Stability threshold of the 2D Couette flow in a homogeneous magnetic field

A planar incompressible and electrically conducting fluid can be described by the 2D Navier-Stokes-MHD system. One simple yet physically relevant laminar state is the Couette flow with a constant homogeneous magnetic field, given by u_E=(y,0), B_E=(b,0) in the domain TxR. The goal is to estimate how large can be a perturbation of this state while still resulting in a solution close to the laminar regime, thereby preventing the onset of turbulence. We prove that Sobolev regular initial perturbations of size O(Re^{-2/3}), with Re being the Reynolds number, remain close to u_E, B_E and exhibit dissipation enhancement. The latter quantifies the convergence towards an x-independent state on a time-scale O(Re^{-1/3}), much faster than the standard diffusive one O(Re^{-1}).