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 $(\beta,0)$ for the 2D MHD equation on $\mathbb{T}\times\mathbb{R}$ with fluid viscosity $\nu$ and magnetic resistivity $\mu$. The nonlinear enhanced dissipation and inviscid damping are also established. In particularly, when $0<\nu\leq\mu^3\leq1$, we get a threshold $\nu^{\frac{1}{2}}\mu^{\frac{1}{3}}$ in $H^N(N\geq4)$. When $0<\mu^3\leq\nu\leq1$, we obtain a threshold $\min\{\nu^{\frac{1}{2}},\mu^{\frac{1}{2}}\}\min\{1,\nu^{-1}\mu^{\frac{1}{3}}\}$, hence improving the results in [19,14,21].