Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T20:50:02.394Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscous profiles of vortex patches

Published online by Cambridge University Press:  08 October 2013

Franck Sueur
Franck Sueur*
Affiliation:
CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France (fsueur@ann.jussieu.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We deal with the incompressible Navier–Stokes equations with vortex patches as initial data. Such data describe an initial configuration for which the vorticity is discontinuous across a hypersurface. We give an asymptotic expansion of the solutions in the vanishing viscosity limit which exhibits an internal layer where the fluid vorticity has a sharp variation. This layer moves with the flow of the Euler equations.

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 1 , January 2015 , pp. 1 - 68
Copyright
©Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abidi, H. and Danchin, R., Optimal bounds for the inviscid limit of Navier–Stokes equations, Asymptot. Anal. 38 (1) (2004), 35–46.Google Scholar
Alinhac, S., Existence d’ondes de raréfaction pour des écoulements isentropiques, in Séminaire E.D.P. Ecole Polytech. XVI (1986–1987).Google Scholar
Alinhac, S., Interaction d’ondes simples pour des équations complètement non-linéaires, Ann. Sci. Éc. Norm. Supér. (4) 21 (1) (1988), 91–132.CrossRefGoogle Scholar
Alinhac, S., Ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, in Journées Équations aux Dérivées Partielles Saint Jean de Monts, VIII (1988).CrossRefGoogle Scholar
Alinhac, S., Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Comm. Partial Differential Equations 14 (2) (1989), 173–230.CrossRefGoogle Scholar
Alinhac, S., Unicité d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Indiana Univ. Math. J. 38 (2) (1989), 345–363.CrossRefGoogle Scholar
Alinhac, S., Remarques sur l’instabilité du problème des poches de tourbillon, J. Funct. Anal. 98 (2) (1991), 361–379.CrossRefGoogle Scholar
Ambrose, D. M. and Masmoudi, N., Well-posedness of 3D vortex sheets with surface tension, Commun. Math. Sci. 5 (2) (2007), 391–430.CrossRefGoogle Scholar
Baouendi, M. S. and Goulaouic, C., Cauchy problems with characteristic initial hypersurface, Comm. Pure Appl. Math. 26 (1973), 455–475.CrossRefGoogle Scholar
Bahouri, H., Chemin, J.-Y. and Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Volume 343, p. xvi+523 (Springer, Heidelberg, 2011).CrossRefGoogle Scholar
Bahouri, H. and Dehman, B., Remarques sur l’apparition de singularités dans les écoulements eulériens incompressibles à donnée initiale höldérienne, J. Math. Pures Appl. (9) 73 (4) (1994), 335–346.Google Scholar
Bertozzi, A. L. and Constantin, P., Global regularity for vortex patches, Comm. Math. Phys. 152 (1) (1993), 19–28.CrossRefGoogle Scholar
Bianchini, S. and Bressan, A., Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. 161 (1) (2005), 223–342.CrossRefGoogle Scholar
Bony, J.-M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. Éc. Norm. Supér. (4) 14 (2) (1981), 209–246.CrossRefGoogle Scholar
Boutet de Monvel, L., Comportement d’un opérateur pseudo-différentiel sur une variété à bord. I. La propriété de transmission, J. Anal. Math. 17 (1966), 241–253.CrossRefGoogle Scholar
Boutet de Monvel, L., Boundary problems for pseudo-differential operators, Acta Math. 126 (1–2) (1971), 11–51.CrossRefGoogle Scholar
