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

In this article, we consider the stability of the 2D magnetohydrodynamics equations close to a combination of Couette flow and a constant magnetic field. We consider the ideal conductor limit for the case when viscosity ν is larger than resistivity κ , ν ⩾ κ > 0 . For this regime, we establish a bound on the Sobolev stability threshold. Furthermore, for κ ⩽ ν 3 this system exhibits instability, which leads to norm inflation of size ν κ − 1 3 .