Perturbation of Stochastic Boussinesq Equations with Multiplicative White Noise

The Boussinesq equation is a mathematics model of thermohydraulics, which consists of equations of fluid and temperature in the Boussinesq approximation.The deterministic case has been studied systematically by many authors (e.g., see [1– 3]). However, in many practical circumstances, small irregularity has to be taken into account.Thus, it is necessary to add to the equation a random force, which is in general a spacetime white noise, as considered recently by many authors for other equations (see [4–11]). The random attractors of boussinesq equations with multiplicative noise have been investigated by [12]. In this paper, We will study the perturbation of stochastic boussinesq equations with multiplicative white noise. We will consider the following stochastic two-dimension a Boussinesq equations perturbed by a multiplicative white noise of Stratonovich form:


Introduction
The Boussinesq equation is a mathematics model of thermohydraulics, which consists of equations of fluid and temperature in the Boussinesq approximation.The deterministic case has been studied systematically by many authors (e.g., see [1][2][3]).However, in many practical circumstances, small irregularity has to be taken into account.Thus, it is necessary to add to the equation a random force, which is in general a spacetime white noise, as considered recently by many authors for other equations (see [4][5][6][7][8][9][10][11]).The random attractors of boussinesq equations with multiplicative noise have been investigated by [12].In this paper, We will study the perturbation of stochastic boussinesq equations with multiplicative white noise.
We will consider the following stochastic two-dimension a Boussinesq equations perturbed by a multiplicative white noise of Stratonovich form: The domain occupied by the fluid is  = (0, 1) × (0, 1), and  1 ,  2 is the canonical basis of R 2 .The unknown V = (V 1 , V 2 ), , and  stand for the velocity vector, temperature, and pressure, respectively. 1 is the temperature at the top,  2 = 1, while  0 =  1 + 1 is the temperature at the boundary below,  2 = 0.The constant numbers  > 0,  > 0, and  > 0 are related to the usual Prandtl, Grashof, and Rayleigh numbers.
We supplement (1) with the following boundary condition: ( When an initial-valued problem is considered, we supplement these equations with V (, 0) = V 0 () , (, 0) =  0 () for  ∈ . ( The existence of a compact random attractor and its Hausdorff, fractal dimension estimates have been investigated by [12].We will solve pathwise (1)- (3).By using the Faedo-Galerkin approximation and a priori estimates, we prove the existence and uniqueness of the global solution and show that the solution continuously depends on the initial value.We also get some regularity results of the solutions.

Mathematical Setting and Basic Estimates
Let and change  to  −  2 +  2 2 /2; then (1) can be rewritten as Let the process be Then  = − ∘ , and if we let we get the new equations (no stochastic differential appears here) div  = 0, with the boundary conditions and the initial value conditions To solve ( 8)-( 12), we consider the Hilbert space  =  1 ×  2 with the scalar products (⋅, ⋅) and norms | ⋅ |, where  2 =  2 () and We also consider the subspace  =  1 ×  2 of , where  2 is the space of functions in  1 () vanishing at  2 = 0 and  2 = 1 and periodic in the direction of  1 . 2 is a Hilbert space for the scalar product and the norm and  1 = { ∈  2 2 : div  = 0}.We also denote by ((⋅, ⋅)) and ‖ ⋅ ‖ the canonical scalar product and norm in  1 and .

The bilinear form
determines a linear isomorphism  from () into  and from  into the dual space   , defined by with () = ( 1 ) × ( 2 ), where Four spaces (), , , and   satisfy and all embedding injections are densely continuous.It is well known that  : () →  is self-adjoint and positive and We also consider the trilinear forms  on  defined by The trilinear form  is continuous on  or even on  1 () 2 ×  1 ().We associate with the form  the bilinear continuous operator  which map  ×  into   and () × () into , defined by Finally, we define the continuous operators Now, we can set (8) in the operator form.If  = {, } is the solution of ( 8) and  = {, } is a test function in , we multiply ( 8) by  and ( 9) by , integrate over , and add the resulting equation.The pressure term disappears and after simplification we find which can be reinterpreted as Note that this equation differs from the determined case, and in determined case, the family () of operator is independent of the time .Initial condition (12) can be reinterpreted as To solve ( 23)-( 24), we also need some Sobolev norm estimates on the bilinear  and the operators  and .
Proof.The proof is the same as the deterministic case (see [10]).
Theorem 4. Assume that  0 ∈ , then there exists a unique solution of (23)-( 24), such that and the mapping  0  → () is continuous from H into D(A), for all  > 0.
Proof.Since  −1 :  → () is a self-adjoint compact operator in , it follows from a classical spectral theorem that there exists a sequence   : 0 <  1 ≤  2 ≤ ⋅ ⋅ ⋅ ,   → ∞ and a family of elements   ∈ () which is completely orthogonal in  such that For each , we look for an approximate solution   of the following form: and initial condition where   is the projector in  (or ) on the space spanned by  1 ,  2 , . . .,   .Since  and   commute, the above equation is also equivalent to where in view of the linearity of   , .
The existence of   on any finite interval [0,   ) follows from standard results of the existence of solutions of ordinary differential equations that   = +∞ is a consequence of these results and of the following priori estimates:   remains bounded in  ∞ (0, ; ) ∩  2 (0, ; ) , ∀ > 0. ( remains bounded in  2 (0, ;   ), which proved (44).

Regularity Results
In this section, we will consider further regularity results for the unique solution.The main result is that  ∈ (), and thus  ∈  2 () (59) which proved the second argument of (51), and thus (51) holds.