Asymptotic stability of Couette flow in a strong uniform magnetic field for the Euler-MHD system

TL;DR

This paper proves the asymptotic stability of Couette flow in a strong magnetic field within the Euler-MHD system, demonstrating vorticity amplification, current decay, and velocity-magnetic damping under certain conditions.

Contribution

It establishes the first rigorous proof of stability and detailed behavior of perturbations for Couette flow in the Euler-MHD system with a strong magnetic field.

Findings

Vorticity grows from order μ to μ^{2/3}

Current density decays polynomially as 1/⟨t⟩^2

Velocity and magnetic field perturbations decay over time

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-$\frac{1}{s}$, $(\frac12<s\leq 1)$ and of size smaller than the resistivity coefficient $\mu$. More precisely, we prove (1) the $\mu^{-\frac13}$-amplification of the perturbed vorticity, namely, the size of the vorticity grows from $\|\omega_{\mathrm{in}}\|_{\mathcal{G}^{\lambda_{0}}}\lesssim \mu$ to $\|\omega_{\infty}\|_{\mathcal{G}^{\lambda'}}\lesssim \mu^{\frac23}$; (2) the polynomial decay of the perturbed current density, namely, $\left\|j_{\neq}\right\|_{L^2}\lesssim \frac{c_0 }{\langle t\rangle^2 }\min\left\{\mu^{-\frac13},\langle t \rangle\right\}$; (3) and the damping for the perturbed velocity and magnetic field, namely, \[ \left\|(u^1_{\neq},b^1_{\neq})\right\|_{L^2}\lesssim \frac{c_0\mu }{\langle t\rangle }\min\left\{\mu^{-\frac13},\langle t \rangle\right\}, \quad \left\|(u^2,b^2)\right\|_{L^2}\lesssim \frac{c_0\mu }{\langle t\rangle^2 }\min\left\{\mu^{-\frac13},\langle t \rangle\right\}. \] We also confirm that the strong uniform magnetic field stabilizes the Euler-MHD system near Couette flow.