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

TL;DR

This paper proves the exponential stability of large solutions to 3D compressible magnetohydrodynamic equations in bounded domains with Navier-slip boundary conditions, improving previous results by removing restrictive assumptions and establishing the first such global stability in this setting.

Contribution

It establishes the first global stability result for large strong solutions of 3D compressible MHD equations in bounded domains, removing previous restrictions on initial data and viscosity conditions.

Findings

Solutions converge exponentially to equilibrium in L^2-norm.

Density and magnetic field converge exponentially in L^∞-norm.

Results extend stability analysis to general bounded domains without previous restrictions.

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 $2\mu>\lambda$ 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.