Global well-posedness for the three-dimensional full compressible viscous non-resistive MHD system

TL;DR

This paper proves the global well-posedness and exponential convergence to steady state for the 3D full compressible viscous non-resistive MHD system around a strong magnetic field, extending previous results to heat-conductive fluids.

Contribution

It extends the analysis of MHD systems to include heat conduction and establishes global existence and stability results using a two-tier energy method reformulated in Lagrangian coordinates.

Findings

Global well-posedness around a strong magnetic field

Exponential convergence to steady state

Stabilization effects of vertical magnetic field

Abstract

In this paper, we consider the three-dimensional full compressible viscous non-resistive MHD system. Global well-posedness is proved for an initial-boundary value problem around a strong background magnetic field. It is also shown that the unique solution converges to the steady state at an almost exponential rate as time tends to infinity. The proof is based on the celebrated two-tier energy method, due to Guo and Tice [\emph{Arch. Ration. Mech. Anal.}, 207 (2013), pp. 459--531; \emph{Anal. PDE.}, 6 (2013), pp. 287--369], reformulated in Lagrangian coordinates. The obtained result may be viewed as an extension of Tan and Wang [\emph{SIAM J. Math. Anal.}, 50 (2018), pp. 1432--1470] to the context of heat-conductive fluids. This in particular verifies the stabilization effects of vertical magnetic field in the full compressible non-resistive fluids.