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 ${\text{Re}}^{-1}$, ${\text{R}}_m^{-1}\ll 1$ 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 $\Vert (u_{in},b_{in}){\Vert }_{H^N}=\epsilon \ll {\text{Re}}^{-1}$, 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 $\mathcal{O}({\text{Re}}^{-1})$ in $H^{N^{\prime }}$ for some $N^{\prime }(\sigma )<N$. Our proof establishes enhanced dissipation estimates describing the decay of the $x$-dependent modes on the timescale $t\sim {\text{Re}}^{1/3}$, 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 $\epsilon \ll {\text{Re}}^{-3/2}$. 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.

41 pages

References in corpus (1)

• Nonlinear transverse cascade and two-dimensional magnetohydrodynamic subcritical turbulence in plane shear flows