Uniqueness of axisymmetric viscous flows originating from circular vortex filaments

Thierry Gallay, Vladimir Sverák

Research output: Contribution to journalArticlepeer-review

Abstract

The incompressible Navier-Stokes equations in R 3 are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large circulation Reynolds number. The emphasis is on uniqueness, as existence has already been established in [10]. The main difficulty which has to be overcome is that the nonlinear regime for such flows is outside of applicability of standard perturbation theory, even for short times. The solutions we consider are archetypal examples of viscous vortex rings, and can be thought of as axisymmetric analogs of the self-similar Lamb-Oseen vortices in two-dimensional flows. Our method provides the leading term in a fixed-viscosity short-time asymptotic expansion of the solution, and may in principle be extended so as to give a rigorous justification, in the axisymmetric situation, of higher-order formal asymptotic expansions that can be found in the literature [7].

Original languageEnglish (US)
Pages (from-to)1025-1071
Number of pages47
JournalAnnales Scientifiques de l'Ecole Normale Superieure
Volume52
Issue number4
DOIs
StatePublished - 2019

Bibliographical note

Funding Information:
Acknowledgements. – This project started during visits of the first named author to the University of Minnesota, whose hospitality is gratefully acknowledged. Our research was supported in part by grants DMS 1362467 and DMS 1159376 from the National Science Foundation (V.S.), and by grant “Dyficolti” ANR-13-BS01-0003-01 from the French Ministry of Research (Th.G.). The authors warmly thank the three anonymous referees for their careful reading of the manuscript and their numerous constructive remarks.

Publisher Copyright:
© 2019 Société Mathématique de France. Tous droits réservés

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