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

Theorems & Definitions (25)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Theorem 1.3
• Remark 1.3
• Lemma 2.1
• proof
• Proposition 3.1
• Proposition 3.2