Boyer, F. and Fabrie, P., Eléments d’analyse pour l’étude de quelques modèles d’écoulements de fluides visqueux incompressibles, Mathématiques et Applications, Volume 52, p. 405 (Springer, 2006).Google Scholar
Brenner, P., The Cauchy problem for symmetric hyperbolic systems in ${L}_{p} $ , Math. Scand. 19 (1966), 27–37.CrossRefGoogle Scholar
Buttke, T. F., A fast adaptive vortex method for patches of constant vorticity in two dimensions, J. Comput. Phys. 89 (1) (1990), 161–186.CrossRefGoogle Scholar
Caflisch, R. E. and Sammartino, M., Vortex layers in the small viscosity limit, in WASCOM 2005—13th Conference on Waves and Stability in Continuous Media, pp. 59–70 (World Sci. Publ., Hackensack, NJ, 2006).Google Scholar
Caflisch, R. E. and Orellana, O. F., Long time existence for a slightly perturbed vortex sheet, Comm. Pure Appl. Math. 39 (6) (1986), 807–838.CrossRefGoogle Scholar
Caflisch, R. E. and Orellana, O. F., Singular solutions and ill-posedness for the evolution of vortex sheets, SIAM J. Math. Anal. 20 (2) (1989), 293–307.CrossRefGoogle Scholar
Carlen, E. A. and Loss, M., Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the 2-D Navier–Stokes equation, Duke Math. J. 81 (1) (1996), 135–157. A celebration of John F. Nash, Jr, 1995.Google Scholar
Charve, F., Asymptotics and vortex patches for the quasigeostrophic approximation, J. Math. Pures Appl. (9) 85 (4) (2006), 493–539.CrossRefGoogle Scholar
Chemin, J.-Y., Calcul paradifférentiel précisé et applications à des équations aux dérivées partielles non semilinéaires, Duke Math. J. 56 (3) (1988), 431–469.CrossRefGoogle Scholar
Chemin, J.-Y., Autour du problème des vortex patches, in Séminaire E.D.P. Ecole Polytech. XI (1989–1990).Google Scholar
Chemin, J.-Y., Persistance des structures géométriques liées aux poches de tourbillon, in Séminaire E.D.P. Ecole Polytech. XIII (1990–1991).Google Scholar
Chemin, J.-Y., Existence globale pour le problème des poches de tourbillon, C. R. Acad. Sci. Paris Sér. I Math. 312 (11) (1991), 803–806.Google Scholar
Chemin, J.-Y., Poches de tourbillon et structure géométrique dans les fluides incompressibles bidimensionnels, in Journées E.D.P. Saint Jean de Monts I (1991).CrossRefGoogle Scholar
Chemin, J.-Y., Sur le mouvement des particules d’un fluide parfait incompressible bidimensionnel, Invent. Math. 103 (3) (1991), 599–629.CrossRefGoogle Scholar
Chemin, J.-Y., Régularité de la trajectoire des particules d’un fluide parfait incompressible remplissant l’espace, J. Math. Pures Appl. (9) 71 (5) (1992), 407–417.Google Scholar
Chemin, J.-Y., Persistance de structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Sci. Éc. Norm. Supér. (4) 26 (4) (1993), 517–542.CrossRefGoogle Scholar
Chemin, J.-Y., Poches de tourbillon à bord singulier, in Séminaire E.D.P. Ecole Polytech. XII (1994–1995).Google Scholar
Chemin, J.-Y., Fluides parfaits incompressibles, Astérisque 230 (1995), 177.Google Scholar
Chemin, J.-Y., Two-dimensional Euler system and the vortex patches problem, in Handbook of mathematical fluid dynamics, Volume III, pp. 83–160 (North-Holland, Amsterdam, 2004).Google Scholar
Chemin, J.-Y., Le système de Navier–Stokes incompressible soixante dix ans après Jean Leray, in Actes des Journées Mathématiques à la Mémoire de Jean Leray, Sémin. Congr., Volume 9, pp. 99–123 (2004).Google Scholar
