Gluing methods for vortex dynamics in Euler flows

TL;DR

This paper introduces a novel gluing method to construct smooth vortex solutions in 2D Euler flows, accurately modeling vortex cores as scaled solutions of Liouville's equation, advancing the understanding of vortex dynamics.

Contribution

It develops a new gluing approach to construct smooth N-vortex solutions, combining desingularization techniques with high-precision vortex core modeling.

Findings

Successfully constructs smooth vortex solutions with concentrated vorticities.

Achieves high-precision modeling of vortex cores using Liouville's equation.

Extends gluing methods to complex vortex dynamics in Euler flows.

Abstract

A classical problem for the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. is that of finding regular solutions with highly concentrated vorticities around $N$ moving {\em vortices}. The formal dynamic law for such objects was first derived in the 19th century by Kirkhoff and Routh. In this paper we devise a {\em gluing approach} for the construction of smooth $N$-vortex solutions. We capture in high precision the core of each vortex as a scaled finite mass solution of Liouville's equation plus small, more regular terms. Gluing methods have been a powerful tool in geometric constructions by {\em desingularization}. We succeed in applying those ideas in this highly challenging setting.