_{m}(t)\in W_{m}$ and then integrate by parts with respect to $x$, to obtain

[TABLE]

Using Lemma 3.7 and the same arguments as in the proof of (5.1), we also have

[TABLE]

This estimate and (15) together imply

[TABLE]

Integrating in time, we get

[TABLE]

Now apply the inequality (8) to complete the proof. ∎

Lemma 5.2**.**

Let $m$ be a positive integer, and let $u_{m}(t)$ be the absolutely continuous solution of (5a) that is guaranteed by Lemma 4.1. Then, for any $t\in[0,T]$,

[TABLE]

where

[TABLE]

Proof.

Taking the inner product of both sides of (7) with $-J^{\alpha}\Delta u_{m}(t)\in W_{m}$ and then integrating by parts with respect to $x$, we obtain

[TABLE]

This inequality and (16) together imply

[TABLE]

Integrating in time and invoking Lemma 3.3, we deduce that

[TABLE]

which, after applying inequality (8) of Lemma 5.1, completes the proof of (17).

Applying (4) with $\phi=J^{\alpha/2}u_{m}$ and $\beta=1-\alpha/2$ gives

[TABLE]

where $z(t)=\|J^{\alpha/2}u_{m}(t)\|_{H^{1}(\Omega)}$. Using Young’s convolution inequality we get

[TABLE]

The inequality (18) now follows immediately from (17). ∎

5.2 The mild solution

Our assumption that $\Omega$ has a $C^{2}$ boundary ensures that if $v\in H^{1}_{0}(\Omega)$ satisfies $\Delta v\in L^{2}(\Omega)$, then $v\in H^{2}(\Omega)$. Moreover, there is a regularity constant $C_{\mathrm{R}}$, depending only on $\Omega$, such that

[TABLE]

Our next result requires a strengthening of the regularity hypothesis on F.

Theorem 5.3**.**

Assume that $u_{0}\in L^{2}(\Omega),\textbf{F}\in W^{2,\infty}((0,T)\times\Omega)$ and g\in L^{2}\bigl{(}0,T;L^{2}(\Omega)\bigr{)}. Then there exists a unique mild solution $u$ of (1) (in the sense of Definition 2.3) such that

[TABLE]

Proof.

In order to prove the existence of a mild solution, we first prove the convergence of the approximate solutions $u_{m}$, and then find the limit of equation (7) as $m$ tends to infinity.

Note first that $\|G_{m}(s)\|\leq\|G(s)\|$ because

[TABLE]

Hence Lemma 5.1 shows that the sequence $\{\partial_{t}(J^{1}u_{m})\}_{m=1}^{\infty}=\{u_{m}\}_{m=1}^{\infty}$ is bounded in L^{2}\bigl{(}0,T;L^{2}(\Omega)\bigr{)}, and Lemma 5.2 shows that the sequence $\{J^{1}u_{m}\}_{m=1}^{\infty}$ is bounded in L^{\infty}\bigl{(}0,T;H^{1}_{0}(\Omega)\bigr{)}. Applying Lemma 3.8 with $B_{0}=H^{1}_{0}(\Omega)$, $B_{1}=B_{2}=L^{2}(\Omega)$, $p=+\infty$ and $r=2$, it follows that there exists a subsequence of $\{J^{1}u_{m}\}_{m=1}^{\infty}$, again denoted by $\{J^{1}u_{m}\}_{m=1}^{\infty}$, and a v\in C\bigl{(}[0,T];L^{2}(\Omega)\bigr{)}, such that

[TABLE]

Furthermore, from the above bounds on $\{J^{1}u_{m}\}_{m=1}^{\infty}$ and well-known results [1, Theorem II.2.7] for weak and weak-$\star$ compactness, by choosing sub-subsequences we get

[TABLE]

