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 ‖⋅‖_𝐿^𝑞, where 𝑞 > 2/(2𝛼 − 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 ≥ 2 − 2𝛼 and 𝛼 ∈ (1/2, 1) is established, which implies the existence of non-trivial invariant measures of the quasi-geostrophic equation driven by nonlinear multiplicative noise.