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Time evolution of a singular vortex patch. (Evolution temporelle d’une poche de tourbillon singulière.) (French) Zbl 0882.35093
The paper deals with the persistence of certain types of degenerated geometries for the 2d incompressible Euler system. One demonstrates that the zones where the field of the velocity is not a Lipschitz field does not contaminate the regular zones. The author proves the persistence of conormal regularity apart from a closed subset. As a consequence one gives the following result for the vortex patches: if the initial boundary is regular apart from a closed subset, it remains regular at any time apart from the closed subset transported by the flow.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
76B47 Vortex flows for incompressible inviscid fluids
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[1] Alinhac S., Annales Scientifiques de l’École Normale Supérieure 21 pp 91– (1988)
[2] Alinhac S., Journal of functronnal analysis 98 pp 361– (1991) · Zbl 0732.35075
[3] Bony J. M., Annales Scientifiques de l’École Normale Supé, 14 pp 209– (1981)
[4] Bony J.M., Propagation des singularités pour les équations aux déivées partielles non linéaires (1979)
[5] Chemin J.Y, Duke Mathematical Journal 56 pp 431– (1998) · Zbl 0676.35009
[6] Chemin J.Y, Inventiones Mathematicne 103 pp 599– (1991) · Zbl 0739.76010
[7] Chemin J.Y, Annales de l’École Normale Supérieure 26 (4) pp 1– (1993)
[8] Danchin R, Fluides parfaits incompressibles 230 (1995)
[9] Dachin R., Analyes numérique et harmonique d’un probleème de mécanique des fluises (1966)
[10] Gérard P., Annales de l’Institut Fourier 37 (3) pp 65– (1987) · Zbl 0617.35079
[11] Majda A., Communications in pure and Applied Mathematics 38 pp 187– (1986) · Zbl 0595.76021
[12] Serfati P.Une preuve directe d’existance des vortex pathes 2D.Notes aux Comptes-rendus de l’Académic des Science de Paris 1994 315 318 318. série 1.
[13] Yudovitch V., Zurnal pychislitel’noj maematiki i matematiceskoj fiziki 3 pp 1032– (1963)
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