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

TL;DR

This paper proves the Sobolev stability of 3D Couette flow in a homogeneous magnetic field under ideal MHD conditions, showing enhanced dissipation and inviscid damping for small perturbations, with magnetic effects suppressing transient growth.

Contribution

It demonstrates the stability of 3D Couette flow in MHD with a magnetic field, improving previous results by allowing larger perturbations thanks to magnetic oscillations.

Findings

Global stability for small initial perturbations in Sobolev space.

Enhanced dissipation at timescale t ~ Re^{1/3}.

Inviscid damping matching linear decay rates.

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.