Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

TL;DR

This paper proves the nonlinear asymptotic stability of planar Couette flow in the 2D Euler equations, demonstrating inviscid damping and convergence of velocity to a shear flow for small Gevrey class perturbations.

Contribution

It provides the first rigorous proof of nonlinear inviscid damping for shear flows near Couette flow in the 2D Euler equations.

Findings

Velocity converges strongly in L^2 to a nearby shear flow.

Vorticity is driven to small scales and weakly converges at infinity.

The result confirms classical linear predictions at the nonlinear level.

Abstract

We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on $\Torus \times \Real$. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L^2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as $t \rightarrow \pm\infty$. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. This convergence was formally derived at the linear level by Kelvin in 1887 and it occurs at an algebraic rate first computed by Orr in 1907; our work appears to be the first rigorous confirmation of this behavior on the nonlinear level.