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.