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 \({\mathbb {T}}\times {\mathbb {R}}\) $\mathbb{T}\times \mathbb{R}$ . The aim is to quantify stability properties of the Couette flow (y, 0) with a constant homogenous magnetic field \((\beta ,0)\) $(\beta ,0)$ when \(|\beta |>1/2\) $|\beta |>1/2$ . The focus lies on the regime with small fluid viscosity \(\nu \) $\nu$ , magnetic resistivity \(\mu \) $\mu$ and we assume that the magnetic Prandtl number satisfies \(\mu ^2\lesssim \textrm{Pr}_{\textrm{m}}=\nu /\mu \le 1\) ${\mu }^2\lesssim {\text{Pr}}_{\text{m}}=\nu /\mu \le 1$ . We establish that small perturbations around this steady state remain close to it, provided their size is of order \(\varepsilon \ll \nu ^\frac{2}{3}\) $\epsilon \ll {\nu }^{\frac{2}{3}}$ in \(H^N\) $H^N$ with N large enough. Additionally, the vorticity and current density experience a transient growth of order \(\nu ^{-\frac{1}{3}}\) ${\nu }^{-\frac{1}{3}}$ while converging exponentially fast to an x-independent state after a time-scale of order \(\nu ^{-\frac{1}{3}}\) ${\nu }^{-\frac{1}{3}}$ . 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.