Global uniform regularity for the 3D compressible MHD equations near a background magnetic field

TL;DR

This paper proves the global regularity of 3D compressible MHD equations with a background magnetic field, showing how magnetic effects stabilize the system and enable vanishing dissipation limits.

Contribution

It introduces a two-tier energy method to establish uniform bounds and decay rates, advancing understanding of magnetic stabilization in compressible MHD.

Findings

Established global-in-time uniform bounds independent of certain viscosities.

Developed a two-tier energy method coupling vertical and horizontal derivatives.

Derived explicit long-time convergence rates to dissipation-free MHD systems.

Abstract

This paper resolves the global regularity problem for the three-dimensional compressible magnetohydrodynamics (MHD) equations in the three-dimensional whole space, in the presence of a background magnetic field. Motivated by geophysical applications, we consider an anisotropic compressible MHD system with weak dissipation in the $x_2$ and $x_3$ directions and small vertical magnetic diffusion. By exploiting the stabilizing effect induced by the background magnetic field and constructing a hierarchy of four energy functionals, we establish global-in-time uniform bounds that are independent of the viscosity in the $x_2$ and $x_3$ directions and the vertical resistivity. A key innovation in our analysis is the development of a two-tier energy method, which couples the boundedness of vertical derivatives with the decay of horizontal derivatives. The analysis of time scale, together with global regularity estimates and sharp decay rates, enable us to rigorously justify the vanishing dissipation limit and derive explicit long-time convergence rates to the compressible MHD system with vanishing dissipation in the $x_2$ and $x_3$ directions and no vertical magnetic diffusion. In the absence of magnetic field and background magnetic field, the global-in-time well-posedness and vanishing viscosity limit for the 3D compressible Navier-Stokes equations with only one direction dissipation remains a challenging open problem. This work reveals the mechanism by which the magnetic field enhances dissipation and stabilizes the fluid dynamics in the global well-posedness and vanishing viscosity limit.