By letting u:=\partial_{t}v\in L^{2}\bigl{(}0,T;L^{2}(\Omega)\bigr{)}, we have $v=J^{1}u$. It remains to prove that $J^{\alpha}u_{m}$ converges weakly to $J^{\alpha}u$ in L^{2}\bigl{(}0,T;H^{2}(\Omega)\bigr{)}. Applying Lemma 3.5 with $\phi=J^{1}u_{m}$ and $\beta=\alpha$, for any $t\in[0,T]$ we deduce that

[TABLE]

This inequality, together with Lemma 5.2, implies that the sequence $\{J^{1+\alpha}u_{m}\}_{m=1}^{\infty}$ is bounded in L^{\infty}\bigl{(}0,T;H^{1}_{0}(\Omega)\bigr{)}. Also, Lemma 5.1 shows that the sequence $\{\partial_{t}(J^{1+\alpha}u_{m})\}_{m=1}^{\infty}=\{J^{\alpha}u_{m}\}_{m=1}^{\infty}$ is bounded in L^{2}\bigl{(}0,T;H^{1}_{0}(\Omega)\bigr{)}. It now follows from Lemma 3.8, again with $B_{0}=H^{1}_{0}(\Omega)$, $B_{1}=B_{2}=L^{2}(\Omega)$, $p=+\infty$ and $r=2$, that there exists a subsequence of $\{J^{1+\alpha}u_{m}\}_{m=1}^{\infty}$ (still denoted by $\{J^{1+\alpha}u_{m}\}_{m=1}^{\infty}$) and \bar{u}\in C\bigl{(}[0,T];L^{2}(\Omega)\bigr{)} such that

[TABLE]

Furthermore, from the upper bound (17) of $\{\partial_{t}(J^{1+\alpha}u_{m})\}_{m=1}^{\infty}$ in L^{2}\bigl{(}0,T;H^{2}(\Omega)\bigr{)}, by choosing a subsequence one gets

[TABLE]

On the other hand, by applying Lemma 3.5 with $\phi=J^{1}(u_{m}-u)$ and $\beta=\alpha$, we deduce that for any $t\in[0,T]$ one has

[TABLE]

Hence, (21) implies that $\lim_{m\to\infty}\|J^{1+\alpha}(u_{m}-u)\|_{L^{\infty}(0,T;L^{2}(\Omega))}=0$. Recalling (23), we have $\bar{u}=J^{1+\alpha}u$. By choosing subsequences, we obtain

[TABLE]

where we used the boundedness of $\{J^{1+\alpha}u_{m}\}_{m=1}^{\infty}$ in L^{\infty}\bigl{(}0,T;H^{1}_{0}(\Omega)\bigr{)} that was already mentioned, and (24).

Multiplying both sides of (7) by a test function $\xi\in C_{c}^{\infty}((0,T)\times\Omega)$, integrating over $(0,T)\times\Omega$ and noting that $\Pi_{m}$ is a self-adjoint operator on $L^{2}(\Omega)$ gives

[TABLE]

where $h_{m}(t):=\int_{0}^{t}\nabla\cdot(\textbf{F}(s)\partial_{t}^{1-\alpha}u_{m}(s))\,ds$. Using (22) and (27), as $m\to\infty$ one has

[TABLE]

To find the limit of the most complicated term $\langle h_{m},\Pi_{m}\xi\rangle_{L^{2}(0,T;L^{2})}$ in (28), we first integrate by parts twice with respect to the time variable:

[TABLE]

It now follows from the boundedness of $\{J^{\alpha}u_{m}\}_{m=1}^{\infty}$ and $\{J^{1+\alpha}u_{m}\}_{m=1}^{\infty}$ in L^{2}\bigl{(}0,T;H^{2}(\Omega)\bigr{)} and L^{\infty}\bigl{(}0,T;H^{1}_{0}(\Omega)\bigr{)}, respectively, that $\{h_{m}\}_{m=1}^{\infty}$ is bounded in L^{2}\bigl{(}0,T;L^{2}(\Omega)\bigr{)}. Hence,

[TABLE]

On the other hand, by using (30) and integration by parts with respect to $x$, we have

[TABLE]

Combining this identity with (25)–(27) gives

[TABLE]

Now invoking (31) yields

[TABLE]

Let $m\rightarrow\infty$ in (28). Using (29) and (32), we deduce that for any $\xi\in C_{c}^{\infty}((0,T)\times\Omega)$ one has

[TABLE]

Since $C_{c}^{\infty}((0,T)\times\Omega)$ is dense in $L^{2}((0,T)\times\Omega)$, the above equation also holds true for any test function $\xi\in L^{2}((0,T)\times\Omega)$. Hence, $u$ satisfies (3) a.e. on $(0,T)\times\Omega$.

