Asymptotic properties of the 2D stochastic fractional Boussinesq equations driven by degenerate noise

Abstract

The current paper is devoted to stochastic fractional Boussinesq equations (FBEs) on the Torus T2=[0,2π]×[0,2π] with degenerate random forcing. We first show the existence and pathwise uniqueness of strong solutions by virtue of the Yamada-Watanabe Theorem, and then we investigate the asymptotic properties of invariant measures in two aspects: the first is, focusing on the stochastic systems, to prove the existence of an invariant measure in the case of multiplicative noise, and then prove that the invariant measure can be unique and exponentially mixing if we are restricted to degenerate additive noise. The second is, from the perspective of stochastic stability, to prove that the zero-noise limits of stationary measures for the perturbed stochastic systems are invariant for the deterministic system and their supports are concentrated on the Birkhoff center.