The stability threshold for 2D MHD equations around Couette with general viscosity and magnetic resistivity

Abstract

We address a threshold problem of the Couette flow $(y,0)$ in a uniform magnetic field $({\hat{I}}^2,0)$ for the 2D MHD equation on $\mathbb{T}\times \mathbb{R}$ with fluid viscosity $\hat{I}½$ and magnetic resistivity $\hat{I}¼$. The nonlinear enhanced dissipation and inviscid damping are also established. In particularly, when $0<\hat{I}½\le \hat{I}{¼}^3\le 1$, we get a threshold $\hat{I}{½}^{\frac{1}{2}}\hat{I}{¼}^{\frac{1}{3}}$ in $H^N(N\ge 4)$. When $0<\hat{I}{¼}^3\le \hat{I}½\le 1$, we obtain a threshold $\min \{\hat{I}{½}^{\frac{1}{2}},\hat{I}{¼}^{\frac{1}{2}}\}\min \{1,\hat{I}{½}^{-1}\hat{I}{¼}^{\frac{1}{3}}\}$, hence improving the results in [19,14,21].