The weak convergence of $u_{m}$ described in (22), and [1, Corollary II.2.8] with (9) together yield $\|u\|_{L^{2}(0,T;L^{2})}^{2}\leq C_{3}\|G\|_{L^{2}(0,T;L^{2})}^{2}$. Similarly, (27) and (17) imply that $\kappa_{\alpha}\|J^{\alpha}\Delta u\|_{L^{2}(0,T;H^{2})}^{2}\leq C_{4}\|G\|_{L^{2}(0,T;L^{2})}^{2}$. Thus, (20) is proved.

The uniqueness of the solution $u$ follows from linearity and (20), because if $u_{0}=0$ and $g=0$, then $G=0$ and hence $u=0$. ∎

6 Existence and uniqueness of the classical solution

Assumption 6.1**.**

In the rest of this paper, we assume that

[TABLE]

Assumption 6.1 is not overly restrictive because (1) is usually considered as a variant of the case $\alpha=1$. We cannot avoid this restriction on $\alpha$ in Sections 6 and 7 since our analysis makes heavy use of $\partial_{t}^{1-\alpha}u$, and for typical solutions $u$ of (1), it will turn out that $\|\partial_{t}^{1-\alpha}u\|_{L^{2}(0,T;L^{2}(\Omega))}<\infty$ only for $1/2<\alpha<1$. To see this heuristically, assume that $u(x,t)=\phi(x)+v(x,t)$, where $v$ vanishes as $t\to 0$ so $\phi(x)$ is the dominant component near $t=0$; then $\partial_{t}^{1-\alpha}u(x,t)=\phi(x)\omega_{\alpha}(t)+\partial_{t}^{1-\alpha}v(x,t)\approx\phi(x)\omega_{\alpha}(t)$ near $t=0$, and $\int_{0}^{T}\omega_{\alpha}^{2}(t)\,dt$ is finite only if $\alpha>1/2$.

6.1 A priori estimates

Since $\partial_{t}^{1-\alpha}1=\omega_{\alpha}(t)$, we can rewrite (5a) in terms of

[TABLE]

as

[TABLE]

We will require the following bound for $u_{0m}$.

Lemma 6.2**.**

If $u_{0}\in H^{2}(\Omega)$, then $\|u_{0m}\|_{H^{2}(\Omega)}\leq C_{\mathrm{R}}\|u_{0}\|_{H^{2}(\Omega)}$ for all $m$, where the constant $C_{R}$ was defined in (19).

Proof.

Write $d^{k}_{0m}=d^{k}_{m}(0)=\langle u_{0},w_{k}\rangle$ for $1\leq k\leq m$, and let $\lambda_{k}>0$ denote the $k$th Dirichlet eigenvalue of the Laplacian so that $-\Delta w_{k}=\lambda_{k}w_{k}$ for all $k$. In this way,

[TABLE]

If $u_{0}\in H^{2}(\Omega)$ then $\Delta u_{0}\in L_{2}(\Omega)$ so, using Parseval’s identity,

[TABLE]

Thus, $\|u_{0m}\|_{H^{2}(\Omega)}\leq C_{\mathrm{R}}\|\Delta u_{0m}\|\leq C_{\mathrm{R}}\|\Delta u_{0}\|\leq C_{\mathrm{R}}\|u_{0}\|_{H^{2}(\Omega)}$. ∎

We now prove upper bounds for $\|\partial_{t}^{1-\alpha}v_{m}(t)\|$ and J^{\alpha}\bigl{(}\|\partial_{t}^{1-\alpha}\nabla v_{m}\|^{2}\bigr{)}(t) for any $t\in[0,T]$. The argument used in the following lemma is based on the proof of [5, Theorem 3.1].

