On the Sobolev stability threshold for shear flows near Couette in 2D MHD equations

In this work, we study the Sobolev stability of shear flows near Couette in the 2D incompressible magnetohydrodynamics (MHD) equations with background magnetic field $(\alpha,0 )^\top$ on $\mathbb {T}\times \mathbb {R}$ . More precisely, for sufficiently large $\alpha$ , we show that when the initial datum of the shear flow satisfies $\left \| U(y)-y\right \|_{H^{N+6}}\ll 1$ , with $N>1$ , and the initial perturbations ${u}_{\mathrm {in}}$ and ${b}_{\mathrm {in}}$ satisfy $\left \| ( {u}_{\mathrm {in}},{b}_{\mathrm {in}}) \right \| _{H^{N+1}}=\epsilon \ll \nu ^{\frac 56+\tilde \delta }$ for any fixed $\tilde \delta >0$ , then the solution of the 2D MHD equations remains $\nu ^{-(\frac {1}{3}+\frac {\tilde \delta }{2})}\epsilon$ -close to $( e^{\nu t \partial _{yy}}U(y),0)^\top$ for all $t>0$ .