Global small solutions to the 3D compressible viscous non-resistive MHD system

TL;DR

This paper proves the global existence and stability of smooth solutions to the 3D compressible viscous non-resistive MHD equations near certain background magnetic fields, addressing a longstanding open problem in mathematical physics.

Contribution

It establishes the first global well-posedness and stability results for this challenging 3D MHD system in the whole space or periodic domain.

Findings

Global existence of smooth solutions near Diophantine magnetic fields

Stability results for solutions without magnetic diffusion

Addresses an open problem in 3D MHD theory

Abstract

Whether or not smooth solutions to the 3D compressible magnetohydrodynamic (MHD) equations without magnetic diffusion are always global in time remains an extremely challenging open problem. No global well-posedness or stability result is currently available for this 3D MHD system in the whole space $\mathbb R^3$ or the periodic box $\mathbb T^3$ even when the initial data is small or near a steady-state solution. This paper presents a global existence and stability result for smooth solutions to this 3D MHD system near any background magnetic field satisfying a Diophantine condition.