Uniqueness of axisymmetric viscous flows originating from circular vortex filaments

Thierry Gallay, Vladimir Sverák

Research output: Contribution to journalArticle

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 - Jan 1 2019

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