On Compressible and Incompressible Vortex Sheets

  • Paolo Secchi
Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)


We introduce the main known results of the theory of incompressible and compressible vortex sheets. Moreover, we present recent results obtained by the author with J.F. Coulombel about compressible vortex sheets in two space dimensions, under a supersonic condition that precludes violent instabilities. The problem is a nonlinear free boundary hyperbolic problem with two difficulties: the free boundary is characteristic and the Lopatinski condition holds only in a weak sense, yielding losses of derivatives. In [18, 20] we prove the existence of such piecewise smooth solutions to the Euler equations close enough to stationary vortex sheets. Since the a priori estimates for the linearized equations exhibit a loss of regularity, our existence result is proved by using a suitable modification of the Nash-Moser iteration scheme.


Weak Solution Euler Equation Energy Estimate Contact Discontinuity Vortex Sheet 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Agemi. The initial boundary value problem for inviscid barotropic fluid motion. Hokkaido Math. J. 10: 156–182, 1981.zbMATHMathSciNetGoogle Scholar
  2. [2]
    S. Alinhac. Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Commun. Partial Diff. Eqs., 14(2):173–230, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    S. Alinhac, P. Gérard. Opérateurs pseudo-différentiels et théorème de Nash-Moser. InterEditions/Editions du CNRS, Paris, Meudon, 1991.zbMATHGoogle Scholar
  4. [4]
    M. Artola, A. Majda. Nonlinear development of instabilities in supersonic vortex sheets. I. The basic kink modes. Phys. D, 28(3):253–281, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    H. Beirão da Veiga. On the barotropic motion of compressible perfect fluids. Ann. Sc. Norm. Sup. Pisa, 8: 317–351, 1981.zbMATHGoogle Scholar
  6. [6]
    G. Birkhoff. Helmholtz and Taylor instability. In Hydrodynamics Instability, Proc. Symp. Appl. Math., A.M.S., Providence, RI, 13: 55–76, 1962.MathSciNetGoogle Scholar
  7. [7]
    A. Blokhin, Y. Trakhinin. Stability of strong discontinuities in fluids and MHD. In Handbook of Mathematical Fluid Dynamics, Vol. I, pages 545–652, North-Holland, 2002.CrossRefMathSciNetGoogle Scholar
  8. [8]
    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.zbMATHMathSciNetGoogle Scholar
  9. [9]
    R.E. Caflish, O.F. Orellana. Long time existence for a slightly perturbed vortex sheet. Commun. Pure Appl. Math., 39:807–838, 1986.CrossRefGoogle Scholar
  10. [10]
    R.E. Caflish, O.F. Orellana. Singular solutions and ill-posedness for the evolution of vortex sheets. SIAM J. Math. Anal., 20:293–307, 1989.CrossRefMathSciNetGoogle Scholar
  11. [11]
    J.-Y. Chemin. Fluides parfaits incompressibles. Astérisque (Vol. 230), Paris, 1995.Google Scholar
  12. [12]
    A. Corli. Asymptotic analysis of contact discontinuities. Ann. Mat. Pura Appl. (4), 173:163–202, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    A. Corli, M. Sablé-Tougeron. Stability of contact discontinuities under perturbations of bounded variation. Rend. Sem. Mat. Univ. Padova, 97:35–60, 1997.zbMATHMathSciNetGoogle Scholar
  14. [14]
    J.-F. Coulombel. Weak stability of nonuniformly stable multidimensional shocks. SIAM J. Math. Anal., 34(1):142–172, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    J.-F. Coulombel. Weakly stable multidimensional shocks. Annales de l’IHP, Analyse Non Linéaire, 21(4):401–443, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    J.-F. Coulombel. Well-posedness of hyperbolic initial boundary value problems. J. Math. Pures Appl., 84(6):786–818, 2005.zbMATHMathSciNetGoogle Scholar
  17. [17]
    J.-F. Coulombel, A. Morando. Stability of contact discontinuities for the nonisentropic Euler equations. Ann. Univ. Ferrara Sez. VII (N.S.), 50:79–90, 2004.MathSciNetzbMATHGoogle Scholar
  18. [18]
    J.-F. Coulombel, P. Secchi. The stability of compressible vortex sheets in two space dimensions. Indiana Univ. Math. J., 53:941–1012, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    J.-F. Coulombel, P. Secchi. On the transition to instability for compressible vortex sheets. Proc. Roy. Soc. Edinburgh, 134A:885–892, 2004.CrossRefMathSciNetGoogle Scholar
  20. [20]
    J.-F. Coulombel, P. Secchi. Nonlinear compressible vortex sheets in two space dimensions, submitted.Google Scholar
  21. [21]
    J.M. Delort. Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc., 4:553–586, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    J. Duchon, R. Robert. Global vortex sheet solutions of Euler equations in the plane. J. Diff. Eqns., 73:215–224, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    L.C. Evans, S. Muller. Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity. J. Amer. Math. Soc., 1:199–219, 1994.CrossRefMathSciNetGoogle Scholar
  24. [24]
    J.A. Fejer, J. W. Miles. On the stability of a plane vortex sheet with respect to three-dimensional disturbances. J. Fluid Mech., 15:335–336, 1963.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    J. Francheteau, G. Métivier. Existence de chocs faibles pour des systèmes quasilinéaires hyperboliques multidimensionnels. Astérisque (Vol. 268), Paris, 2000.Google Scholar
  26. [26]
