TY - JOUR
T1 - Ergodicity of Galerkin approximations of surface quasi-geostrophic equations and Hall-magnetohydrodynamics system forced by degenerate noise
AU - Yamazaki, Kazuo
N1 - Funding Information:
The author would like to express deep gratitude to the Fields Institute at which the author obtained the inspiration for this project during his brief visit in June 2019. The author thanks Prof. Adam Larios and Dr. Lizheng Tao for valuable discussion. Finally, the author also expresses deep gratitude to the editor and the anonymous referees for their valuable comments that have improved this manuscript significantly.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2022/3
Y1 - 2022/3
N2 - 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.
AB - 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.
KW - Ergodicity
KW - Hall-magnetohydrodynamics system
KW - Harris’ condition; Hörmander’s condition
KW - Surface quasi-geostrophic equations
UR - http://www.scopus.com/inward/record.url?scp=85124719090&partnerID=8YFLogxK
U2 - 10.1007/s00030-022-00753-8
DO - 10.1007/s00030-022-00753-8
M3 - Article
AN - SCOPUS:85124719090
SN - 1021-9722
VL - 29
JO - Nonlinear Differential Equations and Applications
JF - Nonlinear Differential Equations and Applications
IS - 2
M1 - 20
ER -