Sobolev Stability for the 2D MHD Equations in the Non-Resistive 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 $\nu {\kappa }^{-\frac{1}{3}}$.

32 pages, 1 figure, comments welcome