Asymptotic stability and large time behavior of some three dimension magnetohydrodynamic equations

Fangfang Jian; Dongxiang Chen; Xiaoli Chen

doi:10.3934/dcds.2023051

Article Contents

2023, Volume 43, Issue 9: 3378-3423. Doi: 10.3934/dcds.2023051

This issue Previous Article Sobolev regularity theory for the non-local elliptic and parabolic equations on $ C^{1,1} $ open sets Next Article Best constants and existence of maximizers for anisotropic weighted Moser-Trudinger inequalities in $ \mathbb{R}^{N} $

Asymptotic stability and large time behavior of some three dimension magnetohydrodynamic equations

School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, China

^*Corresponding author: Xiaoli Chen
^*Corresponding author: Xiaoli Chen

Received: December 2022

Revised: March 2023

Early access: May 12, 2023

Published online: May 12, 2023

This work is partially supported by [NSF of China under Grant 11971209 and 11961032 and the foundation of the Education Division in Jiangxi Province.].

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Abstract

In this paper, we investigate the stability issue and large time behavior of three dimensional magnetohydrodynamic equations with velocity damping and magnetic diffusion only in the $ x_1 $-direction near a background magnetic field. Due to the lack of the magnetic diffusion in two direction, this problem becomes more challenging. The classical anisotropic Sobolev techniques to deal with MHD equations with mixed dissipation fail here. Fortunately, combining anisotropic Sobolev techniques and some interpolation methods, we can establish the asymptotic stability and explicit decay rates of the solutions to the three dimensional MHD equations mentioned above. The most difficult is to obtain the $ L^1 $-norm of $ \|\nabla_h u\|_{L^\infty} $ in time.

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Mathematics Subject Classification: Primary: 35Q35, 35B25; Secondary: 76D03, 76E25.

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