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Global weak solutions and incompressible limit to the isentropic compressible magnetohydrodynamic equations in 2D bounded domains with ripped density and large initial data

Shuai Wang, Guochun Wu, Xin Zhong

Abstract

In our previous work (arXiv:2510.00812), we have shown the global existence and incompressible limit of weak solutions to the isentropic compressible magnetohydrodynamic equations involving ripped density and large initial energy in the whole plane. In this paper we generalize such results to the case of two-dimensional bounded convex domains under Navier-slip boundary conditions. When comparing to the known results for global solutions of the initial-boundary value problem, we obtain uniform a priori estimates independent of the bulk viscosity coefficient.

Global weak solutions and incompressible limit to the isentropic compressible magnetohydrodynamic equations in 2D bounded domains with ripped density and large initial data

Abstract

In our previous work (arXiv:2510.00812), we have shown the global existence and incompressible limit of weak solutions to the isentropic compressible magnetohydrodynamic equations involving ripped density and large initial energy in the whole plane. In this paper we generalize such results to the case of two-dimensional bounded convex domains under Navier-slip boundary conditions. When comparing to the known results for global solutions of the initial-boundary value problem, we obtain uniform a priori estimates independent of the bulk viscosity coefficient.

Paper Structure

This paper contains 10 sections, 16 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Strategy of the proof
  5. Preliminaries
  6. Auxiliary results and inequalities
  7. Uniform estimates for $F$, $\boldsymbol\omega$, $\nabla\mathbf{u}$, and $\dot{\mathbf{u}}$
  8. A priori estimates
  9. Proof of Theorem \ref{['t1.1']}
  10. Proof of Theorem \ref{['t1.2']}

Key Result

Theorem 1.1

Let c1 and c2 be satisfied, there exists a positive number $D$ depending only on $\Omega$, $\hat{\rho}$, $a$, $\gamma$, $\nu$, and $\mu$ such that if then the initial-boundary value problem a1--a3 admits a global weak solution $(\rho,\mathbf{u},\mathbf{B})$ in the sense of Definition d1.1 satisfying where $\sigma\triangleq\min\{1,t\}$.

Theorems & Definitions (30)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 20 more