On the Sobolev stability threshold of 3D Couette flow in a homogeneous magnetic field

Abstract

We study the stability of the Couette flow $(y,0,0)^T$ in the 3D incompressible magnetohydrodynamic (MHD) equations for a conducting fluid on $\mathbb{T} \times \mathbb{R} \times \mathbb{T} $ in the presence of a homogeneous magnetic field $\alpha(\sigma, 0, 1)$. We consider the inviscid, ideal conductor limit $\textbf{Re}^{-1}$, $\textbf{R}_m^{-1} \ll 1$ and prove that for strong and suitably oriented background fields the Couette flow is asymptotically stable to perturbations small in the Sobolev space $H^N$. More precisely, we show that if $\alpha$ and $N$ are sufficiently large, $\sigma \in \mathbb{R} \setminus \mathbb{Q}$ satisfies a generic Diophantine condition, and the initial perturbations $u_{in}$ and $b_{in}$ to the Couette flow and magnetic field, respectively, satisfy $\|(u_{in},b_{in})\|_{H^N} = \epsilon \ll \textbf{Re}^{-1}$, then the resulting solution to the 3D MHD equations is global in time and the perturbation $(u(t,x+yt,y,z),b(t,x+yt,y,z))$ remains $\mathcal{O}(\textbf{Re}^{-1})$ in $H^{N'}$ for some $N'(\sigma) < N$. Our proof establishes enhanced dissipation estimates describing the decay of the $x$-dependent modes on the timescale $t \sim \textbf{Re}^{1/3}$, as well as inviscid damping of the velocity and magnetic field that agrees with the optimal decay rate for the linearized system. In the Navier-Stokes case, high regularity control on the perturbation in a coordinate system adapted to the mixing of the Couette flow is known only under the stronger assumption $\epsilon \ll \textbf{Re}^{-3/2}$. The improvement in the MHD setting is possible because the magnetic field induces time oscillations that partially suppress the lift-up effect, which is the primary transient growth mechanism for the Navier-Stokes equations linearized around the Couette flow.