Cheverry, C., Propagation of oscillations in real vanishing viscosity limit, Comm. Math. Phys. 247 (3) (2004), 655–695.CrossRefGoogle Scholar
Cohen, A. and Danchin, R., Multiscale approximation of vortex patches, SIAM J. Appl. Math. 60 (2) (2000), 477–502.CrossRefGoogle Scholar
Constantin, P., Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations, Comm. Math. Phys. 104 (2) (1986), 311–326.CrossRefGoogle Scholar
Constantin, P. and Titi, E. S., On the evolution of nearly circular vortex patches, Comm. Math. Phys. 119 (2) (1988), 177–198.CrossRefGoogle Scholar
Constantin, P. and Wu, J., Inviscid limit for vortex patches, Nonlinearity 8 (5) (1995), 735–742.CrossRefGoogle Scholar
Coulombel, J.-F. and Secchi, P., Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Éc. Norm. Supér. (4) 41 (1) (2008), 85–139.CrossRefGoogle Scholar
Danchin, R., Poches de tourbillon visqueuses, in Séminaire E.D.P. Ecole Polytech. IX (1995–1996).Google Scholar
Danchin, R., Évolution d’une singularité de type cusp dans une poche de tourbillon, in Journées E.D.P. Saint-Jean-de-Monts V (1997).CrossRefGoogle Scholar
Danchin, R., Évolution temporelle d’une poche de tourbillon singulière, Comm. Partial Differential Equations 22 (5–6) (1997), 685–721.CrossRefGoogle Scholar
Danchin, R., Poches de tourbillon visqueuses, J. Math. Pures Appl. (9) 76 (7) (1997), 609–647.CrossRefGoogle Scholar
Danchin, R., Evolution of a cusp-like singularity in a vortex patch, in Hyperbolic problems: theory, numerics, applications, Volume I, (Zürich, 1998), Internat. Ser. Numer. Math., Volume 129, pp. 189–194 (Birkhäuser, Basel, 1999).CrossRefGoogle Scholar
Danchin, R., Évolution d’une singularité de type cusp dans une poche de tourbillon, Rev. Mat. Iberoam. 16 (2) (2000), 281–329.CrossRefGoogle Scholar
Delort, J.-M., Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc. 4 (3) (1991), 553–586.CrossRefGoogle Scholar
Depauw, N., Poche de tourbillon pour Euler 2D dans un ouvert à bord, in Journées E.D.P. Saint Jean de Monts III (1998).CrossRefGoogle Scholar
Depauw, N., Poche de tourbillon pour Euler 2D dans un ouvert à bord, J. Math. Pures Appl. (9) 78 (3) (1999), 313–351.CrossRefGoogle Scholar
Dispa, S., Intrinsic characterizations of Besov spaces on Lipschitz domains, Math. Nachr. 260 (2003), 21–33.CrossRefGoogle Scholar
Dong, H. and Du, D., On the local smoothness of solutions of the Navier–Stokes equations, J. Math. Fluid Mech. 9 (2) (2007), 139–152.CrossRefGoogle Scholar
Duchon, J. and Robert, R., Global vortex sheet solutions of Euler equations in the plane, J. Differential Equations 73 (2) (1988), 215–224.CrossRefGoogle Scholar
Dutrifoy, A., On 3-D vortex patches in bounded domains, Comm. Partial Differential Equations 28 (7–8) (2003), 1237–1263.CrossRefGoogle Scholar
Frazier, M. and Jawerth, B., A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1) (1990), 34–170.CrossRefGoogle Scholar
Friedrichs, K. O., The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc. 55 (1944), 132–151.CrossRefGoogle Scholar
Gallagher, I. and Gallay, T., Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity, Math. Ann. 332 (2) (2005), 287–327.CrossRefGoogle Scholar
Gallay, T. and Wayne, C. E., Global stability of vortex solutions of the two-dimensional Navier–Stokes equation, Comm. Math. Phys. 255 (1) (2005), 97–129.CrossRefGoogle Scholar
Gamblin, P., Système d’Euler incompressible et régularité microlocale analytique, in Séminaire E.D.P. Ecole Polytech. XX (1992–1993).Google Scholar
