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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
##### Keywords:
Euler flow; vortex dynamics; regular solutions
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##### References:
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