Lemma 6.3**.**

Let $m$ be a positive integer. Let $v_{m}(t)$ be the absolutely continuous solution of (33) that is guaranteed by Lemma 4.1. Then, for almost all $t\in[0,T]$,

[TABLE]

and

[TABLE]

where

[TABLE]

Proof.

For notational convenience, set $z_{m}(t):=\partial_{t}^{1-\alpha}v_{m}(t)\in W_{m}$. Taking the inner product of both sides of (33) with $z_{m}(t)$ and integrating by parts with respect to $x$, we obtain

[TABLE]

By the Cauchy–Schwarz and arithmetic–geometric inequalities, one has

[TABLE]

and, using Lemma 3.7,

[TABLE]

Substituting these bounds into (36) and then applying Lemma 6.2, we obtain

[TABLE]

But $v_{m}(0)=0$, so $z_{m}=\partial_{t}^{1-\alpha}v_{m}={}^{C}\partial_{t}^{1-\alpha}v_{m}=J^{\alpha}(v_{m}^{\prime})$ and thus $\langle v_{m}^{\prime},z_{m}\rangle=\langle v_{m}^{\prime},J^{\alpha}(v_{m}^{\prime})\rangle$. Applying $J^{\alpha}$ to both sides of (37) and invoking Lemma 3.2 to handle the first term, we get

[TABLE]

By the Cauchy–Schwarz and arithmetic-geometric mean inequalities,

[TABLE]

and if $0\leq s<t$, then $(t-s)^{\alpha-1}=(t-s)^{1-\alpha}(t-s)^{2\alpha-2}\leq t^{1-\alpha}(t-s)^{2\alpha-2}$, so

[TABLE]

For the final term in (38), we have

[TABLE]

with

[TABLE]

Hence, (38) yields

[TABLE]

Discard the $\kappa_{\alpha}$ term and then apply the fractional Gronwall inequality (Lemma 3.1) to get (34). Finally, after substituting the bound (34) into the right-hand side of (39), it is straightforward to deduce (35). ∎

The next corollary follows easily from Lemma 6.3.

Corollary 6.4**.**

[TABLE]

Here, for $i=8,9$, the constants $C_{i}=C_{i}(\alpha,T,\kappa_{\alpha},\|\textbf{F}\|_{1,\infty})$ blow up as $\alpha\to(1/2)^{+}$ but are bounded as $\alpha\to 1^{-}$.

Corollary 6.4 implies an $L^{2}(\Omega)$ bound on $u_{m}(t)$, which we give in Corollary 6.5.

Corollary 6.5**.**

With $C_{8}$ as in Corollary 6.4, one has

[TABLE]

Proof.

As $u_{m}(t)$ is absolutely continuous, we have $v_{m}(t)=(J^{1}u_{m}^{\prime})(t)=J^{1-\alpha}(J^{\alpha}u_{m}^{\prime})(t)$, where we used [2, Theorem 2.2]. Thus [2, Theorem 2.22] can be invoked, which yields

[TABLE]

Set $z_{m}(t)=\partial_{t}^{1-\alpha}v_{m}(t)$, so $v_{m}(t)=J^{1-\alpha}z_{m}(t)$. Now Lemma 3.4 and Corollary 6.4 give

[TABLE]

As $u_{m}(t)$ is continuous, the inequality (40a) is valid for all $t$.

Next, using (40a) and the semigroup property $\omega_{\alpha}\ast\omega_{\beta}=\omega_{\alpha+\beta}$, we get

[TABLE]

which gives (40b). ∎

In the next lemma, we also provide upper bounds for $\{v_{m}\}_{m}$ in $W^{1,2}(0,T;L^{2})\cap L^{2}(0,T;H^{2})$ and \big{\{}\partial_{t}^{1-\alpha}\Delta v_{m}\big{\}}_{m} in $L^{2}(0,T;L^{2})$. Recall that the constant $\rho_{\alpha}>0$ was defined in Lemma 3.6.

