EXISTENCE, UNIQUENESS AND REGULARITY OF THE SOLUTION OF THE TIME-FRACTIONAL FOKKER–PLANCK EQUATION WITH GENERAL FORCING

. A time-fractional Fokker–Planck initial-boundary value problem is considered, with diﬀerential operator u t −∇· ( ∂ 1 − α t κ α ∇ u − F ∂ 1 − α t u ), where 0 < α < 1. The forcing function F = F ( t,x ), which is more diﬃcult to analyse than the case F = F ( x ) investigated previously by other authors. The spatial domain Ω ⊂ R d , where d ≥ 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 (Ω). For 1 / 2 < α < 1 and u 0 ∈ H 2 (Ω) ∩ H 10 (Ω), 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.


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: u(0, x) = u 0 (x) for x ∈ Ω, (1b) u(t, x) = 0 for x ∈ ∂Ω and 0 < t < T, where κ α > 0 is constant and Ω is an open bounded domain with C 2 boundary in R d for some d ≥ 1. In (1a), one has 0 < α < 1 and ∂ 1−α t is the standard Riemann-Liouville fractional derivative operator defined by ∂ 1−α t u = (J α u) t , where J β denotes the Riemann-Liouville fractional integral operator of order β, viz., 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, 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 hererelying 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 α → 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 < α < 1, with constants that blow up as α → 1/2 (cf. the comment following Assumption 6.1). By contrast, the results in [11] hold for the full range of values 0 < α < 1, but with constants that blow up as α → 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 α ∈ (0, 1) and u 0 ∈ L 2 (Ω); • By imposing a further condition on u 0 and restricting α 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 (Ω) and H 2 (Ω), needed for the error analysis of numerical methods for solving (1); see, e.g., [5,6,14].

Notation and definitions
Throughout the paper, we often suppress the spatial variables and write v or v(t) instead of v(t, ·) for various functions v. We also use the notation v ′ for the time derivative. Let · denote the L 2 (Ω) norm defined by v 2 = v, v , where ·, · is the L 2 (Ω) inner product. Let · H r (Ω) and | · | H r (Ω) be the standard Sobolev norm and seminorm on the Hilbert space of functions whose rth-order derivatives lie in L 2 (Ω). We borrow some standard notation from parabolic partial differential equations, e.g., C([0, T ]; L 2 (Ω)).
Assume throughout the paper that the forcing function F = (F 1 , . . . , F d ) T ∈ W 1,∞ ((0, T ) × Ω) and that its divergence ∇ · F is continuous on [0, T ] × Ω. Then F is continuous on [0, T ] × Ω and we set 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 Ω, κ α , 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.
The Mittag-Leffler function E α (z) that is used in the fractional Gronwall inequality of Lemma 3.1 is defined by for z ∈ 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

Technical preliminaries
This section provides some properties of fractional integrals that will be needed in our analysis.
The following lemmas will be used several times in in our analysis.

Proof. Minkowski's integral inequality gives
To prove the second inequality, apply Hölder's inequality to (4) to obtain for any β > 1/2, which completes the proof of this lemma.
The following estimate involving the force F is used several times in our analysis.
We now recall a fundamental compactness result that will be used several times in the proofs of our main results.

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., [ Let Π m be the orthogonal projector from L 2 (Ω) onto W m defined by: for each v ∈ L 2 (Ω), one has The projections of the source term and initial data are denoted by g m (t) := Π m g(t) and u 0m := Π m u 0 .
We aim to choose the functions d k m so that for k = 1, 2, . . . , m and t ∈ (0, T ] one has Existence and uniqueness of a solution to (5) are guaranteed by the following lemma. 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) : . Formally integrating this equation in time we obtain the Volterra integral equation [6, p.1768]: where It is shown in [6] that (6) has a unique solution u m ∈ C [0, T ]; H 1 0 (Ω) . Now, g ∈ L 1 0, T ; L 2 (Ω) implies that g m ∈ L 1 0, T ; L 2 (Ω) , and it follows that G m : [0, T ] → L 2 (Ω) is absolutely continuous. Furthermore, Theorem 2.5 of [2] implies (using the continuity of u m ) that t → t s=0 K m (t, s)u m (s) ds is absolutely continuous. Hence, (6) shows that u m : [0, T ] → L 2 (Ω) 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 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.

