Sobolev Stability for the 2D MHD Equations in the Non-Resistive Limit

TL;DR

This paper investigates the stability of 2D MHD equations near Couette flow with a constant magnetic field, establishing Sobolev stability bounds in the ideal conductor limit and identifying conditions for instability and norm inflation.

Contribution

It provides the first Sobolev stability threshold for the 2D MHD equations in the non-resistive limit and characterizes the instability regime when resistivity is sufficiently small.

Findings

Established Sobolev stability bounds for the 2D MHD system.

Identified instability conditions leading to norm inflation.

Demonstrated the impact of resistivity on stability in the ideal conductor limit.

Abstract

In this article, we consider the stability of the 2D magnetohydrodynamics (MHD) equations close to a combination of Couette flow and a constant magnetic field. We consider the ideal conductor limit for the case when viscosity $\nu$ is larger than resistivity $\kappa$, $\nu\ge \kappa>0$. For this regime, we establish a bound on the Sobolev stability threshold. Furthermore, for $\kappa\le \nu^3$ this system exhibits instability, which leads to norm inflation of size $\nu \kappa^{-\frac 1 3 }$.