Gluing methods for 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 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 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 desingularization. We succeed in applying those ideas in this highly challenging setting.