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On initial value and terminal value problems for subdiffusive stochastic Rayleigh-Stokes equation

In this paper, we study two stochastic problems for time-fractional Rayleigh-Stokes equation including the initial value problem and the terminal value problem. Here, two problems are perturbed by Wiener process, the fractional derivative are taken in the sense of Riemann-Liouville, the source function and the time-spatial noise are nonlinear and satisfy the globally Lipschitz conditions. We attempt to give some existence results and regularity properties for the mild solution of each problem.

• The initial value problem: Consider the following time-fractional Rayleigh-Stokes stochastic equation where J = (0, T ), κ is a positive constant and 0 < α < 1, ∆ is the Laplacian operator. This equation is subjected to the Dirichlet boundary condition and the initial value condition Here, ∂ α /∂t α stands for the Riemann-Liouville fractional derivative (4) whereupon Γ is the Gamma function, see [23]. {W(t, .)} t∈J represents a standard Wiener process (see Subsection 2.2), the generalized derivativeẆ(t) = ∂ ∂t W(t) describes a white noise, the nonlinear term F and the time-spatial-noise σ will be specified later. The problem of finding u satisfying (1), (2), (3) is called an initial value problem.
• The terminal value problem: Let the initial condition (3) be replaced by the final value condition u(T, x) = ϕ(x), x ∈ X .
Then, the problem of finding u satisfying (1), (2), (5) is called a final value problem. The Rayleigh-Stokes problem arises in modeling the behavior of some non-Newtonian fluids. The Riemann-Liouville derivative in this model has been found to be flexible in describing viscoelastic behaviors of the flow [1,7,26]. Let us propose some details on the history of two problems we are interested in and some related studies. We begin with the deterministic model of the initial value problem (when the term σ(t, u)Ẇ(t) disappears). For the homogeneous problem, the Sobolev regularities for both smooth and nonsmooth initial data was established in [1]. For the linear case, Shen [26] used fractional Laplace transform to contribute the exact solution. In [34], the exact solutions was constructed by using the eigenfunction expansion on a rectangular domain. The readers can refer to some other studies on the exact solutions for the Rayleigh-Stokes problem in [10], [31]. In [33], Zaky extended and developed Legendre-tau algorithms for solving one-dimensional and two-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade fluid. Some other works [4,5,6,12] used different numerical methods to study the Rayleigh-Stokes problem with a heated generalized second grade fluid. In contract to the initial value problem, the study of the terminal value problem is still limited. We can list here some papers concerned with the deterministic model of this kind of problem [19,20,28].
To the best of our knowledge, both the initial value problem and terminal value problem for the stochastic Rayleigh-Stokes equation driven by Wiener process have not been investigated in the literature. This motivated us to study the existence, uniqueness and regularity of each of two stochastic problems. It is the fact that the analysis technique used in the deterministic case cannot be applied in the stochastic case. The problems of finding suitable spaces for the solutions in the stochastic case is our challenges since the solutions are more complex and it is required to cleverly improve the estimates for the solution operators.
The main contributions of this paper are as follows. In Section 2, we provide some preliminary results including some solution spaces, fractional calculus and stochastic analysis techniques. The existence, uniqueness result and some regularity properties of the mild solution to the initial value problem (1),(2),(3) is investigated in Section 3. In Section 4, we continue to study the existence, uniqueness, regularity of the mild solution to the terminal value problem (1),(2),(5).

Preliminaries.
2.1. Hilbert scale spaces. For convenience, we denote by , we denote the Lebesgue and the Sobolev-Slobodeckij spaces respectively, and Since X is a bounded domain with sufficiently smooth boundary and A is self-adjoint on L 2 x , it is well-known there exists a sequence of eigenfunctions φ j ∈ D(A) and a sequence of corresponding eigenvalues λ j such that Aφ j (x) = λ j φ j (x) on X . Additionally, the sequence {λ j } j=1,2,... is positive, non-decreasing, and tends to infinity as j → ∞, {φ j } j≥1 forms an orthonormal basis of L 2 x . Next, we introduce the Hilbert scale spaces which will be defined by basing on where ., . is the usual inner product of L 2 x . If ξ = 0, one can seeḢ 0 x ≡ L 2 x . Leṫ H −ξ be the dual space ofḢ ξ . The fractional operator (see [3]) A ξ :Ḣ ξ/2 →Ḣ −ξ/2 can be defined as

