Existence and uniqueness of invariant measures of 3D stochastic MHD-α model driven by degenerate noise

ABSTRACT

In this paper, we establish the existence and uniqueness of invariant measures of the 3D stochastic magnetohydrodynamic-α model (MHD-α) driven by degenerate additive noise. We firstly study the Feller property of solutions and establish the existence of invariant measures by utilizing the classical Krylov–Bogoliubov theorem. Then, we prove the uniqueness of invariant measures for the corresponding transition semigroup by utilizing the notion of asymptotic strong Feller proposed by Hairer and Mattingly [Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann Math (2). 2006;164(3):993–1032]. The proof not only requires the investigation of degenerate noise, but also the study of highly nonlinear, unbounded drifts.

Keywords:

• MHD-α model
• invariant measure
• degenerate noise
• asymptotic strong Feller

2010 Mathematics Subject Classifications:

• Primary 60H15
• Secondary 35B40
• 37A25

Acknowledgments

This work was partly supported by National Natural Science Foundation of China (NSFC) (No. 11801032), Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences(No. 2008DP173182), China Postdoctoral Science Foundation funded project (No. 2018M641204).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partly supported by National Natural Science Foundation of China (NSFC) [grant number 11801032], NSFC [grant number 11971227], and Beijing Institute of Technology Research Fund Program for Young Scholars. Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences [grant number 2008DP173182], China Postdoctoral Science Foundation funded project [grant number 2018M641204].