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× R T × R . The aim is to quantify stability properties of the Couette flow ( y , 0) with a constant homogenous magnetic field (β ,0) ( β , 0 ) when |β |>1/2 | β | > 1 / 2 . The focus lies on the regime with small fluid viscosity ν ν , magnetic resistivity μ μ and we assume that the magnetic Prandtl number satisfies μ ² Prₘ=ν /μ ≤ 1 μ 2 ≲ Pr m = ν / μ ≤ 1 . We establish that small perturbations around this steady state remain close to it, provided their size is of order ε ν ²/3 ε ≪ ν 2 3 in HN H N with N large enough. Additionally, the vorticity and current density experience a transient growth of order ν -1/3 ν - 1 3 while converging exponentially fast to an x -independent state after a time-scale of order ν -1/3 ν - 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.