2.2.
Stochastic processes on the Hilbert scale spaces. Let Ω, F, P, {F t } t∈J be a filtered complete probability space, which satisfies that the filtration {F t } t∈J is a right continuous increasing family and that F 0 contains all P-null sets of F. Throughout this paper, for each ξ ≥ 0, notation {W ξ (t, .)} t∈J describes aḢ ξ x -valued Wiener process which is defined on Ω, F, P, {F t } t∈J with a finite trace covariance According to the above setting, {W ξ (t, .)} t∈J can be expanded in the form j=1 is a sequence of mutually independent one-dimension standard Wiener processes.
Let us recall the definition of the expectation. For a random variable χ : Ω →Ḣ ξ x and a real number q ≥ 2, we note that E χ qḢ x . Let us define by ξ is a Hillbert-Schmidt operator fromḢ ξ x toḢ ξ x where its norm is given by In the case ξ = ξ , we will write The following lemma is useful to estimate the stochastic integrals, and is a straightforward consequence of the Burkholder-Davis-Gundy inequality proved in Da Prato and Zabczyk [2, Lemma 7.2, page 182].

2.3.
Solution spaces and assumptions on nonlinearities. Let ξ ≥ 0 and q ≥ 2. By L ∞ t L q ωḢ ξ x := L ∞ (0, T ; L q (Ω,Ḣ ξ x )), we denote the space of all F t -adapted measurable processes w from J to L q ωḢ ξ For ρ > 0, we introduce the following space We define by C t L q ωḢ ξ x := C(J; L q ωḢ ξ x ) the space of all continuous F t -adapted measurable processes w from J to L q ωḢ ξ x such that w CtL q ωḢ ξ x := sup t∈J w(t) L q ωḢ ξ x < ∞.
Similarly, the notation C t, * L q ωḢ ξ x := C(J * ; L q ωḢ ξ x ), presents the set of all continuous F t -adapted measurable processes w from J * to L q ωḢ ξ x . By C θ t, * L q ωḢ ξ x := C θ (J * ; L q ωḢ ξ x ), we introduce the following Hölder continuous space of exponent θ By Z θ t, * L q ωḢ ξ x := Z θ (J * ; L q ωḢ ξ x ), we denote the space of all processes w in the space C t, * L q ωḢ ξ x such that lim exists finitely and Next, let us introduce the following definitions for the nonlinearity and the timespatial noise.
Definition 2.2 (Globally Lipschitz nonlinearity). Let q ≥ 2 and ξ ≥ 0. The nonlinearity F is said to be globally Lipschitz and written by F ∈ L glo (L q ωḢ ξ x and there exists a positive function ωḢ ξ x and t ∈ J.
Definition 2.3 (Globally Lipschitz time-spatial-noise). Let q ≥ 2 and ξ ≥ 0. The time-spatial-noise σ is said to be globally Lipschitz and written by σ ∈ L glo (L q ωḢ ξ x ; and there exists a positive function for all w † , w ‡ in L q ωḢ ξ x and t ∈ J.
3. Existence, regularity of the solution of the initial value problem.

3.1.
Mild solution for initial value problem. In this subsection, we give the definition of mild solution to Problem (1)-(3). To do this, let us first consider the following deterministic problem We now express u(t, .) in the form x . Due to the arguments in [1], the coefficients u j (t) have the following expression whereupon For fixed s, F j (s, ., u(s, .)) denotes by the j-th coefficient of For the sake of convenience, for t ∈ J, we define Then, we have the following representation for the solution to Problem (6) Motivated by (10), we state the definitions of mild solution to the initial value problem (1)-(3) as follows: for almost all ω for fixed t.

