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Global solutions to 3D compressible MHD equations with partial magnetic diffusion

Jiahong Wu, Xiaoping Zhai

TL;DR

The paper addresses global well-posedness for the 3D compressible viscous MHD equations on $\mathbb{R}^3$ with partial magnetic diffusion acting horizontally, a setting where density has no dissipative mechanism. It develops a rigorous a priori framework based on anisotropic Sobolev inequalities and energy methods, including a reformulation with $I(a)$ and $J(a)$ to control nonlinearities and a coupling mechanism between density and velocity that stabilizes the density. The authors prove the existence of a unique global strong solution for small initial data in $H^3$, with precise regularity and dissipative bounds, by combining basic $L^2$ energy estimates and higher-order $H^3$ estimates, and then extending the local solution via a continuation argument. This work advances understanding of compressible MHD with partial diffusion in unbounded domains and highlights the effectiveness of anisotropic analysis in overcoming the lack of full diffusion.

Abstract

The global existence of strong solutions to the compressible viscous magnetohydrodynamic (MHD) equations in $\mathbb{R}^3$ remains a significant open problem. When there is no magnetic diffusion, even small data global well-posedness is unknown. This study investigates the Cauchy problem in $\mathbb{R}^3$ for the compressible viscous MHD equations with horizontal magnetic diffusion. Using various anisotropic Sobolev inequalities and sharp estimates, we establish the existence of global solutions under small initial data within the Sobolev space framework.

Global solutions to 3D compressible MHD equations with partial magnetic diffusion

TL;DR

The paper addresses global well-posedness for the 3D compressible viscous MHD equations on with partial magnetic diffusion acting horizontally, a setting where density has no dissipative mechanism. It develops a rigorous a priori framework based on anisotropic Sobolev inequalities and energy methods, including a reformulation with and to control nonlinearities and a coupling mechanism between density and velocity that stabilizes the density. The authors prove the existence of a unique global strong solution for small initial data in , with precise regularity and dissipative bounds, by combining basic energy estimates and higher-order estimates, and then extending the local solution via a continuation argument. This work advances understanding of compressible MHD with partial diffusion in unbounded domains and highlights the effectiveness of anisotropic analysis in overcoming the lack of full diffusion.

Abstract

The global existence of strong solutions to the compressible viscous magnetohydrodynamic (MHD) equations in remains a significant open problem. When there is no magnetic diffusion, even small data global well-posedness is unknown. This study investigates the Cauchy problem in for the compressible viscous MHD equations with horizontal magnetic diffusion. Using various anisotropic Sobolev inequalities and sharp estimates, we establish the existence of global solutions under small initial data within the Sobolev space framework.
Paper Structure (5 sections, 1 theorem, 86 equations)

This paper contains 5 sections, 1 theorem, 86 equations.

Table of Contents

  1. Introduction and main result
  2. Proof of Theorem \ref{['dingli1']}
  3. Basic energy estimates
  4. Higher order energy estimates
  5. Completing the proof of Theorem \ref{['dingli1']}

Key Result

Theorem 1.1

For any $(\rho_0-\bar{\rho}, \mathbf{u} _0, \mathbf{B} _0) \in H^3(\mathbb{R}^3)$, there exists a constant $\varepsilon>0$ such that if then system m1 with long has a unique global strong solution $(\rho-\bar{\rho}, \mathbf{u} , \mathbf{B} )$ such that Moreover, for a constant $C>0$ and any $t > 0$,

Theorems & Definitions (3)

  • Theorem 1.1
  • proof
  • proof