Lemma 6.6**.**

Let $m$ be a positive integer. Let $v_{m}(t)$ be the absolutely continuous solution of (33) that is guaranteed by Lemma 4.1. Then for almost all $t\in[0,T]$, one has

[TABLE]

where

[TABLE]

Proof.

Take the inner product of both sides of (33) with $-\Delta v_{m}\in W_{m}$ and integrate by parts with respect to $x$ to get

[TABLE]

Integrating in time and noting that, by Lemma 3.6,

[TABLE]

we obtain

[TABLE]

with a free parameter $\epsilon>0$. Choosing $\epsilon=\kappa_{\alpha}\rho_{\alpha}t^{\alpha-1}/6$ and recalling Lemma 3.7 yields

[TABLE]

by Lemma 6.2. Invoking Corollary 6.4, we have

[TABLE]

and the bound (41) follows.

In a similar fashion, we next take the inner product of both sides of (33) with $v^{\prime}_{m}\in W_{n}$ and integrate by parts with respect to $x$ to obtain

[TABLE]

Choosing $\epsilon=1/6$ and invoking Lemma 3.7 gives

[TABLE]

Integrating both sides of the inequality in time and invoking Lemma 3.2, we deduce that

[TABLE]

The second result (42) now follows from (6.1), (45) and Lemma 6.2.

Using similar arguments, take the inner product of both sides of (33) with $-\partial_{t}^{1-\alpha}\Delta v_{m}\in W_{m}$, integrate by parts with respect to $x$ and note that $\partial_{t}^{1-\alpha}\Delta v_{m}=J^{\alpha}\Delta v^{\prime}_{m}$ to obtain

[TABLE]

Now integrate in time, invoking Lemma 3.2 and using (6.1), to deduce that

[TABLE]

and (43) now follows by (6.1) and Lemma 6.2, which completes the proof of the lemma. ∎

Inequality (6.1) may also be derived (with a different constant factor) by applying (19) to (43).

We remark that the function $\alpha\mapsto\rho_{\alpha}$ is monotone increasing for $\alpha\in(0,1)$, with $\rho_{\alpha}\to 1$ as $\alpha\to 1$. Thus, $\rho_{1/2}<\rho_{\alpha}<1$ for $1/2<\alpha<1$, with $\rho_{1/2}=\sqrt{2\pi/27}=0.48240\ldots$.

6.2 The classical solution

In this section, by using the method of compactness, we show that there is a subsequence of $\{v_{m}\}_{m}$ such that the sum of its limit and the inital data satisfies equation (1) almost everywhere.

Theorem 6.7**.**

Assume that $\alpha\in(1/2,1)$, $u_{0}\in H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$, $\textbf{F}\in W^{1,\infty}((0,T)\times\Omega)$ and $g\in L^{2}(0,T;L^{2})$. Then there exists a unique classical solution of (1), in the sense of Definition 2.4, such that

[TABLE]

where

[TABLE]

Proof.

From Corollary 6.5 and Lemma 6.6 we obtain

[TABLE]

which shows that the sequence $\{v_{m}\}_{m=1}^{\infty}$ is bounded in $L^{\infty}(0,T;H^{1})\cap L^{2}(0,T;H^{2}\cap H^{1}_{0})$ and that the sequence $\{v^{\prime}_{m}\}_{m=1}^{\infty}$ is bounded in $L^{2}(0,T;L^{2})$. Since the embeddings $H^{2}\hookrightarrow H^{1}\hookrightarrow L^{2}$ are compact, it follows from Lemma 3.8 that there exists a subsequence of $\{v_{m}\}_{m=1}^{\infty}$ (still denoted by $\{v_{m}\}_{m=1}^{\infty}$) such that

[TABLE]

