Long time dynamics of stochastic fractionally dissipative quasi-geostrophic equations with stochastic damping

Abstract

A stochastic fractionally dissipative quasi-geostrophic equation with stochastic damping is considered in this paper. First, we show that the null solution is exponentially stable in the sense of $q^{-}$-th moment of $\|\cdot\|_{L^{q}}$, where $q > 2/(2\alpha - 1)$ and $q^{-}$ denotes the number strictly less than $q$ but close to it, and from this fact we further prove that the sample paths of solutions converge to zero almost surely in $L^{q}$ as time goes to infinity. In particular, a simple example is used to interpret the intuition. Then the uniform boundedness of pathwise solutions in $H^{s}$ with $s \geq 2 - 2\alpha$ and $\alpha \in (1/2, 1)$ is established, which implies the existence of non-trivial invariant measures of the quasi-geostrophic equation driven by nonlinear multiplicative noise.