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Gluing methods for vortex dynamics in Euler flows. (English) Zbl 1439.35382
Summary: 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 G. Kirchhoff [Vorlesungen über mathematische Physik. I. Mechanik. Leipzig. Teubner (1876; JFM 08.0542.01)] and E. J. Routh [Proc. Lond. Math. Soc. 12, 73–89 (1881; JFM 13.0741.01)]. 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.

MSC:
35Q31 Euler equations
76B47 Vortex flows for incompressible inviscid fluids
35B65 Smoothness and regularity of solutions to PDEs
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