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

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.

Hölder regularity for collapses of point vortices

“…Then, there exists a collapse which has the 1/(α + 1)-Hölder regularity stated by Theorem 1.3, proving then that this exponent is optimal. Concerning the optimality of the 1/2-Hölder regularity for the Euler point-vortex dynamics in bounded domains Ω, it is a consequence of the results established by [14].…”

“…We consider the α-point-vortex dynamic (10). The case α = 1 (corresponding to Euler point-vortices in the plane) being already well-known [1,2,14], we fully concentrate here on the case α = 1 .…”

“…The existence of collapses for the Euler point-vortex problem in the plane has been obtained independently by [1,16,24]. Concerning the existence of collapses of vortices for all bounded smooth domains, it has been obtained recently by [14]. In [21], they prove that when the domain is a disc, the collapses are improbable in the sense that the set of initial datum leading to a collapse in finite time has a vanishing Lebesgue measure.…”