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Stability and exponential decay for magnetohydrodynamic equations

Part of: Equations of mathematical physics and other areas of application Partial differential equations Incompressible viscous fluids

Published online by Cambridge University Press:  25 April 2022

Wen Feng ,
Farzana Hafeez ,
Jiahong Wu  and
Dipendra Regmi
Wen Feng
Affiliation:
Department of Mathematics, Niagara University, NY 14109, United States, (wfeng@niagara.edu)
Farzana Hafeez
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, United States, (farzana.hafeez@okstate.edu; jiahong.wu@okstate.edu)
Jiahong Wu
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, United States, (farzana.hafeez@okstate.edu; jiahong.wu@okstate.edu)
Dipendra Regmi
Affiliation:
Department of Mathematics, University of North Georgia-Gainesville Campus, Oakwood, GA 30566, United States, (dipendra.regmi@ung.edu)
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Abstract

This paper focuses on a 2D magnetohydrodynamic system with only horizontal dissipation in the domain $\Omega = \mathbb {T}\times \mathbb {R}$ with $\mathbb {T}=[0,\,1]$ being a periodic box. The goal here is to understand the stability problem on perturbations near the background magnetic field $(1,\,0)$. Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in $H^{1}$ as $t\to \infty$. As a consequence, the solution converges to its horizontal average asymptotically.

Keywords

Magnetohydrodynamic equationspartial dissipationstabilitywave equations

MSC classification

Primary: 35A01: Existence problems
Secondary: 35B35: Stability 35Q35: PDEs in connection with fluid mechanics 76D09: Viscous-inviscid interaction
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 3 , June 2023 , pp. 853 - 880
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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