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

TL;DR

This paper investigates the stability threshold for 2D magnetohydrodynamic (MHD) equations around Couette flow with general viscosity and magnetic resistivity, establishing enhanced dissipation, inviscid damping, and improved thresholds.

Contribution

It derives new stability thresholds for 2D MHD equations with general viscosity and resistivity, extending and improving previous results in the literature.

Findings

Established nonlinear enhanced dissipation and inviscid damping.

Derived stability thresholds depending on viscosity and resistivity regimes.

Improved previous stability results for the 2D MHD equations.

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].