Ergodicity of Galerkin approximations of surface quasi-geostrophic equations and Hall-magnetohydrodynamics system forced by degenerate noise

We study the two-dimensional surface quasi-geostrophic equations and the three-dimensional Hall-magnetohydrodynamics system forced by degenerate noise. In comparison with a vorticity formulation of the Navier–Stokes equations, the non-linear term of the surface quasi-geostrophic equations is more si...

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Bibliographic Details
Published inNonlinear differential equations and applications Vol. 29; no. 2
Main Author Yamazaki, Kazuo
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.03.2022
Springer Nature B.V
Subjects
Analysis
Approximation
Galerkin method
Magnetohydrodynamics
Mathematical analysis
Mathematics
Mathematics and Statistics
Noise
Vorticity
37L55
60H30
35Q35
Ergodicity
Surface quasi-geostrophic equations
Harris’ condition; Hörmander’s condition
Hall-magnetohydrodynamics system
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Summary:We study the two-dimensional surface quasi-geostrophic equations and the three-dimensional Hall-magnetohydrodynamics system forced by degenerate noise. In comparison with a vorticity formulation of the Navier–Stokes equations, the non-linear term of the surface quasi-geostrophic equations is more singular by one derivative. In comparison with the magnetohydrodynamics system, the Hall term of the Hall-magnetohydrodynamics system is also more singular by one derivative. We prove the existence and uniqueness of an invariant measure for the Galerkin approximations of the surface quasi-geostrophic equations and the Hall-magnetohydrodynamics system, both forced by degenerate noise which consists of only a few modes.
ISSN:1021-9722
1420-9004
DOI:10.1007/s00030-022-00753-8