Large time behavior and asymptotic stability of the two-dimensional Euler and linearized Euler equations

TL;DR

This paper investigates the long-term behavior and stability of 2D Euler and linearized Euler flows near parallel jet profiles with stationary streamlines, revealing new decay phenomena and confirming stability through theoretical and numerical analysis.

Contribution

It introduces the study of asymptotic stability near stationary streamlines and uncovers a new vorticity depletion mechanism affecting velocity decay.

Findings

Velocity decays with power laws at large times.

Vorticity depletion occurs at stationary streamlines.

Theoretical predictions match numerical simulations.

Abstract

We study the asymptotic behavior and the asymptotic stability of the two-dimensional Euler equations and of the two-dimensional linearized Euler equations close to parallel flows. We focus on spectrally stable jet profiles $U(y)$ with stationary streamlines $y_{0}$ such that $U'(y_{0})=0$, a case that has not been studied previously. We describe a new dynamical phenomenon: the depletion of the vorticity at the stationary streamlines. An unexpected consequence, is that the velocity decays for large times with power laws, similarly to what happens in the case of the Orr mechanism for base flows without stationary streamlines. The asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations. We argue on the asymptotic stability of these flow velocities even in the absence of any dissipative mechanisms.