The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field

Abstract

We address a stability threshold problem of the Couette flow $(y,0,0)$ in a uniform magnetic fleld $\alpha(\sigma,0,1)$ with $\sigma\in\mathbb{Q}$ for the 3D MHD equations on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. Previously, the authors in \cite{L20,RZZ25} obtained the threshold $\gamma=1$ for $\sigma\in\mathbb{R}\backslash\mathbb{Q}$ satisfying a generic Diophantine condition, where they also proved $\gamma = 4/3$ for a general $\sigma\in\mathbb{R}$. In the present paper, we obtain the threshold $\gamma=1$ in $H^N(N>13/2)$, hence improving the above results when $\sigma$ is a rational number. The nonlinear inviscid damping for velocity $u^2_{\neq}$ is also established. Moreover, our result shows that the nonzero modes of magnetic field has an amplification of order $\nu^{-1/3}$ even on low regularity, which is very different from the case considered in \cite{L20,RZZ25}.