Asymptotic Stability of Couette Flow in a Strong Uniform Magnetic Field for the Euler-MHD System

Abstract

In this paper, we prove the asymptotic stability of Couette flow in a strong uniform magnetic field for the Euler-MHD system, when the perturbations are in Gevrey-\(\frac{1}{s}\), \((\frac{1}{2}<s\leqq 1)\) and of size smaller than the resistivity coefficient \(\mu \). More precisely, we prove that

1. (1)

   the \(\mu ^{-\frac{1}{3}}\)-amplification of the perturbed vorticity, namely, the size of the vorticity grows from \(\Vert \omega _{\textrm{in}}\Vert _{\mathcal {G}^{\lambda _{0}}}\lesssim \mu \) to \(\Vert \omega _{\infty }\Vert _{\mathcal {G}^{\lambda '}}\lesssim \mu ^{\frac{2}{3}}\);

2. (2)

   the polynomial decay of the perturbed current density, namely, \(\left\| j_{\ne }\right\| _{L^2}\lesssim \frac{c_0 }{\langle t\rangle ^2 }\min \left\{ \mu ^{-\frac{1}{3}},\langle t \rangle \right\} \);

3. (3)

   and the damping for the perturbed velocity and magnetic field, namely,

   $$\begin{aligned} \left\| (u^1_{\ne },b^1_{\ne })\right\| _{L^2}\lesssim \frac{c_0\mu }{\langle t\rangle }\min \left\{ \mu ^{-\frac{1}{3}},\langle t \rangle \right\} , \quad \left\| (u^2,b^2)\right\| _{L^2}\lesssim \frac{c_0\mu }{\langle t\rangle ^2 }\min \left\{ \mu ^{-\frac{1}{3}},\langle t \rangle \right\} . \end{aligned}$$

We also confirm that the strong uniform magnetic field stabilizes the Euler-MHD system near Couette flow.