Stability threshold of Couette flow for the 3D MHD equations

Abstract

In this paper, we consider the stability of 3D Couette flow ${(y,0,0)}^{\top }$ in a uniform background magnetic field $\alpha {(\sigma ,0,1)}^{\top }$. In particular, the MHD equations on $T\times R\times T$ that we are concerned with are of different viscosity coefficient ν and magnetic diffusion coefficient μ. It is shown that if the background magnetic field $\alpha {(\sigma ,0,1)}^{\top }$ with $\sigma \in R﹨Q$ satisfying a generic Diophantine condition is so strong that $\left|\alpha \right|\gg \frac{\nu +\mu }{\sqrt{\nu \mu }}$, and the initial perturbations $u_{\mathrm{in}}$ and $b_{\mathrm{in}}$ satisfy ${\Vert (u_{\mathrm{in}},b_{\mathrm{in}})\Vert }_{H^{N+2}}\ll \min \{\nu ,\mu \}$ for sufficiently large N, then the resulting solution remains close to the steady state in $L^2$ at the same order for all time. Compared with the result of Liss [Comm. Math. Phys., 377(2020), 859–908], we use a more general energy method to address the physically relevant case $\nu \ne \mu$ based on some new observations.