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.
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Acknowledgements
We are grateful to Robert L. Jerrard for many useful discussions on the two-dimensional Euler flow, in particular about its linearization around a Liouville profile. M. Musso thanks the Center of Mathematical Modeling at Universidad de Chile for its hospitality while part of this work was concluded. J. Dávila has been supported by grants Fondecyt 1170224 and PAI AFB-170001, Chile. M. del Pino has been supported by a UK Royal Society Research Professorship and Grant PAI AFB-170001, Chile. M. Musso has been supported by grant FONDECYT 1160135, Chile. The research of J. Wei is partially supported by NSERC of Canada.
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Davila, J., Del Pino, M., Musso, M. et al. Gluing Methods for Vortex Dynamics in Euler Flows. Arch Rational Mech Anal 235, 1467–1530 (2020). https://doi.org/10.1007/s00205-019-01448-8
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DOI: https://doi.org/10.1007/s00205-019-01448-8