April, 2024 Long time dynamics of stochastic fractionally dissipative quasi-geostrophic equations with stochastic damping
Tongtong LIANG, Yejuan WANG
Author Affiliations +
J. Math. Soc. Japan 76(2): 563-591 (April, 2024). DOI: 10.2969/jmsj/90479047


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.

Funding Statement

This work was supported by the Innovative Groups of Basic Research in Gansu Province under grant 22JR5RA391, the Major Science and Technology Projects in Gansu Province-Leading Talents in Science and Technology under grant 23ZDKA0005, the National Natural Science Foundation of China under grant 41875084, the Fundamental Research Funds for the Central Universities under grant lzujbky-2023-it37, and Innovative Star of Gansu Province under grant 2023CXZX-052.


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Tongtong LIANG. Yejuan WANG. "Long time dynamics of stochastic fractionally dissipative quasi-geostrophic equations with stochastic damping." J. Math. Soc. Japan 76 (2) 563 - 591, April, 2024. https://doi.org/10.2969/jmsj/90479047


Received: 2 November 2022; Published: April, 2024
First available in Project Euclid: 26 December 2023

Digital Object Identifier: 10.2969/jmsj/90479047

Primary: 60H15
Secondary: 37L15 , 37L40

Keywords: almost sure exponential stability , invariant measure , moment exponential stability , stochastic damping , Stochastic quasi-geostrophic equation

Rights: Copyright ©2024 Mathematical Society of Japan


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Vol.76 • No. 2 • April, 2024
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