Global existence and stability of solutions for the 2D non-resistive compressible MHD system

TL;DR

This paper proves the global existence and stability of solutions for the 2D non-resistive compressible MHD equations using pure energy estimates, without requiring initial data in $L^1$ or negative Sobolev spaces.

Contribution

It introduces new techniques such as effective viscous flux quantities and a pseudo-negative-derivative method to establish stability solely through $H^s$ energy estimates.

Findings

Established global classical solutions near equilibrium in 2D non-resistive compressible MHD.

Developed a novel approach bypassing traditional decay assumptions like $L^1$ integrability.

Closed higher-order energy estimates within standard Sobolev spaces.

Abstract

This paper investigates the non-resistive compressible magnetohydrodynamic (MHD) equations in $\mathbb{R}^2$. We establish the global existence and stability of classical solutions for initial data sufficiently close to a constant equilibrium state. A distinguishing feature of our result is that global stability is derived solely from pure $H^s$ energy estimate and intrinsic $L^2$ time-decay mechanism, thereby bypassing the traditional initial data requirement of $L^1$ integrability or negative-order Sobolev norm regularity. To achieve this goal, firstly we introduce some quantities motivated by effective viscous flux, which intrinsically couples density and magnetic field perturbation. Secondly, to overcome the critical time-decay obstacle arising from the absence of negative-index regularity, we develop a novel pseudo-negative-derivative technique. Moreover, we regard the wildest nonlinear term as a whole and abandon obtaining time decay estimate for each item. These approaches enable us to close the higher-order energy estimate entirely within standard Sobolev spaces.