A Sharp Uniqueness-Nonuniqueness Dichotomy for Neutral Vortex Arrays in $2$D Euler Equations

Abstract

We establish a sharp uniqueness--nonuniqueness criterion for the two-dimensional incompressible Euler equations within a class of stationary arrays of disjoint neutral vortices. Let $\Theta$ be any finite nondecreasing envelope for the vorticity $L^p$ norms, and set $I_{\Theta }={\int }_{p_0}^{\infty }\Theta (p{)}^{-1}dp$. If $I_{\Theta }=\infty$, every array in the corresponding class admits a unique Euler evolution, even among bounded energy-admissible weak velocity solutions. If $I_{\Theta }<\infty$, there exists a compactly supported array in the same class admitting two distinct unforced Lagrangian Euler evolutions from the same initial datum. The criterion holds for arbitrary nondecreasing envelopes $\Theta$.

arXiv comment45 pages, 2 figures