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Global Well-posedness for 2D non-resistive MHD equations in half-space

Zhaoyun Zhang, Xiaopeng Zhao

Abstract

This paper focuses on the initial boundary value problem of two-dimensional non-resistive MHD equations in a half space. We prove that the MHD equations have a unique global strong solution around the equilibrium state $(0,\bf{e_1})$ for Dirichlet boundary condition of velocity and modified Neumann boundary condition of magnetic.

Global Well-posedness for 2D non-resistive MHD equations in half-space

Abstract

This paper focuses on the initial boundary value problem of two-dimensional non-resistive MHD equations in a half space. We prove that the MHD equations have a unique global strong solution around the equilibrium state for Dirichlet boundary condition of velocity and modified Neumann boundary condition of magnetic.
Paper Structure (4 sections, 7 theorems, 135 equations)

This paper contains 4 sections, 7 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Result
  3. Global well-posedness
  4. Time asymptotic estimates

Key Result

Lemma 2.1

(1-1)2011B2014P Let $f\in H^1({\mathbb{R}}^2_+)$. Then, for any $2<p<\infty$ and there is $C=C(p)$, (1-2)2011B For $m\geq 1$ is an integer and $p\in [1,+\infty)$, we have and (2)1951H For any $f=(f^1,f^2)\in H^2({\mathbb{R}}^2_+)$ with $\nabla\cdot f=0$ in ${\mathbb{R}}^2_+$ and $f^2=0$ on $\partial {\mathbb{R}}^2_+$. There exists a positive constant C such that

Theorems & Definitions (9)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.1
  • Remark 2.1
  • Theorem 3.1
  • Proposition 3.1
  • proof