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On the stability and exponential decay of the 3D MHD system with mixed partial dissipation near a equilibrium state

Xuemin Deng, Yuelong Xiao, Aibin Zang

Abstract

A main result of this paper establishes the global stability of the 3D MHD equations with mixed partial dissipation near a background magnetic field in the domain $Ω=\mathbb{T}^2\times\mathbb{R}$ with $\mathbb{T}^2=[0, 1]^2$. More precisely, each velocity equation lacks its own directional dissipation, and the magnetic equation lacks vertical dissipation in the MHD system. The key point to obtain the stability result is that we decompose the solution $(u,b)$ into the zeroth horizontal mode and the non-zeroth modes and complete the desired bound with the strong Poincaré type inequalities in the treatment of several nonlinear terms. Then we focus on the large-time behavior of the solution, where the non-zeroth modes decay exponentially in $H^2$, and the solution converges to its zeroth horizontal mode.

On the stability and exponential decay of the 3D MHD system with mixed partial dissipation near a equilibrium state

Abstract

A main result of this paper establishes the global stability of the 3D MHD equations with mixed partial dissipation near a background magnetic field in the domain with . More precisely, each velocity equation lacks its own directional dissipation, and the magnetic equation lacks vertical dissipation in the MHD system. The key point to obtain the stability result is that we decompose the solution into the zeroth horizontal mode and the non-zeroth modes and complete the desired bound with the strong Poincaré type inequalities in the treatment of several nonlinear terms. Then we focus on the large-time behavior of the solution, where the non-zeroth modes decay exponentially in , and the solution converges to its zeroth horizontal mode.
Paper Structure (4 sections, 7 theorems, 139 equations)

This paper contains 4 sections, 7 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['theorem1t']}.
  4. Proof of Theorem \ref{['theorem2t']}.

Key Result

Theorem 1.1

Consider the system $(smhe)$. Let the initial data $(u_{in}, b_{in})\in H^3(\Omega)$ and $div~u_{in}=div~b_{in}=0$ such that for some sufficiently small $\epsilon>0$. Then there exists a unique global solution $(u, b)$ satisfying where the constant $C>0$ is independent of $\epsilon$ and $t$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • proof