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-$\frac{1}{s}$, $(\frac{1}{2}<s\le 1)$ and of size smaller than the resistivity coefficient $\mu$. More precisely, we prove (1) the ${\mu }^{-\frac{1}{3}}$-amplification of the perturbed vorticity, namely, the size of the vorticity grows from $\Vert {\omega }_{\mathrm{in}}{\Vert }_{{\mathcal{G}}^{{\lambda }_0}}\lesssim \mu$ to $\Vert {\omega }_{\infty }{\Vert }_{{\mathcal{G}}^{{\lambda }^{\prime }}}\lesssim {\mu }^{\frac{2}{3}}$; (2) the polynomial decay of the perturbed current density, namely, ${\Vert j_{\ne }\Vert }_{L^2}\lesssim \frac{c_0}{⟨t{⟩}^2}\min \left\{{\mu }^{-\frac{1}{3}},⟨t⟩\right\}$; (3) and the damping for the perturbed velocity and magnetic field, namely, \[ \left\|(u^1_{\neq},b^1_{\neq})\right\|_{L^2}\lesssim \frac{c_0μ}{\langle t\rangle }\min\left\{μ^{-\frac13},\langle t \rangle\right\}, \quad \left\|(u^2,b^2)\right\|_{L^2}\lesssim \frac{c_0μ}{\langle t\rangle^2 }\min\left\{μ^{-\frac13},\langle t \rangle\right\}. \] We also confirm that the strong uniform magnetic field stabilizes the Euler-MHD system near Couette flow.

74 pages