Long time evolution of concentrated Euler flows with planar symmetry

TL;DR

This paper investigates the long-term behavior of concentrated vorticity in planar symmetric Euler flows, revealing conditions under which the vorticity remains concentrated for extended periods, supported by a simplified model.

Contribution

It provides new insights into the persistence of vorticity concentration in Euler flows with planar symmetry and introduces a toy model to illustrate this phenomenon.

Findings

Vorticity concentration can persist for long times in certain conditions.

A toy model replicates the long-time concentration behavior.

The study offers theoretical understanding of vorticity evolution in symmetric flows.

Abstract

We study the time evolution of an incompressible Euler fluid with planar symmetry when the vorticity is initially concentrated in small disks. We discuss how long this concentration persists, showing that in some cases this happens for quite long times. Moreover, we analyze a toy model that shows a similar behavior and gives some hints on the original problem.