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The incompressible Navier-Stokes equations in 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].
Nous montrons que les équations de Navier-Stokes incompressibles dans 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].
@article{ASENS_2019__52_4_1025_0,
author = {Gallay, Thierry and \v{S}ver\'ak, Vladim{\'\i}r},
title = {Uniqueness of axisymmetric viscous flows originating from circular vortex filaments},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
pages = {1025--1071},
publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
volume = {Ser. 4, 52},
number = {4},
year = {2019},
doi = {10.24033/asens.2402},
mrnumber = {4038451},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.24033/asens.2402/}
}
TY - JOUR AU - Gallay, Thierry AU - Šverák, Vladimír TI - Uniqueness of axisymmetric viscous flows originating from circular vortex filaments JO - Annales scientifiques de l'École Normale Supérieure PY - 2019 SP - 1025 EP - 1071 VL - 52 IS - 4 PB - Société Mathématique de France. Tous droits réservés UR - http://geodesic.mathdoc.fr/articles/10.24033/asens.2402/ DO - 10.24033/asens.2402 LA - en ID - ASENS_2019__52_4_1025_0 ER -
%0 Journal Article %A Gallay, Thierry %A Šverák, Vladimír %T Uniqueness of axisymmetric viscous flows originating from circular vortex filaments %J Annales scientifiques de l'École Normale Supérieure %D 2019 %P 1025-1071 %V 52 %N 4 %I Société Mathématique de France. Tous droits réservés %U http://geodesic.mathdoc.fr/articles/10.24033/asens.2402/ %R 10.24033/asens.2402 %G en %F ASENS_2019__52_4_1025_0
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://geodesic.mathdoc.fr/articles/10.24033/asens.2402/
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