Furthermore, from the upper bounds of $\{v_{m}\}_{m=1}^{\infty}$ we have

[TABLE]

By virtue of Lemma 3.5, the strong convergence in (48) implies that

[TABLE]

This, together with Corollary 6.4 and (43), yields

[TABLE]

Multiplying both sides of (33) by a test function $\xi\in L^{2}(0,T;L^{2})$, integrating over $(0,T)\times\Omega$ and noting that $\Pi_{m}$ is a self-adjoint operator on $L^{2}(\Omega)$, we deduce that

[TABLE]

Now let $m\to\infty$ in this equation and recall (6.2) and (50). We get

[TABLE]

for all $\xi\in L^{2}(0,T;L^{2})$, where $u:=v+u_{0}$. From (48)–(50), we have

[TABLE]

Hence, it follows from (51) that $u$ satisfies (1) a.e. in $(0,T)\times\Omega$.

Taking the limit as $m\to\infty$ in (47), we obtain (46). The uniqueness of the solution $u$ follows from (46), which completes the proof of the theorem. ∎

Remark 6.8**.**

It follows from the uniqueness in Theorems 5.3 and 6.7 that the mild solution will become the classical solution when $\alpha\in(1/2,1)$ and $u_{0}\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$. Furthermore, the continuous dependence of both the mild and classical solutions on the initial data $u_{0}$ follows from (20) and (46).

7 Regularity of the classical solution

Recall that $1/2<\alpha<1$ and that in general $C=C(\Omega,\kappa_{\alpha},\textbf{F},T)$. From Lemma 7.2 onwards, we allow $C=C(\Omega,\kappa_{\alpha},\textbf{F},T,q)$, where $q$ appears in the statements of our results below.

From Theorem 6.7, for almost every $(t,x)\in(0,T)\times\Omega$ the solution $u(t,x)$ satisfies (1). Using the identity $\partial_{t}^{1-\alpha}u=(J^{\alpha}u)^{\prime}(t)=(J^{\alpha}u^{\prime})(t)+u(0)\omega_{\alpha}(t)$, we rewrite (1) as

[TABLE]

From this equation and the fact that $J^{\alpha}\phi(0)=0$ for any function $\phi\in L^{2}(0,T;L^{2})$, we deduce that $u^{\prime}(t)=O(t^{\alpha-1})$ when $t$ is close to [math]. By letting $z(t,x):=tu^{\prime}(t,x)$, we have $z(0)=0$. The regularity of $z$ is examined in the following lemma.

Lemma 7.1**.**

Assume that $\int_{0}^{T}\|tg^{\prime}(t)\|^{2}\,dt$ is finite. Then, the function $z$ defined above satisfies

[TABLE]

for some constant $C_{13}$.

Proof.

For any $t>0$, multiplying both sides of (52) by $t$ and using the elementary identity

[TABLE]

we obtain a differential equation for $z$:

[TABLE]

Differentiating both sides of this equation with respect to $t$ and noting that $J^{\alpha}z^{\prime}=\partial_{t}^{1-\alpha}z$, we have

[TABLE]

where

[TABLE]

Applying Lemma 3.7 and letting $g_{1}(t)=tg^{\prime}(t)$, we find that

[TABLE]

with Lemma 5.1 and Theorem 6.7 implying that

[TABLE]

Thus, applying Theorem 6.7 to equation (55) with initial data $z(0)=0$, we deduce the bound (53). ∎

From Theorem 6.7, for almost every $(t,x)\in(0,T)\times\Omega$ we have the identity

[TABLE]

where $f:=g-\nabla\cdot(\textbf{F}\partial_{t}^{1-\alpha}u)\in L^{2}(0,T;H^{1})$. The regularity of solutions to problem (56) subject to the initial condition $u_{0}\in H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$ was studied in [9]. In order to apply [9, Theorem 5.7], we need at least an upper bound for $\int_{0}^{t}s\|f^{\prime}(s)\|ds$ which is proved in the following lemma. Here and subsequently, notation such as $f^{\prime}$ and $f^{(j)}$ indicates time derivatives, and we denote higher-order fractional derivatives by $\partial^{j-\alpha}_{t}u:=\partial_{t}^{j-1}\partial_{t}^{1-\alpha}u=(J^{\alpha}u)^{(j)}$ for $j\in\{1,2,3,\ldots\}$ and $0<\alpha<1$.

