Stability of 2D MHD with Only Horizontal Dissipation near a Background Magnetic Field on $\mathbb{R}^2$

Abstract

Whether the two-dimensional Navier--Stokes equations with dissipation in only one direction are stable in $H^2$ on the whole plane is a well-known open problem: without transverse dissipation, a cubic interaction fails to be time-integrable. We show that a magnetic field removes this obstruction. For the 2D incompressible MHD equations on $\mathbb R^2$ in which both the viscosity $ν$ and the magnetic diffusivity $μ$ act only in the horizontal direction, we prove the global nonlinear $H^2$ stability of every background field $γe_2$ aligned with the non-dissipative direction, with a stability threshold that is explicit in $ν$, $μ$ and $γ$. The stabilizing mechanism is genuinely nonlinear: at the linear level the background field produces no damping at all in the vertical direction. Instead, in Elsässer variables the obstruction becomes an interaction between Alfvén waves travelling in opposite directions, and we identify nonlinear dissipative quantities that measure the overlap of these waves and are time-integrable although neither wave decays. We then sharpen the threshold by a space--time div--curl estimate for balance laws in the $(t,x_2)$-plane. For $ν=μ\leq1\leqγ$, the admissible perturbations are of size $ν^{2/3}γ^{1/3}$, improving the direct scale $ν$ and growing without bound as the background field strengthens.