Properties of solution operators.
We begin with the following lemma which shows basic bounds for the functions R α,j , j ≥ 1.
The following lemma gives the appropriate estimates for the solution operators. From now on, if an operator G : L q ωḢ ξ → L q ωḢ ξ , for ξ, ξ ≥ 0, satisfies we will write G L q ωḢ ξ →L q ωḢ ξ for short.
Proof. Let v ∈ L q ωḢ ξ x and v j := v, φ j . By applying Lemma 3.2 and the fact that This completes the proof.
• Firstly, we prove that (21) holds. By choosing δ = 2 and = 1 , one arrives at It follows that there exists c α,4 > 0 such that • Next, we prove that (22) holds. By choosing δ = 1, = 1 , one also have By two latter estimates, we deduce that there exists c α,5 > 0 such that The proof is completed. x . The second theorem investigates the regularity of the solution on the space C γ t, * L q ωḢ ξ x , for any positive constant γ satisfying γ < min{1 − α, 1 q }.
Step 1. Firstly, it is required to show that Φ is well-defined on L ∞ t L q ωḢ ξ x . The first term G α (t)ψ(x) can be estimated by using Lemma 3.3 as x . For the second term, by applying Lemma 3.3, F (·, ·, 0) = 0, and F ∈ L glo (L q ωḢ ξ x ), one can see where we have used E F (s, ·, w(s, ·)) Using the fact that We continue to estimate the last term by considering two following cases.
Step 2. Next, we will apply Banach's fixed point theorem to show that Φ has a unique fixed point. For w ∈ L ∞ t,ρ L q ωḢ ξ x , we define Φ 1 w(t, x) := For w † , w ‡ ∈ L ∞ t,ρ L q ωḢ ξ x , by a similar way as in (23), it is obvious that We continue with the second operator Φ 2 in the two following cases.
Remark 1. In Theorem 3.5, the function σ is defined by Definition 2.3. Here, we can find a constant C ρ such that C ρ → 0 as ρ → ∞, and e −ρt (31)). Therewith, the global existence has been obtained consequently. However, we note that the above result does not hold true if σ is locally Lipschitz continuous. For example, let us consider q = 2 for simplicity and assume that σ satisfies the following locally Lipschitz continuity Let U R be the closed ball of center at zero and radius R in L ∞ t,ρ L q ωḢ ξ x . For w † , w ‡ taken in this ball, the following chain can be obtained by making uses of Lemma

