Stability threshold of Couette flow for the 3D MHD equations

In this paper, we consider the stability of 3D Couette flow (y, 0, 0)(inverted perpendicular) in a uniform background magnetic field alpha(sigma, 0, 1)(inverted perpendicular). In particular, the MHD equations on T x R x T that we are concerned with are of different viscosity coefficient nu and magnetic diffusion coefficient mu. It is shown that if the background magnetic field alpha(sigma, 0, 1)(inverted perpendicular) with sigma is an element of R\Q satisfying a generic Diophantine condition is so strong that |alpha| >> nu+mu/root nu mu, and the initial perturbations u(in) and b(in) satisfy ||(u(in), b(in))||HN+2 << min {nu, mu} for sufficiently large N, then the resulting solution remains close to the steady state in L-2 at the same order for all time. Compared with the result of Liss [Comm. Math. Phys., 377(2020), 859-908], we use a more general energy method to address the physically relevant case nu not equal mu based on some new observations. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.