An Expanding Self-Similar Vortex Configuration for the 2D Euler Equations

TL;DR

This paper constructs solutions to the 2D Euler equations where vorticity concentrates around points following expanding spiral trajectories, linking continuous vorticity distributions to point-vortex dynamics.

Contribution

It introduces a novel class of solutions with vorticity concentrated near points following self-similar expanding spirals, connecting continuous vorticity fields to point-vortex models.

Findings

Solutions exhibit vorticity concentration around moving points

Points follow expanding self-similar spiral trajectories

Vorticity support shrinks to points as epsilon approaches zero

Abstract

This paper addresses the long-time dynamics of solutions to the 2D incompressible Euler equations. We construct solutions with continuous vorticity $\omega_{\varepsilon}(x,t)$ concentrated around points $\xi_{j}(t)$ that converge to a sum of Dirac delta masses as $\varepsilon\to0$. These solutions are associated with the Kirchhoff-Routh point-vortex system, and the points $\xi_{j}(t)$ follow an expanding self similar trajectory of spirals, with the support of the vorticities contained in balls of radius $3\varepsilon$ around each $\xi_{j}$.