Gamblin, P., Système d’Euler incompressible et régularité microlocale analytique, Ann. Inst. Fourier (Grenoble) 44 (5) (1994), 1449–1475.CrossRefGoogle Scholar
Gamblin, P. and Saint Raymond, X., Le problème des poches de tourbillon en dimension trois d’espace, C. R. Acad. Sci. Paris Sér. I Math. 315 (13) (1992), 1385–1388.Google Scholar
Gamblin, P. and Saint Raymond, X., On three-dimensional vortex patches, Bull. Soc. Math. France 123 (3) (1995), 375–424.CrossRefGoogle Scholar
Germain, P., Pavlović, N. and Staffilani, G., Regularity of solutions to the Navier–Stokes equations evolving from small data in ${\mathrm{BMO} }^{- 1} $ , Int. Math. Res. Not. IMRN (21)(2007).CrossRefGoogle Scholar
Glass, O., Sueur, F. and Takahashi, T., Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid, Ann. Sci. Éc. Norm. Supér. 45 (1) (2012), 1–51.CrossRefGoogle Scholar
Grenier, E., Boundary layers, in Handbook of mathematical fluid dynamics, Volume III, pp. 245–309 (North-Holland, Amsterdam, 2004).Google Scholar
Grubb, G. and Hörmander, L., The transmission property, Math. Scand. 67 (2) (1990), 273–289.CrossRefGoogle Scholar
Guès, O., Développement asymptotique de solutions exactes de systèmes hyperboliques quasilinéaires, Asymptot. Anal. 6 (3) (1993), 241–269.Google Scholar
Gues, O., Viscous boundary layers and high frequency oscillations, in Singularities and oscillations (Minneapolis, MN, 1994/1995), IMA Vol. Math. Appl., Volume 91, pp. 61–77 (Springer, New York, 1997).CrossRefGoogle Scholar
Guès, O., Métivier, G., Williams, M. and Zumbrun, K., Boundary layer and long time stability for multidimensional viscous shocks, Discrete Contin. Dyn. Syst. 11 (1) (2004), 131–160.CrossRefGoogle Scholar
Guès, O., Métivier, G., Williams, M. and Zumbrun, K., Multidimensional viscous shocks. II. The small viscosity limit, Comm. Pure Appl. Math. 57 (2) (2004), 141–218.CrossRefGoogle Scholar
Guès, O., Métivier, G., Williams, M. and Zumbrun, K., Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Ration. Mech. Anal. 175 (2) (2005), 151–244.CrossRefGoogle Scholar
Guès, O., Métivier, G., Williams, M. and Zumbrun, K., Multidimensional viscous shocks. I. Degenerate symmetrizers and long time stability, J. Amer. Math. Soc. 18 (1) (2005), 61–120 (electronic).CrossRefGoogle Scholar
Guès, O., Métivier, G., Williams, M. and Zumbrun, K., Navier–Stokes regularization of multidimensional Euler shocks, Ann. Sci. Éc. Norm. Supér. (4) 39 (1) (2006), 75–175.CrossRefGoogle Scholar
Guès, O., Métivier, G., Williams, M. and Zumbrun, K., Nonclassical multidimensional viscous and inviscid shocks, Duke Math. J. 142 (1) (2008), 1–110.CrossRefGoogle Scholar
Gues, O. and Rauch, J., A transmission strategy for hyperbolic internal waves of small width, in Séminaire E.D.P. Ecole Polytech. XIII (2005–2006).CrossRefGoogle Scholar
Gues, O. and Rauch, J., Nonlinear asymptotics for hyperbolic internal waves of small width, J. Hyperbolic Differ. Equ. 3 (2) (2006), 269–295.CrossRefGoogle Scholar
Guès, O. and Sueur, F., On 3D domain walls for the Landau Lifshitz equations, Dyn. Partial Differ. Equ. 4 (2) (2007), 143–165.CrossRefGoogle Scholar
Guès, O. and Williams, M., Curved shocks as viscous limits: a boundary problem approach, Indiana Univ. Math. J. 51 (2) (2002), 421–450.CrossRefGoogle Scholar
