Viscous profiles of vortex patches

Sueur, Franck

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2008-2009), Exposé no. 13, 11 p.

[1] S. Alinhac. Interaction d’ondes simples pour des équations complètement non-linéaires. Ann. Sci. École Norm. Sup. (4), 21(1):91–132, 1988. | Numdam | MR | Zbl

[2] M. S. Baouendi and C. Goulaouic. Cauchy problems with characteristic initial hypersurface. Comm. Pure Appl. Math., 26:455–475, 1973. | MR | Zbl

[3] M. Beals. Self-spreading and strength of singularities for solutions to semilinear wave equations. Ann. of Math.,118(1):87–214, 1983. | MR | Zbl

[4] J.-M. Bony. Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. (4), 14(2):209–246, 1981. | Numdam | MR | Zbl

[5] T. F. Buttke. A fast adaptive vortex method for patches of constant vorticity in two dimensions. J. Comput. Phys., 89(1):161–186, 1990. | MR | Zbl

[6] J.-Y. Chemin. Calcul paradifférentiel précisé et applications à des équations aux dérivées partielles non semilinéaires. Duke Math. J., 56(3):431–469, 1988. | MR | Zbl

[7] J.-Y. Chemin. Sur le mouvement des particules d’un fluide parfait incompressible bidimensionnel. Invent. Math., 103(3):599–629, 1991. | MR | Zbl

[8] J.-Y. Chemin. Persistance de structures géométriques dans les fluides incompressibles bidimensionnels. Ann. Sci. École Norm. Sup. (4), 26(4):517–542, 1993. | Numdam | MR | Zbl

[9] J.-Y. Chemin. Two-dimensional Euler system and the vortex patches problem. In Handbook of mathematical fluid dynamics. Vol. III, pages 83–160. North-Holland, Amsterdam, 2004. | MR

[10] N. Depauw. Poche de tourbillon pour Euler 2D dans un ouvert à bord. J. Math. Pures Appl. (9), 78(3):313–351, 1999. | MR | Zbl

[11] A. Dutrifoy. On 3-D vortex patches in bounded domains. Comm. Partial Differential Equations, 28(7-8):1237–1263, 2003. | MR | Zbl

[12] P. Gamblin and X. Saint Raymond. On three-dimensional vortex patches. Bull. Soc. Math. France, 123(3):375–424, 1995. | Numdam | MR | Zbl

[13] L. Gårding. Hyperbolic equations in the twentieth century, Matériaux pour l’histoire des mathématiques au XX${}^{\mathrm{e}}$ siècle (Nice, 1996), Sémin. Congr., 3:37–68, 1998. | MR | Zbl

[14] G. Grubb and L. Hörmander. The transmission property. Math. Scand., 67(2):273–289, 1990. | MR | Zbl

[15] L. Hörmander. The analysis of linear partial differential operators. I-IV. Classics in Mathematics. Springer-Verlag, Berlin, 2003. | MR | Zbl

[16] C. Huang. Remarks on regularity of non-constant vortex patches. Commun. Appl. Anal., 3(4):449–459, 1999. | MR | Zbl

[17] C. Huang. Singular integral system approach to regularity of 3D vortex patches. Indiana Univ. Math. J., 50(1):509–552, 2001. | MR | Zbl

[18] A. Majda. Vorticity and the mathematical theory of incompressible fluid flow. Comm. Pure Appl. Math., 39(S, suppl.):S187–S220, 1986. Frontiers of the mathematical sciences: 1985 (New York, 1985). | MR | Zbl

[19] P. Serfati. Une preuve directe d’existence globale des vortex patches $2$D. C. R. Acad. Sci. Paris Sér. I Math., 318(6):515–518, 1994. | MR | Zbl

[20] F. Sueur. Vorticity internal transition layers for the Navier-Stokes equations. Preprint, available on arXiv.

[21] F. Sueur. Vorticity internal transition layers for the Navier-Stokes equations. Journées “Équations aux Dérivées Partielles”, Evian 2008.

[22] V. I. Yudovich. Non-stationary flows of an ideal incompressible fluid. Ž. Vyčisl. Mat. Fiz., 3:1032–1066, 1963. | MR | Zbl

[23] N. J. Zabusky. Contour dynamics for the Euler equations in two dimensions. J. Comput. Phys., 30(1):96–106, 1979. | MR | Zbl

[24] P. Zhang and Q. J. Qiu. Propagation of higher-order regularities of the boundaries of $3$-D vortex patches. Chinese Ann. Math. Ser. A, 18(3):381–390, 1997. | MR | Zbl