Local and global existence of pathwise solution for the stochastic Boussinesq equations with multiplicative noises

Abstract

Considering the stochastic Boussinesq equations in Td with the nonlinear multiplicative noises, we establish the local existence of pathwise solutions. Furthermore, we establish the global existence of pathwise solution when the noises are non-degenerate, which show that the non-degenerate multiplicative noises would provide a regularizing effect: the global existence of solution occurs with high probability if the initial data are sufficiently small, or if the noise coefficients are sufficiently large.