Hmidi, T., Régularité höldérienne des poches de tourbillon visqueuses, C. R. Math. Acad. Sci. Paris 339 (10) (2004), 705–708.CrossRefGoogle Scholar
Hmidi, T., Régularité höldérienne des poches de tourbillon visqueuses, J. Math. Pures Appl. (9) 84 (11) (2005), 1455–1495.CrossRefGoogle Scholar
Hmidi, T., Poches de tourbillon singulières dans un fluide faiblement visqueux, Rev. Mat. Iberoam. 22 (2) (2006), 489–543.CrossRefGoogle Scholar
Hörmander, L., The analysis of linear partial differential operators. I–IV, in Classics in Mathematics. (Springer-Verlag, Berlin, 2003).Google Scholar
Hou, T. Y., Lowengrub, J. S. and Shelley, M. J., The long-time motion of vortex sheets with surface tension, Phys. Fluids 9 (7) (1997), 1933–1954.CrossRefGoogle Scholar
Huang, C., Remarks on regularity of non-constant vortex patches, Commun. Appl. Anal. 3 (4) (1999), 449–459.Google Scholar
Huang, C., Singular integral system approach to regularity of 3D vortex patches, Indiana Univ. Math. J. 50 (1) (2001), 509–552.CrossRefGoogle Scholar
Huang, C., Global smooth nonconstant vortex patches in bounded domains, Far East J. Math. Sci. (FJMS) 22 (2) (2006), 227–238.Google Scholar
Judovič, V. I., Non-stationary flows of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz. 3 (1963), 1032–1066.Google Scholar
Junca, S., Geometric optics with critical vanishing viscosity for one-dimensional semilinear initial value problems, Rev. Mat. Iberoam. 24 (2) (2008), 549–566.CrossRefGoogle Scholar
Kato, T., Nonstationary flows of viscous and ideal fluids in ${\mathbf{R} }^{3} $ , J. Funct. Anal. 9 (1972), 296–305.CrossRefGoogle Scholar
Kato, T., Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Volume 448, pp. 25–70 (Springer, Berlin, 1975).Google Scholar
Kato, T., On the smoothness of trajectories in incompressible perfect fluids, in Nonlinear wave equations (Providence, RI, 1998), Contemp. Math., Volume 263, pp. 109–130 (Amer. Math. Soc., Providence, RI, 2000).CrossRefGoogle Scholar
Lebeau, G., Régularité du problème de Kelvin–Helmholtz pour l’équation d’Euler 2d, ESAIM Control Optim. Calc. Var. 8 (2002), 801–825 (electronic). A tribute to J. L. Lions.CrossRefGoogle Scholar
Lions, J.-L. and Magenes, E., Problèmes aux limites non homogènes, tome I. (Dunod, Paris, 1968).Google Scholar
Liu, T.-P. and Xin, Z., Pointwise decay to contact discontinuities for systems of viscous conservation laws, Asian J. Math. 1 (1) (1997), 34–84.CrossRefGoogle Scholar
Majda, A., Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math. 39 (S, suppl.) (1986), S187–S220. Frontiers of the mathematical sciences: 1985 (New York, 1985).CrossRefGoogle Scholar
Majda, A. J. and Bertozzi, A. L., Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, Volume 27 (Cambridge University Press, Cambridge, 2002).Google Scholar
Masmoudi, N., Remarks about the inviscid limit of the Navier–Stokes system, Comm. Math. Phys. 270 (3) (2007), 777–788.CrossRefGoogle Scholar
McShane, E. J., Extension of range of functions, Bull. Amer. Math. Soc. 40 (12) (1934), 837–842.CrossRefGoogle Scholar
Métivier, G., The mathematics of nonlinear optics. Guy Métivier’s web page.Google Scholar
Métivier, G., Ondes soniques, in Séminaire E.D.P. Ecole Polytech. XVII (1987–1988).Google Scholar
Métivier, G., Ondes soniques, J. Math. Pures Appl. (9) 70 (2) (1991), 197–268.Google Scholar
Métivier, G., Stability of multidimensional shocks, in Advances in the theory of shock waves, Progr. Nonlinear Differential Equations Appl., Volume 47, pp. 25–103 (Birkhäuser Boston, Boston, MA, 2001).CrossRefGoogle Scholar