and the condition (33)
By taking the essential supremum of e −ρs w † (s, ·) − w ‡ (s, ·) L 2 ωḢ ξ x on (0, T ), we imply from the above chain that A direct computation shows that e −2ρt t 0 e 4ρs ds equals (e 2ρt − e −2ρt )/(4ρ), and so it tends to positive infinity as ρ approaches positive infinity. Therefore, we cannot derive the contraction of the mapping Φ analogously Proof of Theorem 3.5.
It is well-known that, constructing global existence of problems that including locally Lipschitz continuous nonlinearity and noise is not an easy task. In the future, we will study global existence of RSE with more a general assumption than (33). Now, we study the regularity of the solution in the space L ∞ t L q ωḢ ξ x . From now on, we use a b to describe that there exists a positive constant C such that a ≤ Cb.
Theorem 3.6. Consider ξ ≥ 0 and q ≥ 2. Assume that ψ, F, σ satisfy the conditions of Theorem 3.5. Then, the following regularity estimate holds Proof. We aim to estimate u(t, ·) L q ωḢ ξ x firstly by using some materials which have been proved in the proof of Theorem 3.5. ds.
Proof. From the integral equation (11), it is obvious that By Lemma 3.4, we directly obtain an upper bound for E 1 (r, t) For E 2 (r, t), we can bound it by using Lemma 3.3 and the condition F ∈ L glo (L q ωḢ ξ x ), F (·, ·, 0) = 0 as For E 3 (r, t), we can bound it by using Lemma 3.4 and the condition F ∈ L glo (L q ωḢ ξ x ), F (·, ·, 0) = 0 as It follows that We continue with the last two terms E 4 (r, t), E 5 (r, t) by considering them in two cases of q.
Remark 2. Now, let us explain the reason why we need the estimate in part a) of Lemma 3.2 instead of the overestimate which was used in [18]. Recall that the estimate in part a) of Lemma 3.2 helps us to obtain the result u ∈ C 2 t, * L q ωḢ ξ x . In contrast to this, by using the overestimate (45), we could not obtain u ∈ C θ t, * L q ωḢ ξ x , with some θ > 0. We can explain the reason as follows. Look at the terms t 0 (t − s) −(α+ 1+ 2) K 1 (s)ds in (40) and t 0 (t − s) −q(α+ 1 + 2) K 2 (s)ds in (44). If the overestimate (45) is used instead of the estimate in part a) of Lemma 3.2, there would be a trouble because the two terms we have mentioned would be changed by t 0 (t−s) −(2−α) K 1 (s)ds and t 0 (t−s) −q(2−α) K 2 (s)ds respectively. Since two latter integrals are not convergent, we could not obtain u ∈ C θ t, * L q ωḢ ξ x , with some θ > 0. Hence, it is required to improve the overestimate (45) by a new one as in part a) of Lemma 3.2.
4. Existence, regularity of the solution of the terminal value problem.
Defining the operator G α (t) as in (9) and , we have the following representation for the solution to Problem (46) Motivated by (48), we give the definitions of mild solution to the terminal value problem (1),(2),(5) as follows: x with some θ ∈ (0, 1), q ≥ 2, ξ ≥ 0, is called a mild solution of Problem (1) if there holds that for almost all ω for fixed t.
Lemma 4.3. Let t, r, τ ∈ J * such that r > t. For any real number ξ, there holds Proof. Since K α,j (s) λ −1 j s −α , thanks to the fundamental theorem of calculus, one deduces Hence, the classical derivative of R α,j (t) finitely exists for t ∈ J * , and is bounded as follows Let v ∈ L q ωḢ ξ x . Applying the latter estimate, where we have used λ −1 j ≤ λ −1 1 . Similarly, we have x . From two latter estimates, we conclude that (50) holds.
This subsection is aimed to contribute the existence and uniqueness of mild solution to Problem (1),(2), (5). To this end, we prepare some useful estimates for the terms appearing in equation (48). Firstly, the following lemma shows upper bounds for two first terms in Z 1−α t, * L q ωḢ ξ x .
• If q = 2, then the Itô isometry yields that E t 0 G α (t − s)σ(s, w(s, ·))dW ξ (s, ·) Since σ ∈ L glo (L q ωḢ ξ x ; L 2 0 (Ḣ ξ x )) and σ(·, 0) = 0, we immediately deduce that E σ(s, w(s, ·)) Furthermore, it follows from w ∈ Z 1−α t, * L q ωḢ ξ x that E w(s, ·) • If q > 2, then using again the consequence of the Burkholder-Davis-Gundy inequality and the same techniques as the previous case, where we recall that σ(·, 0) = 0, σ ∈ L glo (L q ωḢ ξ x ; L 2 0 (Ḣ ξ x )). Multiplying both sides of the latter estimates with t 1−α , where we recall that F ∈ L glo (L q ωḢ ξ x ) and σ ∈ L glo (L q ωḢ ξ x ; L 2 0 (Ḣ ξ x )). Finally, the existence and uniqueness of a mild solution u in Z 1−α t, * L 2 ωḢ ξ x is obtained by the Banach contraction principle. For the case q > 2, Π 2 < 1, we note that the proof can be similarly based on Part a where it is necessary to estimate the Itô integral (59) in the case q > 2 (instead of q = 2) by using Part b of Lemma 4.5.

5.
Conclusion. In this study, we consider the initial value problem and the terminal value problem for a stochastic time-fractional Rayleigh-Stokes equation, where the source function and the time-spatial noise are nonlinear. By using some useful stochastic analysis techniques and fractional calculus, we obtain the existence, uniqueness of the mild solution of each problem. Furthermore, some regularity properties and continuity results for the solutions are also proposed.