Global well-posedness for 2D non-resistive compressible MHD system in periodic domain

TL;DR

This paper proves the global existence, stability, and decay of smooth solutions for the 2D non-resistive compressible MHD system in a periodic domain, using energy estimates and exploiting hidden dissipation mechanisms.

Contribution

It is the first to analyze this system on bounded domains and to establish global well-posedness using pure energy methods.

Findings

Global existence of smooth solutions near background magnetic field

Decay rates and stability over large times

First analysis of such system with pure energy estimates in bounded domains

Abstract

This paper focuses on the 2D compressible magnetohydrodynamic (MHD) equations without magnetic diffusion in a periodic domain. We present a systematic approach to establishing the global existence of smooth solutions when the initial data is close to a background magnetic field. In addition, stability and large-time decay rates are also obtained. When there is no magnetic diffusion, the magnetic field and the density are governed by forced transport equations and the problem considered here is difficult. This paper implements several key observations and ideas to maximize the enhanced dissipation due to hidden structures and interactions. In particular, the weak smoothing and stabilization generated by the background magnetic field and the extra regularization in the divergence part of the velocity field are fully exploited. Compared with the previous works, this paper appears to be the first to investigate such system on bounded domains and the first to solve this problem by pure energy estimates, which help reduce the complexity in other approaches. In addition, this paper combines the well-posedness with the precise large-time behavior, a strategy that can be extended to higher dimensions.