Free access
Issue
ESAIM: M2AN
Volume 40, Number 2, March-April 2006
Page(s) 225 - 237
DOI http://dx.doi.org/10.1051/m2an:2006012
Published online 21 June 2006
  1. G.R. Baker and J.T. Beale, Vortex blob methods applied to interfacial motion. J. Comput. Phys. 196 (2004) 233–258. [CrossRef] [MathSciNet]
  2. R. Caflisch and O. Orellana, Long time existence for a slightly perturbed vortex sheet. Comm. Pure Appl. Math. 39 (1986) 807–838. [CrossRef] [MathSciNet]
  3. R. Caflisch and O. Orellana, Singularity solutions and ill-posedness for the evolution of vortex sheets. SIAM J. Math. Anal. 20 (1989) 293–307. [CrossRef] [MathSciNet]
  4. J.-Y. Chemin, A remark on the inviscid limit for two-dimensional incompressible fluids. Comm. Partial Differential Equations 21 (1996) 1771–1779. [CrossRef] [MathSciNet]
  5. A. Chorin and P. Bernard, Discretization of a vortex sheet with an example of roll-up. J. Comput. Phys. 13 (1973) 423–429. [CrossRef]
  6. J.-M. Delort, Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc. 4 (1991) 553–586. [MathSciNet]
  7. R. DiPerna and A. Majda, Concentrations and regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. XL (1987) 301–345.
  8. R. DiPerna and A. Majda, Reduced Hausdorff dimension and concentration-cancellation for two-dimensional incompressible flow. J. Am. Math. Soc. 1 (1988) 59–95. [CrossRef]
  9. J. Duchon and R. Robert, Global vortex sheet solutions of Euler equations in the plane. Comm. Partial Differential Equations 73 (1988) 215–224.
  10. D. Ebin, Ill-posedness of the Rayleigh-Taylor and Helmholtz problem for incompressible fluids. Comm. Partial Differential Equations 73 (1988) 1265–1295. [CrossRef]
  11. L.C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics 74 A.M.S., Providence, RI (1990).
  12. C. Greengard and E. Thomann, On DiPerna-Majda concentration sets for two-dimensional incompressible flow. Comm. Pure Appl. Math. 41 (1988) 295–303. [CrossRef] [MathSciNet]
  13. R. Krasny, Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65 (1986) 292–313. [CrossRef]
  14. R. Krasny, Computation of vortex sheet roll-up in the Trefftz plane. J. Fluid Mech. 184 (1987) 123–155. [CrossRef]
  15. R. Krasny and M. Nitsche, The onset of chaos in vortex sheet flow. J. Fluid Mech. 454 (2002) 47–69. [CrossRef] [MathSciNet]
  16. G. Lebeau, Régularité du problème de Kelvin-Helmholtz pour l'équation d'Euler 2D. ESAIM: COCV 8 (2002) 801–825. [CrossRef] [EDP Sciences]
  17. J.G. Liu and Z.P. Xin, Convergence of vortex methods for weak solutions to the 2D Euler equations with vortex sheet data. Comm. Pure Appl. Math. XLVIII (1995) 611–628.
  18. M.C. Lopes Filho, H.J. Nussenzveig Lopes and Y.X. Zheng, Convergence of the vanishing viscosity approximation for superpositions of confined eddies. Commun. Math. Phys. 201 (1999) 291–304. [CrossRef]
  19. M.C. Lopes Filho, H.J. Nussenzveig Lopes and E. Tadmor, Approximate solutions of the incompressible Euler equations with no concentrations. Ann. I. H. Poincaré-An. 17 (2000) 371–412. [CrossRef]
  20. M.C. Lopes Filho, H.J. Nussenzveig Lopes and Z.P. Xin, Existence of vortex sheets with reflection symmetry in two space dimensions. Arch. Rational Mech. Anal. 158 (2001) 235–257. [CrossRef]
  21. M.C. Lopes Filho, H.J. Nussenzveig Lopes and M.O. Souza, On the equation satisfied by a steady Prandtl-Munk vortex sheet. Comm. Math. Sci. 1 (2003) 68–73.
  22. A. Majda, Remarks on weak solutions for vortex sheets with a distinguished sign. Indiana U. Math J. 42 (1993) 921–939. [CrossRef]
  23. A. Majda, G. Majda and Y.X. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations. I. Temporal development and non-unique weak solutions in the single component case. Physica D 74 (1994) 268–300. [CrossRef] [MathSciNet]
  24. M. Nitsche, M.A. Taylor and R. Krasny, Comparison of regularizations of vortex sheet motion, Proc. 2nd MIT Conf. Comput. Fluid and Solid Mech., K.J. Bathe Ed., Elsevier, Cambridge, MA (2003).
  25. W.R.C. Phillips and D.I. Pullin, On a generalization of Kaden's problem. J. Fluid Mech. 104 (1981) 45–53. [CrossRef]
  26. D.I. Pullin, On similarity flows containing two branched vortex sheets, in Mathematical Aspects of Vortex Dynamics, R. Caflisch Ed., SIAM (1989) 97–106.
  27. V. Scheffer, An inviscid flow with compact support in space-time. J. Geom. Anal. 3 (1993) 343–401. [MathSciNet]
  28. S. Schochet, The weak vorticity formulation of the 2D Euler equations and concentration-cancellation. Comm. P.D.E. 20 (1995) 1077–1104. [CrossRef]
  29. S. Schochet, Point-vortex method for periodic weak solutions of the 2-D Euler equations. Comm. Pure Appl. Math. XLIX (1996) 911–965.
  30. A. Shnirelman, On the non-uniqueness of weak solutions of the Euler equations. Comm. Pure Appl. Math. L (1997) 1261–1286.
  31. P.L. Sulem, C. Sulem, C. Bardos and U. Frisch, Finite time analyticity for the two and three dimensional Kelvin-Helmholtz instability. Comm. Math. Phys. 80 (1981) 485–516. [CrossRef] [MathSciNet]
  32. G. Tryggvason, W. Dahn and K. Sbeih, Fine structure of rollup by viscous and inviscid simulation. J. Fluids Eng.-T ASME 113 (1991) 31–36. [CrossRef]
  33. I. Vecchi and S.J. Wu, On L1-vorticity for 2-D incompressible flow. Manuscripta Math. 78 (1993) 403–412. [CrossRef] [MathSciNet]
  34. M. Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Sci. École Norm. S. Ser. 4 32 (1999) 769–812.
  35. S.J. Wu, Recent progress in mathematical analysis of vortex sheets, in Proceedings of the ICM, Beijing (2002) Vol. III, 233–242.
  36. V. Yudovich, Non-stationary flow of an ideal incompressible liquid (in Russian), Zh. Vych. Mat. 3 (1963) 1032–1066.
  37. V. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal, incompressible fluid. Math. Res. Lett. 2 (1995) 27–38. [MathSciNet]

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