Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise

We present the time-spatial regularity of the nonlocal stochastic convolution for Caputo-type time fractional nonlocal Ornstein-Ulenbeck equations, and establish the existence and uniqueness of mild solutions for time fractional and space nonlocal stochastic Boussinesq equations driven by Gaussian white noise.

There are several papers on the well-posedness for time fractional partial differential equations. Lions studied the weak solution of fractional Navier-Stokes equation( [7]), Shinbrot studied the fractional derivatives of solutions of the Navierstokes equations in 1971, see [9] for details. Recently, there are some papers on the Caputo-type time fractional Navier-Stokes equation Carvalho-Neto and Planas in [2] established the existence and uniqueness of mild solutions of time fractional Navier-Stokes equation (2). Zhou and Peng in [12] and [13] studied the weak solution for equation (2) in Besov space. Very recently, Wang, Xu & Kloeden investigated the asymptotic behavior of a stochastic lattice system with a Caputo fractional time derivative in [10]. Zeng and Yang [11] studied the mild solution of the time fractional Navier-Stokes equations driven by fractional Brownian motions. To the best of our knowledge, there is a few paper to study the regularity of time-space fractional partial differential equations. The novelty in this paper is to establish the time regularity and space regularity for the nonlocal stochastic convolution, which can be used in other time and space fractional stochastic fluid equations. To the end, the restriction are imposed on the order of fractional derivative α and the order of spatial nonlocal effects β 1 , β 2 . We also prove the existence and uniqueness of mild solutions of (1) by the the Banach fixed point Theorem. Moreover, the dependence of the order of time-fractional derivative, the order of the space-fractional derivative and the regularity of the initial data are revealed.
The rest of the paper is organized as follows. Some basic concepts, the function setting and the definition of mild solution of (1) are presented in section 2. In section 3, the regularity of the nonlocal stochastic convolution are established. The existence and uniqueness of mild solutions for time fractional nonlocal stochastic Boussinesq equations are established in section 4.

2.
Preliminaries. In this section, we present the definitions of fractional operators and definition of mild solutions for stochastic systems (1), which is taken from [11].
Next we give some notations of Banach spaces which the mild solutions exist in.
where φ = (u, θ) ∈ H. For simplicity, we use the notations u , θ and φ to represent the norm for space H 1 , H 2 and H respectively. Let Then (e m,n , λ m,n ) are the eigenvectors and eigenvalues of (−∆) with Dirichlet boundary conditions. Denote A = −∆ and the operator A σ 2 is well defined in the space of functions In what follows, we define the bilinear operators B 1 and B 2 by Then Equation (1) can be rewritten in the following abstract evolution equation where I 1−α t is the (1 − α)-order Riemann-Liouville fractional integral operator. Define the Mittag-Leffler families operators based on the analytic semigroup S(t) generated by the space fractional operator A β 2 : Then we can define the mild solution of (7) through the Mittag-Leffler families operators. 4. An F t -adapted random field {(u(t, x), θ(t, x)), t ≥ 0, x ∈ T 2 } is said to be a mild solution of (7) with initial value (u 0 , θ 0 ) if the following integral equation is fulfilled : 3. Regularity of the nonlocal stochastic convolution. Here, we consider the nonlocal stochastic convolution where where e k , k ∈ N is an orthonormal basis of L 2 (0, 1) and β k , k ∈ N is a family of independent real-valued Brownian motions.
Next, we study the space regularity of equation (9).
Proof. It follows from the fact e m,n L ∞ < 1 that

TIANLONG SHEN, JIANHUA HUANG AND CAIBIN ZENG
Proof. Taking any (v, η) in B T R , and define and We deduce that It is clear that It follows from Hölder inequality and ∇e m,n L ∞ ≤ Cλ .

TIME-SPACE FRACTIONAL STOCHASTIC BOUSSINESQ EQUATIONS 1531
Then Similarly, and Similarly to I 1 and I 2 By the estimates of I 1 -I 3 and J 1 -J 2 , we infer that Now and