A priori estimates
In order to prove a priori estimates, we consider the integrated form of equation (5a): where G m (t) = Π m G(t) as in (6). Let C P denote the Poincaré constant for Ω, viz., v 2 ≤ C P ∇v 2 for v ∈ H 1 0 (Ω). 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 ∈ [0, T ] one has where Proof. Taking the inner product of both sides of (7) with J α u m (t) ∈ W m then integrating by parts with respect to x, we obtain Integrating by parts with respect to the time variable, and using Minkowski's integral inequality and Hölder's inequality, we have It follows from (10) and (11) that Integrating in time and invoking Lemma 3.3, we deduce that But Lemma 3.4 gives us Thus, setting ψ m (t) := J 1 ( J α/2 u m 2 )(t), we deduce from (12) that Applying Lemma 3.1, one obtains This inequality and (13) together yield Now (8) follows immediately on recalling (12)- (14) and the Poincaré inequality.
In a similar fashion, we take the inner product of both sides of (7) with u m (t) ∈ W m and then integrate by parts with respect to x, to obtain Using Lemma 3.7 and the same arguments as in the proof of (11), we also have This estimate and (15) together imply Integrating in time, we get Now apply the inequality (8) to complete the proof. and where Proof. Taking the inner product of both sides of (7) with −J α ∆u m (t) ∈ W m and then integrating by parts with respect to x, we obtain This inequality and (16) together imply Integrating in time and invoking Lemma 3.3, we deduce that which, after applying inequality (8) of Lemma 5.1, completes the proof of (17). Applying (4) with φ = J α/2 u m and β = 1 − α/2 gives where z(t) = J α/2 u m (t) H 1 (Ω) . Using Young's convolution inequality we get The inequality (18) now follows immediately from (17).

The mild solution
Our assumption that Ω has a C 2 boundary ensures that if v ∈ H 1 0 (Ω) satisfies ∆v ∈ L 2 (Ω), then v ∈ H 2 (Ω). Moreover, there is a regularity constant C R , depending only on Ω, such that Our next result requires a strengthening of the regularity hypothesis on F. Theorem 5.3. Assume that u 0 ∈ L 2 (Ω), F ∈ W 2,∞ ((0, T ) × Ω) and g ∈ L 2 0, T ; L 2 (Ω) . Then there exists a unique mild solution u of (1) (in the sense of Definition 2.3) such that 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) Furthermore, from the above bounds on {J 1 u m } ∞ m=1 and well-known results [1, Theorem II.2.7] for weak and weak-⋆ compactness, by choosing sub-subsequences we get By letting u := ∂ t v ∈ L 2 0, T ; L 2 (Ω) , we have v = J 1 u. It remains to prove that J α u m converges weakly to J α u in L 2 0, T ; H 2 (Ω) . Applying Lemma 3.5 with φ = J 1 u m and β = α, for any t ∈ [0, T ] we deduce that This inequality, together with Lemma 5.2, implies that the sequence On the other hand, by applying Lemma 3.5 with φ = J 1 (u m − u) and β = α, we deduce that for any t ∈ [0, T ] one has Hence, (21) implies that lim m→∞ J 1+α (u m −u) L ∞ (0,T ;L 2 (Ω)) = 0. Recalling (23), we haveū = J 1+α u. By choosing subsequences, we obtain where we used the boundedness of {J 1+α u m } ∞ m=1 in L ∞ 0, T ; H 1 0 (Ω) that was already mentioned, and (24).
Assumption 6.1 is not overly restrictive because (1) is usually considered as a variant of the case α = 1. We cannot avoid this restriction on α in Sections 6 and 7 since our analysis makes heavy use of ∂ 1−α t u, and for typical solutions u of (1), it will turn out that ∂ 1−α t u L 2 (0,T ;L 2 (Ω)) < ∞ only for 1/2 < α < 1. To see this heuristically, assume that u(x, t) = φ(x)+v(x, t), where v vanishes as t → 0 so We will require the following bound for u 0m .
Proof. Write d k 0m = d k m (0) = u 0 , w k for 1 ≤ k ≤ m, and let λ k > 0 denote the kth Dirichlet eigenvalue of the Laplacian so that −∆w k = λ k w k for all k. In this way, If u 0 ∈ H 2 (Ω) then ∆u 0 ∈ L 2 (Ω) so, using Parseval's identity, We now prove upper bounds for and where .
Taking the inner product of both sides of (33) with z m (t) and integrating by parts with respect to x, we obtain By the Cauchy-Schwarz and arithmetic-geometric inequalities, one has and, using Lemma 3.7, Substituting these bounds into (36) and then applying Lemma 6.2, we obtain v ′ m , z m + But v m (0) = 0, so z m = ∂ 1−α . Applying J α to both sides of (37) and invoking Lemma 3.2 to handle the first term, we get By the Cauchy-Schwarz and arithmetic-geometric mean inequalities, For the final term in (38), we have Hence, (38) yields Discard the κ α 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.
and v m As u m (t) is continuous, the inequality (40a) is valid for all t.
In a similar fashion, we next take the inner product of both sides of (33) with v ′ m ∈ W n and integrate by parts with respect to Choosing ǫ = 1/6 and invoking Lemma 3.7 gives Integrating both sides of the inequality in time and invoking Lemma 3.2, we deduce that The second result (42) now follows from (44), (45) and Lemma 6.2.