A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution

Transactions of the American Mathematical Society

Volumn 359, Issue 9, 2007, Pages 4125-4142

  Filho, Milton C Lopes ^a   Lopes, Helena J Nussenzveig ^a   Schochet, Steven ^b  

^a UNIVERSITY OF CAMPINAS   (Brazil)

^b TEL AVIV UNIVERSITY   (Israel)

Author keywords

Ideal flow; Incompressible flow; Vortex sheets; Weak solutions

Indexed keywords

EID: 47049087667     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-07-04309-7     Document Type: Article

References (41)

1. G.K. Batchelor. An introduction to fluid dynamics. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1999. MR1744638 (2000j:76001)
2. D. Benedetto and M. Pulvirenti. From vortex layers to vortex sheets. SIAM J. Appl. Math., 52(4):1041-1056, 1992. MR1174045 (93g:76027) (Pubitemid 23572887)
3. G. Birkhoff. Helmholtz and Taylor instability. Proc. Sympos. Appl. Math., Vol. XIII, pages 55-76. American Mathematical Society, Providence, R.I., 1962. MR0137423 (25:875)
4. R. Caflisch and O. Orellana. Long time existence for a slightly perturbed vortex sheet. Comm. Pure Appl. Math., 39(6):807-838, 1986. MR0859274 (87m:76018)
5. R. Caflisch and O. Orellana. Singular solutions and ill-posedness for the evolution of vortex sheets. SIAM J. Math. Anal., 20(2):293-307, 1989. MR0982661 (90d:76020)
6. G. David. Operateurs integraux singuliers sur certaines courbes du plan complexe. Ann. Sci. Ecole Norm. Sup. (4), 17(1):157-189, 1984. MR0744071 (85k:42026)
7. J.-M. Delort. Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc., 4(3):553-586, 1991. MR1102579 (92f:76019)
8. R. DiPerna. A. Majda. Concentrations in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math., 40(3):301-345, 1987. MR0882068 (88e:35149)
9. R. DiPerna and A. Majda. Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys., 108(4):667-689, 1987. MR0877643 (88a:35187)
10. R. DiPerna and A. Majda. Reduced Hausdorff dimension and concentration-cancellation for two-dimensional incompressible flow. J. Amer. Math. Soc., 1(1):59-95, 1988. MR0924702 (89e:35126)
11. J. Duchon and R. Robert. Global vortex sheet solutions of Euler equations in the plane. J. Differential Equations, 73(2):215-224, 1988. MR0943940 (89h:35262)
12. L.C. Evans. Weak convergence methods for nonlinear partial differential equations. CBMS Lecture Notes v. 74, American Math. Soc., Providence, 1990. MR1034481 (91a:35009)
13. L.C. Evans and S. Miiller. Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity. J. Amer. Math. Soc., 7(1):199-219, 1994. MR1220787 (94h:35205)
14. H. Kaden. Aufwicklung einer unstabilen unstetigkeitsflasche. Ing. Arch., 2:140-168, 1931.
15. T. Kambe. Spiral vortex solution of Birkhoff-Rott equation. Phys. D, 37(1-3):463-473, 1989. MR1024399 (90k:76031) (Pubitemid 20615759)
16. R. Krasny. Desingularization of periodic vortex sheet roll-up. J. Comput. Phys., 65:292-313, 1986.
17. R. Krasny. A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech., 167:65-93, 1986. MR0851670 (87g:76028) (Pubitemid 16603670)
18. R. Krasny. Computation of vortex sheet roll-up in the Trefftz plane. J. Fluid Mech., 184:123-155, 1987. (Pubitemid 18536609)
19. G. Lebeau. Regularite du probleme de Kelvin-Helmholtz pour l'equation d'Euler 2d. ESAIM Control Optim. Calc. Var., 8:801-825 (electronic), 2002. MR1932974 (2004a:76011)
20. J.-G. Liu and Z. Xin. Convergence of vortex methods for weak solutions to the 2-D Euler equations with vortex sheet data. Comm. Pure Appl. Math., 48(6):611-628, 1995. MR1338471 (96c:76016)
21. M.C. Lopes Filho, H.J. Nussenzveig Lopes, and M.O. Souza. On the equation satisfied by a steady Prandtl-Munk vortex sheet. Commun. Math. Sci., 1(1):68-73, 2003. MR1979844 (2004c:76029)