Lemma 7.2**.**

Let $u$ be the solution of (1) and $f:=g-\nabla\cdot(\textbf{F}\partial_{t}^{1-\alpha}u)$. Then, for $q\in\{0,1,2,\dots\}$, $\textbf{F}\in W^{q,\infty}(0,T;L^{2}(\Omega))$, and for any $t\in(0,T]$, there is a constant $C=C(\Omega,\kappa_{\alpha},\textbf{F},q)$ such that

[TABLE]

Proof.

Inequality (57) holds for $q=0$ by virtue of (46) (with $t$ playing the role of $T$) because

[TABLE]

For the case $q=1$, we note first that

[TABLE]

By (54) we have $(J^{\alpha}z)(t)=t(J^{\alpha}u^{\prime})(t)-\alpha(J^{\alpha+1}u^{\prime})(t)$, and differentiating with respect to $t$ gives

[TABLE]

where we used the identities $(J^{\alpha}u^{\prime})(t)=\partial_{t}^{1-\alpha}u-u_{0}\omega_{\alpha}(t)$ and $(\alpha-1)\omega_{\alpha}(t)=t\omega_{\alpha-1}(t)$. Thus,

[TABLE]

Hence, by Theorem 6.7 and Lemma 7.1 (with $t$ again playing the role of $T$),

[TABLE]

implying that the desired inequality (57) holds for $q=1$.

Multiply both sides of (58) by $t$ and then differentiate with respect to $t$, obtaining

[TABLE]

Since $z$ satisfies (55) — an equation similar to (1a) but with a different source $\bar{G}$ and with $z(0)=0$ — we get an estimate for $z$ corresponding to (59):

[TABLE]

This inequality, together with (46), (53), (59) and (60), yields

[TABLE]

which implies the desired inequality (57) for $q=2$.

The general case follows by iterating the arguments above; cf. [11]. ∎

We can now prove regularity estimates for the classical solution $u$.

Theorem 7.3**.**

Let $g_{j}(t):=t^{j}g^{(j)}(t)$ for $j=1,2,3,\dots$ For $q\in\{1,2,3,\dots\},\textbf{F}\in W^{q,\infty}(0,T;L^{2}(\Omega))$ and for any $t\in(0,T]$,

[TABLE]

and

[TABLE]

Proof.

By (56), it follows from [9, Theorem 4.4] with $r=2$ and $\nu=\alpha$, and from [9, Theorem 5.6] with $r=0$, $\mu=2$ and $\nu=\alpha$, that

[TABLE]

Similarly, from [9, Theorem 4.4] with $r=2$ and $\nu=\alpha$, and from [9, Theorem 5.4] with $r=\mu=0$, and

[TABLE]

The theorem follows by Lemma 7.2 since \int_{0}^{t}s^{j}\|f^{(j)}(s)\|\,ds\leq t^{1/2}\bigl{(}\int_{0}^{t}s^{2j}\|f^{(j)}(s)\|^{2}\,ds\bigr{)}^{1/2}. ∎

Corollary 7.4**.**

Let $\eta>1/2$. If $\|g^{(j)}(t)\|\leq Mt^{\eta-1-j}$ for $0\leq j\leq q+1,\textbf{F}\in W^{q,\infty}(0,T;L^{2}(\Omega))$ and $t\in(0,T]$, then

[TABLE]

Proof.

The assumption on $g$ ensures that $\|g_{j}\|\leq Mt^{\eta-1/2}$. ∎

The alternative and longer analysis in [11, Theorems 6.2 and 6.3] shows that these bounds can be improved to

[TABLE]

for any $\alpha\in(0,1)$ and $\eta>0$.