Transition and Stability of 3D MHD Around Couette Flow

TL;DR

This paper investigates the stability and transition phenomena of 3D incompressible MHD equations near Couette flow with a perpendicular magnetic field, revealing sharp thresholds and destabilizing effects in Sobolev spaces.

Contribution

It identifies the nonlinear transient-growth regime and establishes optimal stability thresholds, highlighting the destabilizing effects of magnetic fields in 3D MHD near Couette flow.

Findings

Sharp Sobolev stability thresholds for 2D MHD around Couette flow.

Existence of a nonlinear transient-growth regime with a critical exponent.

Destabilizing effects of magnetic fields in 3D compared to 2D cases.

Abstract

We study the three-dimensional incompressible magnetohydrodynamic (MHD) equations near Couette flow with a constant magnetic field perpendicular to the shear plane. Couette flow induces mixing and generates magnetic induction, while the constant magnetic field stabilizes $z$-dependent modes. In contrast, the $z$-averaged magnetic field exhibits algebraic growth. Letting $\mu$ denote the inverse fluid and magnetic Reynolds numbers, we analyze how $\mu$ governs stability thresholds in Sobolev spaces. We identify a nonlinear transient-growth regime characterized by the sharp threshold $5/6\le \gamma\le 1 $. For $x$-average-free initial data of size $\mu^\gamma$, solutions are nonlinearly stable; however, for certain initial data, the solution departs from the linear dynamics at rate $\mu^{\gamma-1}$ due to a first-order nonlinear instability. The exponent $\gamma = 5/6$ is optimal for the associated energy functional and cannot be improved in Sobolev spaces without secondary transient-growth mechanisms. Below this threshold, solutions necessarily transition away from the linear dynamics at a minimal rate. As a consequence, the $3d$ results yield sharp Sobolev stability thresholds for the $2d$ MHD equations around Couette flow without a constant magnetic field. In particular, the threshold is strictly larger than in prior $2d$ results with a constant field, revealing destabilizing effects normally suppressed by a constant magnetic field. Crucially, this stabilization is restricted to the direction of the magnetic field.