Asymptotic stability of shear flows for 2D Euler equations at Yudovich regularity

TL;DR

This paper proves the nonlinear asymptotic stability of shear flows in the 2D Euler equations within an infinite channel at Yudovich regularity, using vorticity advection to infinity as a stabilizing mechanism.

Contribution

It introduces a new stabilization mechanism for 2D Euler shear flows at minimal regularity in an infinite channel setting.

Findings

Decay of perturbations on compact subsets

Weak convergence of vorticity to zero

Stability at Yudovich class regularity

Abstract

The nonlinear asymptotic stability of shear flows in the 2D Euler equations has traditionally been linked to inviscid damping in the periodic setting. Since Gevrey regularity is required to suppress the ``echo'' phenomenon, asymptotic stability is known to be impossible in Sobolev spaces. In this paper, we identify a distinct stabilizing mechanism available in the infinite channel: the advection of vorticity to spatial infinity. We establish nonlinear asymptotic stability for the 2D Euler equations in the infinite channel $\mathbb{R}\times[0,1]$ at the minimal regularity of the Yudovich class ($L^{\infty}$ vorticity). Specifically, for a class of non-negative shear flows with a curvature bound, any $L^\infty$-small, compactly supported vorticity perturbation leads to decay on compact subsets and weak convergence to zero.