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Stability for the 2D incompressible MHD equations with only magnetic diffusion

Xiaoping Zhai

Abstract

This paper presents a global stability result on perturbations near a background magnetic field to the 2D incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion on the periodic domain. The stability result provides a significant example for the stabilizing effects of the magnetic field on electrically conducting fluids. In addition, we obtain an explicit large-time decay rate of the solutions.

Stability for the 2D incompressible MHD equations with only magnetic diffusion

Abstract

This paper presents a global stability result on perturbations near a background magnetic field to the 2D incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion on the periodic domain. The stability result provides a significant example for the stabilizing effects of the magnetic field on electrically conducting fluids. In addition, we obtain an explicit large-time decay rate of the solutions.
Paper Structure (6 sections, 4 theorems, 52 equations)

This paper contains 6 sections, 4 theorems, 52 equations.

Table of Contents

  1. Introduction and main result
  2. The proof of the theorem
  3. $L^2$ energy estimate
  4. High order energy estimate
  5. A key Lemma
  6. Complete the proof of the main theorem

Key Result

Theorem 1.1

Assume $\mathbf{n}$ satisfies the Diophantine condition (diufantu). Let $\alpha>0, \beta>0$ be two arbitrarily fixed constants. For any ${N}\ge (2\beta+3)r+\alpha+2\beta+5$ with $r>1$, and $( \mathbf{u} _0, \mathbf{B} _0)\in H^{N}( \mathbb{T} ^2)$ with If there exists a small constant $\varepsilon$ such that Then the system m admits a global solution $( \mathbf{u} , \mathbf{B} )\in C([0,\infty

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.4
  • Lemma 2.8
  • ...and 1 more