    M. Grassin Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math. J., 47:1397–1432, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    M. Grassin, D. Serre Existence de solutions globales et régulières aux équations d’Euler pour un gaz parfait isentropique. C. R. Acad. Sci. Paris Sér. I Math., 325(7):721–726, 1997.zbMATHMathSciNetGoogle Scholar
  28. [28]
    E. Harabetian. A convergent series expansion for hyperbolic systems of conservation laws. Trans. Amer. Math. Soc., 294 (1986), no. 2, 383–424.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    D. Hateau. Instabilité des feuilles de tourbillon avec conduction de chaleur. C. R. Acad. Sci. Paris Sér. I Math., 330(7):629–633, 2000.zbMATHMathSciNetGoogle Scholar
  30. [30]
    T. Kato. The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Rat. Mech. Anal. 58:181–205, 1975.zbMATHCrossRefGoogle Scholar
  31. [31]
    R. Krasny. On singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167: 65–93, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    R. Krasny. Computing vortex sheet motion. In Proc. Int. Congress Math., Kyoto 1990 Math. Soc. Japan, Tokyo, 2: 1573–1583, 1991.Google Scholar
  33. [33]
    H.-O. Kreiss. Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math., 23:277–298, 1970.CrossRefMathSciNetGoogle Scholar
  34. [34]
    P.D. Lax. Hyperbolic systems of conservation laws. II. Commun. Pure Appl. Math., 10:537–566, 1957.zbMATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    G. Lebeau. Régularité du problème de Kelvin-Helmholtz pour l’équation d’Euler 2d. ESAIM Control Optim. Calc. Var., 8:801–825, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    H. Lindblad. Well-posedness for the motion of an incompressible liquid with free surface boundary. Annals of Math., 162 no. 1, 2005.Google Scholar
  37. [37]
    H. Lindblad. Well-posedness for the motion of a compressible liquid with free surface boundary. Commun. Math. Phys., to appear.Google Scholar
  38. [38]
    T.-L. Liu, T. Yang. Compressible Euler equations with vacuum. J. Diff. Eqns. 137: 223–237, 1997.CrossRefMathSciNetGoogle Scholar
  39. [39]
    A. Majda. The stability of multidimensional shock fronts. Mem. Amer. Math. Soc., 41(275):iv+95, 1983.MathSciNetGoogle Scholar
  40. [40]
    A. Majda. The existence of multidimensional shock fronts. Mem. Amer. Math. Soc., 43(281):v+93, 1983.MathSciNetGoogle Scholar
  41. [41]
    A. Majda. Remarks on weak solutions for vortex sheets with a distinguished sign. Indiana Univ. Math. J., 42:921–939, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    A. Majda, A. Bertozzi. Vorticity and incompressible flow. Cambridge University Press, Cambridge, 2002.zbMATHGoogle Scholar
  43. [43]
    A. Majda, S. Osher. Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary. Commun. Pure Appl. Math., 28(5):607–675, 1975.zbMATHCrossRefMathSciNetGoogle Scholar
  44. [44]
    C. Marchioro, M. Pulvirenti. Mathematical theory of incompressible nonviscous fluids. Springer, 1994.Google Scholar
  45. [45]
    D.I. Meiron, G.R. Baker, S.A. Orszag. Analytic structure of vortex sheet dynamics. Part I. J. Fluid Mech., 114: 283, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  46. [46]
    G. Métivier. Stability of multidimensional shocks. In Advances in the theory of shock waves, PNDEA, Birkhäuser, 25–103, 2001.Google Scholar
  47. [47]
    Y. Meyer. Remarques sur un théorème de J.-M. Bony. Rend. Circ. Mat. Palermo (2), suppl. 1:1–20, 1981.Google Scholar
  48. [48]
    J.W. Miles. On the disturbed motion of a plane vortex sheet. J. Fluid Mech., 4:538–552, 1958.zbMATHCrossRefMathSciNetGoogle Scholar
  49. [49]
    D.W. Moore. The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. Roy. Soc. London Ser. A, 365: 105–119, 1979.zbMATHMathSciNetCrossRefGoogle Scholar
  50. [50]
    V. Scheffer. An inviscid flow with compact support in space-time. J. Geom. Anal. 3: 343–401, 1993.zbMATHMathSciNetGoogle Scholar
  51. [51]
    D. Serre. Solutions classiques globales des équations d’Euler pour un fluid parfait compressible. Ann. Inst. Fourier, 47:139–153, 1997.zbMATHMathSciNetGoogle Scholar
  52. [52]
    D. Serre. Systems of conservation laws. 2. Cambridge Univ. Press, Cambridge, 2000.zbMATHGoogle Scholar
  53. [53]
    A. Shnirelman. On the non-uniqueness of weak solutions of the Euler equations. Commun. Pure Appl. Math. L: 1261–1286, 1997.CrossRefMathSciNetGoogle Scholar
  54. [54]
    T. Sideris. Formation of singularities in three-dimensional compressible flow. Commun. Math. Phys. 10: 475–485, 1985.CrossRefMathSciNetGoogle Scholar
  55. [55]
    C. Sulem, P.L. Sulem, C. Bardos, U. Frisch. Finite time analyticity for the two and three-dimensional Kevin-Helmholtz instability. Commun. Math. Phys., 80:485–516, 1981.zbMATHCrossRefMathSciNetGoogle Scholar
  56. [56]
    P. Woodward. Simulation of the Kelvin-Helmholtz instability of a supersonic slip surface with a piecewise parabolic method. L.L.L. preprint, 1984.Google Scholar
  57. [57]
    S. Wu. Recent progress in mathematical analysis of vortex sheets I.C.M. 2002, Vol. III, 233–242.Google Scholar
  58. [58]
    T. Yang, C.J. Zhu. Non-existence of global smooth solutions to symmetrizable nonlinear hyperbolic systems. Proc. R. Soc. Edinb. 133A: 719–728, 2003.MathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Paolo Secchi
    • 1
  1. 1.Dipartimento di MatematicaFacoltà di IngegneriaBresciaItaly

Personalised recommendations