22. M.C. Lopes Filho, H.J. Nussenzveig Lopes, and E. Tadmor. Approximate solutions of the incompressible Euler equations with no concentrations. Ann. Inst. H. PoincareAnal. Non Lineaire, 17(3):371-412, 2000. MR1771138 (2001h:35152)
23. M.C. Lopes Filho, H.J. Nussenzveig Lopes, andZ. Xin. Existenceofvortexsheetswith reflection symmetry in two space dimensions. Arch. Ration. Mech. Anal., 158(3):235-257, 2001. MR1842346 (2002e:35197) (Pubitemid 33292867)
24. A. Majda. The interaction ofnonlinear analysis and modern applied mathematics. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pages 175-191, Tokyo, 1991. Math. Soc. Japan. MR1159212 (93g:35003a)
25. A. Majda. Remarks on weak solutions for vortex sheets with a distinguished sign. Indiana Univ. Math. J., 42(3):921-939, 1993. MR1254126 (94k:35236)
26. A. Majda. and A. Bertozzi. Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, Cambridge Univ. Press, Cambridge, 2002. MR1867882 (2003a:76002)
27. C. Marchioro and M. Pulvirenti. Mathematical theory of incompressible nonviscous fluids, volume 96 of Applied Mathematical Sciences. Springer-Verlag, New York, 1994. MR1245492 (94k:76001)
28. D. Meiron, G. Baker, and S. Orszag. Analytic structure of vortex sheet dynamics. I. Kelvin-Helmholtz instability. J. Fluid Mech., 114:283-298, 1982. MR0647268 (83c:76017) (Pubitemid 12515292)
29. D.W. Moore. The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. Roy. Soc. London Ser. A, 365(1720):105-119, 1979. MR0527594 (80b:76006)
30. M. Munk. Isoperimetrische Aufgaben aus der Theorie des Fluges. Inaug.-dissertation, Gottingen, 1919.
31. H.J. Nussenzveig Lopes. A refined estimate of the size of concentration sets for 2D incompressible inviscid flow. Indiana Univ. Math. J., 46(1):165-182, 1997. MR1462801 (98h:35191)
32. S. Osher and R. Fedkiw. Level set methods and dynamic implicit surfaces, volume 153 of Applied Mathematical Sciences. Springer-Verlag, New York, 2003. MR1939127 (2003j:65002)
33. L. Prandtl. Uber die Entstehung von Wirbeln in der idealen Flussigkeit, mit Anwendung auf die Tragflugeltheorie und andere Aufgaben, in Von Karman and Levi-Civita (eds.) Vortrage aus dem Gebiete der Hydro-und Aerodynamik, Springer, Berlin, 1922.
34. D.I. Pullin, J.D. Buntine, and P.G. Saffman. On the spectrum of a stretched spiral vortex. Phys. Fluids, 6(9):3010-3027, 1994. MR1290064 (95f:76047)
35. N. Rott. Diffrraction of a weak shock with vortex generation. J. Fluid Mech., 1:111-128, 1956. MR0082327 (18:534e)
36. P.G. Saffman. Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York, 1992. MR1217252 (94c:76015)
37. S. Schochet. The weak vorticity formulation of the 2-D Euler equations and concentration-cancellation. Comm. Partial Differential Equations, 20(5-6):1077-1104, 1995. MR1326916 (96h:35155)
38. S. Schochet. The point-vortex method for periodic weak solutions of the 2-D Euler equations. Comm. Pure Appl. Math., 49(9):911-965, 1996. MR1399201 (97e:35137) (Pubitemid 126354437)
39. C. Sulem, P.-L. Sulem, C. Bardos, and U. Frisch. Finite time analyticity for the two-and three-dimensional Kelvin-Helmholtz instability. Comm. Math. Phys., 80(4):485-516, 1981. MR0628507 (83d:76012)
40. S. Wu. Recent progress in mathematical analysis of vortex sheets. In Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), pages 233-242, Beijing, 2002. Higher Ed. Press. MR1957535 (2003m:76015)
41. Y. Zheng. Concentration-cancellation for the velocity fields in two-dimensional incompressible fluid flows. Comm. Math. Phys., 135(3):581-594, 1991. MR1091579 (92e:35131)