Existence and uniqueness of invariant measures of 3D stochastic MHD-α model driven by degenerate noise. Issue 2 (22nd January 2022)

• Record Type: Journal Article
• Authors: Zhang, Rangrang
• Is Part Of: Applicable analysis. Volume 101:Issue 2(2022)
• Page Start: 629
• Page End: 654
• Publication Date: 2022-01-22
• Subjects: MHD-α model -- invariant measure -- degenerate noise -- asymptotic strong Feller; Primary 60H15 -- Secondary 35B40 -- 37A25; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2020.1757077
• Languages: English
• ISSNs: 0003-6811

Abstract

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.