Transition and Stability of 3D MHD Around Couette Flow

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.