Uniqueness of axisymmetric viscous flows originating from circular vortex filaments
[Unicité des écoulements visqueux axisymétriques issus de filaments tourbillonnaires circulaires]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 4, pp. 1025-1071.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Nous montrons que les équations de Navier-Stokes incompressibles dans ℝ3 possèdent une unique solution axisymétrique sans swirl lorsque le tourbillon initial est un filament circulaire dont le nombre de Reynolds de circulation peut être arbitrairement grand. L'accent est mis ici sur l'unicité, car l'existence a déjà été établie dans [10]. La difficulté principale à surmonter est que, pour de tels écoulements, le régime non linéaire ne peut être décrit par une théorie de perturbation standard, même pour des temps petits. Les solutions que nous construisons sont des exemples typiques d'anneaux tourbillonnaires visqueux, et peuvent être considérées comme l'analogue axisymétrique des tourbillons autosimilaires de Lamb-Oseen que l'on rencontre dans les écoulements plans. Notre méthode fournit le terme dominant d'un développement asymptotique de la solution à temps petits, la viscosité étant fixée, et peut en principe se généraliser à des ordres plus élevés et donner ainsi une justification complète, dans le cadre axisymétrique, des développements asymptotiques formels que l'on trouve dans la littérature [7].

The incompressible Navier-Stokes equations in ℝ3 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].

Publié le :
DOI : 10.24033/asens.2402
Classification : 35Q30, 35B07, 35A02, 76D05, 76D17, 76N10.
Keywords: Équations de Navier-Stokes, filaments de tourbillon, solutions axisymétriques, unicitÃé
Mot clés : Navier-Stokes equations, vortex filaments, axially symmetric solutions, uniqueness 
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     title = {Uniqueness of axisymmetric viscous flows  originating from circular  vortex filaments},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
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Gallay, Thierry; Šverák, Vladimír. Uniqueness of axisymmetric viscous flows  originating from circular  vortex filaments. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 4, pp. 1025-1071. doi : 10.24033/asens.2402. http://www.numdam.org/articles/10.24033/asens.2402/

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