Invariant measure of stochastic Boussinesq equation with zero viscosity in Banach space

In this paper, we investigate the stochastic Boussinesq equations driven by Gaussian noise on a two-dimensional domain . By the regularity estimation on the high-order Sobolev space, we prove the existence of weak solutions in non-separable Banach space. We also show that the Markov semigroup generated by the solution of stochastic Boussinesq equations is also weak Feller. Endowed with the weak-★ topology on , we prove the existence of the invariant measure by Krylov–Bogoliubov theorem.