On the Sobolev Stability Threshold of 3D Couette Flow in a Uniform Magnetic Field

Abstract

We study the stability of the Couette flow $(y,0,0)^T$ in the presence of a uniform magnetic field $\alpha (\sigma , 0, 1)$ on ${{\mathbb {T}}}\times {{\mathbb {R}}}\times {{\mathbb {T}}}$ using the 3D incompressible magnetohydrodynamics (MHD) equations. We consider the inviscid, ideal conductor limit $\mathbf{Re} ^{-1}$, $\mathbf{R }_m^{-1} \ll 1$ 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 $\mathbf{Re} ^{-1} = \mathbf{R }_m^{-1} \in (0,1]$, $\alpha > 0$ and $N > 0$ are sufficiently large, $\sigma \in {{\mathbb {R}}}{\setminus } {\mathbb {Q}}$ satisfies a generic Diophantine condition, and the initial perturbations $u_{\text{ in }}$ and $b_{\text{ in }}$ to the Couette flow and magnetic field, respectively, satisfy $\Vert u_{\text{ in }}\Vert _{H^N} + \Vert b_{\text{ in }}\Vert _{H^N} = \epsilon \ll \mathbf{Re} ^{-1}$, then the resulting solution to the 3D MHD equations is global in time and the perturbations $u(t,x+yt,y,z)$ and $b(t,x+yt,y,z)$ remain ${\mathcal {O}}(\mathbf{Re} ^{-1})$ in $H^{N'}$ for some $1 \ll N'(\sigma ) < N$. Our proof establishes enhanced dissipation estimates describing the decay of the x-dependent modes on the timescale $t \sim \mathbf{Re} ^{1/3}$, as well as inviscid damping of the velocity and magnetic field with a rate that agrees with the prediction of the linear stability analysis. 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 $\epsilon \ll \mathbf{Re} ^{-3/2}$ (Bedrossian et al. in Ann. Math. 185(2): 541–608, 2017). 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 Couette flow.