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

Abstract

We consider a 2D incompressible and electrically conducting fluid in the domain $${\mathbb {T}}\times {\mathbb {R}}$$ T × R . The aim is to quantify stability properties of the Couette flow (y, 0) with a constant homogenous magnetic field $$(\beta ,0)$$ ( β , 0 ) when $$|\beta |>1/2$$ | β | > 1 / 2 . The focus lies on the regime with small fluid viscosity $$\nu $$ ν , magnetic resistivity $$\mu $$ μ and we assume that the magnetic Prandtl number satisfies $$\mu ^2\lesssim \textrm{Pr}_{\textrm{m}}=\nu /\mu \le 1$$ 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 $$\varepsilon \ll \nu ^\frac{2}{3}$$ ε ≪ ν 2 3 in $$H^N$$ H N with N large enough. Additionally, the vorticity and current density experience a transient growth of order $$\nu ^{-\frac{1}{3}}$$ ν - 1 3 while converging exponentially fast to an x-independent state after a time-scale of order $$\nu ^{-\frac{1}{3}}$$ 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.