On the dynamics of point vortices for the 2D Euler equation with $L^p$ vorticity

TL;DR

This paper investigates how solutions to the 2D Euler equations with $L^p$ vorticity evolve, showing they stay concentrated around points that approximate the classical vortex system, extending understanding of vortex dynamics with less regular vorticity.

Contribution

It demonstrates that for $L^p$ vorticity with $p>2$, vortex regions remain localized and closely follow the Helmholtz--Kirchhoff point vortex dynamics, even with minimal regularity.

Findings

Vortex regions stay concentrated around moving points.

These points approximate solutions to the classical vortex system.

Results hold for vorticity in $L^p$ with $p>2$.

Abstract

We study the evolution of solutions to the 2D Euler equations whose vorticity is sharply concentrated in the Wasserstein sense around a finite number of points. Under the assumption that the vorticity is merely $L^p$ integrable for some $p>2$, we show that the evolving vortex regions remain concentrated around points, and these points are close to solutions to the Helmholtz--Kirchhoff point vortex system.