Study on the well-posedness of stochastic boussinesq equations with navier boundary conditions

Due to the Navier boundary conditions (first introduced in [28]) which allow the fluid to slip along the boundary are more consistent with the actual physical background, we consider stochastic Boussinesq equations in a 2D-bounded domain with Navier type boundary conditions in this paper. First, we obtain a priori estimates of the weak solutions of stochastic Boussinesq equations in L2 and L4 via Ito's formula and the BurkHolder-Davis-Gundy's inequality. Then, we establish the existence and uniqueness of the weak solu-tions which base on a priori estimates and monotonicity arguments. Different from the periodic conditions and the Dirichlet conditions, the Navier boundary conditions generate boundary terms on velocity by integration by parts, and we attempt to over come the difficulties cased by boundary terms. Based on the theory of the existence of equations, we can gain a better understanding of the essence and mathematical structure, thereby guiding the strategy and technique selection for numerical solving.