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].
Categories: math.AP