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

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 $\mathrm{\parallel }{\omega }_{\mathrm{i}\mathrm{n}}{\mathrm{\parallel }}_{{\mathcal{G}}^{{\lambda }_0}}\lesssim \mu$ to $\mathrm{\parallel }{\omega }_{\mathrm{\infty }}{\mathrm{\parallel }}_{{\mathcal{G}}^{{\lambda }^{\mathrm{\prime }}}}\lesssim {\mu }^{\frac{2}{3}}$; (2) the polynomial decay of the perturbed current density, namely, ${\parallel j_{\mathrm{\ne }}\parallel }_{L^2}\lesssim \frac{c_0}{⟨t{⟩}^2}\min \{{\mu }^{-\frac{1}{3}},⟨t⟩\}$; (3) and the damping for the perturbed velocity and magnetic field, namely, $${\parallel (u_{\mathrm{\ne }}^1,b_{\mathrm{\ne }}^1)\parallel }_{L^2}\lesssim \frac{c_0\mu }{⟨t⟩}\min \{{\mu }^{-\frac{1}{3}},⟨t⟩\},{\parallel (u^2,b^2)\parallel }_{L^2}\lesssim \frac{c_0\mu }{⟨t{⟩}^2}\min \{{\mu }^{-\frac{1}{3}},⟨t⟩\}\mathrm{.}$$ We also confirm that the strong uniform magnetic field stabilizes the Euler-MHD system near Couette flow.