Sobolev stability for the 2D MHD equations in the non-resistive limit

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 $\nu\kappa^{-\frac{1}{3}}$.