Miura, H. and Sawada, O., On the regularizing rate estimates of Koch–Tataru’s solution to the Navier–Stokes equations, Asymptot. Anal. 49 (1–2) (2006), 1–15.Google Scholar
Rankine, W., On the thermodynamic theory of waves of finite longitudinal disturbance, Philos. Trans. R. Soc. Lond. 160 (1870), 277–288.Google Scholar
Rauch, J., Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc. 291 (1) (1985), 167–187.CrossRefGoogle Scholar
Rempel, S. and Schulze, B.-W., Index theory of elliptic boundary problems. (North Oxford Academic Publishing Co. Ltd., London, 1985), Reprint of the 1982 edition.Google Scholar
Rousset, F., Characteristic boundary layers in real vanishing viscosity limits., J. Differential Equations 210 (1) (2005), 25–64.CrossRefGoogle Scholar
Rychkov, V. S., On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains, J. Lond. Math. Soc. (2) 60 (1) (1999), 237–257.CrossRefGoogle Scholar
Sablé-Tougeron, M., Ondes de gradients multidimensionnelles, Mem. Amer. Math. Soc. 106 (511) (1993), viii+93.Google Scholar
Serfati, P., Une preuve directe d’existence globale des vortex patches 2D, C. R. Acad. Sci. Paris Sér. I Math. 318 (6) (1994), 515–518.Google Scholar
Serre, D., Systems of conservation laws. 1. (Cambridge University Press, Cambridge, 1999), Hyperbolicity, entropies, shock waves, Translated from the 1996 French original by I. N. Sneddon.CrossRefGoogle Scholar
Sueur, F., Approche visqueuse de solutions discontinues de systèmes hyperboliques semilinéaires, Ann. Inst. Fourier (Grenoble) 56 (1) (2006), 183–245.CrossRefGoogle Scholar
Sueur, F., Couches limites semilinéaires, Ann. Fac. Sci. Toulouse Math. (6) 15 (2) (2006), 323–380.CrossRefGoogle Scholar
Sueur, F., Couches limites: un problème inverse, Comm. Partial Differential Equations 31 (1–3) (2006), 123–194.CrossRefGoogle Scholar
Sulem, C., Sulem, P.-L., Bardos, C. and Frisch, U., Finite time analyticity for the two- and three-dimensional Kelvin–Helmholtz instability, Comm. Math. Phys. 80 (4) (1981), 485–516.CrossRefGoogle Scholar
Swann, H. S. G., The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in ${R}_{3} $ , Trans. Amer. Math. Soc. 157 (1971), 373–397.Google Scholar
Triebel, H., Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers, Rev. Mat. Complut. 15 (2) (2002), 475–524.CrossRefGoogle Scholar
Triebel, H., Theory of function spaces. III, Monographs in Mathematics, Volume 100 (Birkhäuser Verlag, Basel, 2006).Google Scholar
Whitney, H., Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1) (1934), 63–89.CrossRefGoogle Scholar
Wu, S., Mathematical analysis of vortex sheets, Comm. Pure Appl. Math. 59 (8) (2006), 1065–1206.CrossRefGoogle Scholar
Yudovich, V. I., Non-stationary flows of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz. 3 (1963), 1032–1066.Google Scholar
Zabusky, N. J., Contour dynamics for the Euler equations in two dimensions, J. Comput. Phys. 30 (1) (1979), 96–106.CrossRefGoogle Scholar
Zhang, P. and Qiu, Q. J., Propagation of higher-order regularities of the boundaries of 3-D vortex patches, Chinese Ann. Math. Ser. A 18 (3) (1997), 381–390.Google Scholar
Zhang, P. and Qiu, Q. J., The three-dimensional revised vortex patch problem for the system of incompressible Euler equations, Acta Math. Sinica 40 (3) (1997), 437–448.Google Scholar