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

Abstract

This paper addresses the long-time dynamics of solutions to the 2D incompressible Euler equations. We construct solutions with continuous vorticity $\ddot{I}{‰}_{\epsilon }(x,t)$ concentrated around points $\hat{I}{¾}_j(t)$ that converge to a sum of Dirac delta masses as $\epsilon \to 0$. These solutions are associated with the Kirchhoff-Routh point-vortex system, and the points $\hat{I}{¾}_j(t)$ follow an expanding self similar trajectory of spirals, with the support of the vorticities contained in balls of radius $3\epsilon$ around each $\hat{I}{¾}_j$.