Stability Threshold of the 2D Couette Flow in a Homogeneous Magnetic Field Using Symmetric Variables

We consider a 2D incompressible and electrically conducting fluid in the domain T x R. The aim is to quantify stability properties of the Couette flow (y, 0) with a constant homogenous magnetic field (beta, 0) when |beta|>1/2. The focus lies on the regime with small fluid viscosity nu, magnetic resistivity mu and we assume that the magnetic Prandtl number satisfies mu(2 )less than or similar to Pr-m = nu/mu <= 1. We establish that small perturbations around this steady state remain close to it, provided their size is of order epsilon << nu(2/3) in H(N )with N large enough. Additionally, the vorticity and current density experience a transient growth of order nu(-1/3)while converging exponentially fast to an x-independent state after a time-scale of order nu(-1/3). The growth is driven by an inviscid mechanism, while the subsequent exponential decay results from the interplay between transport and diffusion, leading to the dissipation enhancement. A key argument to prove these results is to reformulate the system in terms of symmetric variables, inspired by the study of inhomogeneous fluid, to effectively characterize the system's dynamic behavior.