Burst of Point Vortices and Non-Uniqueness of 2D Euler Equations

TL;DR

This paper constructs solutions to the 2D Euler equations where point vortices suddenly burst or collapse, demonstrating non-uniqueness and energy dissipation in weak solutions.

Contribution

It provides a rigorous construction of vortex burst and collapse solutions, showing non-uniqueness and energy dissipation in 2D Euler equations.

Findings

Existence of vortex burst solutions in Euler system.

Persistence of self-similar vortex behaviors under perturbations.

Examples of non-unique weak solutions with energy dissipation.

Abstract

We give a rigorous construction of solutions to the Euler point vortices system in which three vortices burst out of a single one in a configuration of many vortices, or equivalently that there exist configurations of arbitrarily many vortices in which three of them collapse in finite time. As an intermediate step, we show that well-known self-similar bursts and collapses of three isolated vortices in the plane persist under a sufficiently regular external perturbation. We also discuss how our results produce examples of non-unique weak solutions to 2-dimensional Euler's equations -- in the sense introduced by Schochet -- in which energy is dissipated.