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Global weak solutions of 3D compressible magnetohydrodynamic equations subject to large external potential forces with discontinuous initial data and vacuum

Geyuan Chen, Xin Zhong

TL;DR

The paper proves the global existence of weak solutions for the 3D isentropic compressible MHD equations in a bounded domain under large external potential forces, allowing vacuum and large oscillations as long as the initial energy is small. It develops novel a priori estimates via the effective viscous flux, handles boundary contributions with Navier-slip conditions, and employs Zlotnik's inequality to obtain uniform density bounds. By constructing smooth approximations and passing to the limit, it extends prior Cauchy results to bounded domains and demonstrates long-time decay toward equilibrium. These results advance the mathematical theory of MHD with external forcing and vacuum in physically relevant geometries, with potential applications to geophysical and astrophysical flows.

Abstract

We investigate the compressible magnetohydrodynamic equations subject to large external potential forces with discontinuous initial data in a three-dimensional bounded domain under Navier-slip boundary conditions. We show the global existence of weak solutions for such an initial-boundary value problem provided the initial energy is suitably small. In particular, the initial data may contain vacuum states and possibly exhibit large oscillations. To overcome difficulties brought by boundary and large external forces, some new estimates based on the effective viscous flux play crucial roles.

Global weak solutions of 3D compressible magnetohydrodynamic equations subject to large external potential forces with discontinuous initial data and vacuum

TL;DR

The paper proves the global existence of weak solutions for the 3D isentropic compressible MHD equations in a bounded domain under large external potential forces, allowing vacuum and large oscillations as long as the initial energy is small. It develops novel a priori estimates via the effective viscous flux, handles boundary contributions with Navier-slip conditions, and employs Zlotnik's inequality to obtain uniform density bounds. By constructing smooth approximations and passing to the limit, it extends prior Cauchy results to bounded domains and demonstrates long-time decay toward equilibrium. These results advance the mathematical theory of MHD with external forcing and vacuum in physically relevant geometries, with potential applications to geophysical and astrophysical flows.

Abstract

We investigate the compressible magnetohydrodynamic equations subject to large external potential forces with discontinuous initial data in a three-dimensional bounded domain under Navier-slip boundary conditions. We show the global existence of weak solutions for such an initial-boundary value problem provided the initial energy is suitably small. In particular, the initial data may contain vacuum states and possibly exhibit large oscillations. To overcome difficulties brought by boundary and large external forces, some new estimates based on the effective viscous flux play crucial roles.

Paper Structure

This paper contains 4 sections, 17 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A prior estimates
  4. Proof of Theorem 1.1

Key Result

Lemma 1.1

Assume that $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary and $\phi$ satisfies then there exists a unique solution $(\rho_s, 0, 0)$ of 1.8 such that where $\underline{\rho}$ and $\bar{\rho}$ are positive constants depending only on $A$, $\gamma$, $\inf\limits_{\Omega} \phi$, and $\sup\limits_{\Omega} \phi$. In addition, if $\phi \in W^{2, q}$ for some $q \in (3, 6)$, then

Theorems & Definitions (31)

  • Lemma 1.1: LY
  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 21 more