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Global stability for the compressible isentropic magnetohydrodynamic equations in 3D bounded domains with Navier-slip boundary conditions

Yang Liu, Guochun Wu, Xin Zhong

TL;DR

The paper establishes global stability for large strong solutions to the 3D compressible isentropic MHD equations in general bounded domains with Navier-slip boundary conditions. It proves exponential decay in $L^2$ for $(\rho-\bar{\rho}_0, \sqrt{\rho}u, H)$ when the density is uniformly bounded in time, and, with a positive lower bound on the initial density, exponential decay in $L^{\infty}$ for $(\rho-\bar{\rho}_0, H)$. The authors develop a Lyapunov framework combining energy methods, the Bogovskii operator, and div-curl estimates, and leverage the effective viscous flux $F$ and the magnetic field structure to control higher-order norms and boundary contributions. This work extends global stability results to 3D bounded domains, removing prior torus-specific assumptions and certain viscous-coefficient restrictions, and provides the first global stability result for large strong MHD solutions in this setting.

Abstract

We study the global stability of large solutions to the compressible isentropic magnetohydrodynamic equations in a three-dimensional (3D) bounded domain with Navier-slip boundary conditions. It is shown that the solutions converge to an equilibrium state exponentially in the $L^2$-norm provided the density is essentially uniform-in-time bounded from above. Moreover, we also obtain that the density and magnetic field converge to their equilibrium states exponentially in the $L^\infty$-norm if additionally the initial density is bounded away from zero. These greatly improve the previous work in (J. Differential Equations 288 (2021), 1-39), where the authors considered the torus case and required the $L^6$-norm of the magnetic field to be uniformly bounded as well as zero initial total momentum and an additional restriction $2μ>λ$ for the viscous coefficients. This paper provides the first global stability result for large strong solutions of compressible magnetohydrodynamic equations in 3D general bounded domains.

Global stability for the compressible isentropic magnetohydrodynamic equations in 3D bounded domains with Navier-slip boundary conditions

TL;DR

The paper establishes global stability for large strong solutions to the 3D compressible isentropic MHD equations in general bounded domains with Navier-slip boundary conditions. It proves exponential decay in for when the density is uniformly bounded in time, and, with a positive lower bound on the initial density, exponential decay in for . The authors develop a Lyapunov framework combining energy methods, the Bogovskii operator, and div-curl estimates, and leverage the effective viscous flux and the magnetic field structure to control higher-order norms and boundary contributions. This work extends global stability results to 3D bounded domains, removing prior torus-specific assumptions and certain viscous-coefficient restrictions, and provides the first global stability result for large strong MHD solutions in this setting.

Abstract

We study the global stability of large solutions to the compressible isentropic magnetohydrodynamic equations in a three-dimensional (3D) bounded domain with Navier-slip boundary conditions. It is shown that the solutions converge to an equilibrium state exponentially in the -norm provided the density is essentially uniform-in-time bounded from above. Moreover, we also obtain that the density and magnetic field converge to their equilibrium states exponentially in the -norm if additionally the initial density is bounded away from zero. These greatly improve the previous work in (J. Differential Equations 288 (2021), 1-39), where the authors considered the torus case and required the -norm of the magnetic field to be uniformly bounded as well as zero initial total momentum and an additional restriction for the viscous coefficients. This paper provides the first global stability result for large strong solutions of compressible magnetohydrodynamic equations in 3D general bounded domains.

Paper Structure

This paper contains 3 sections, 11 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.1

Assume that the initial data $(\rho_0\ge 0, u_0, H_0)$ satisfies Let $(\rho, u, H)$ be a global strong solution to the problem a1--a4 verifying that for some positive constant $\hat{\rho}$. Then there exist two positive constants $C_1$ and $\eta_1$, which are dependent on $\hat{\rho}$ and $K$, but independent of $t$, such that If additionally $\inf\limits_{x\in\Omega}\rho_0(x)\ge \rho_*>0$, the

Theorems & Definitions (20)

  • Definition 1.1: Strong solutions
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 10 more