A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution
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- by Milton C. Lopes Filho, Helena J. Nussenzveig Lopes and Steven Schochet PDF
- Trans. Amer. Math. Soc. 359 (2007), 4125-4142 Request permission
Abstract:
In this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible Euler equations and as a (weak) solution of the Birkhoff-Rott equations. We begin by discussing the classical Birkhoff-Rott equations with respect to arbitrary parametrizations of the sheet. We introduce a notion of weak solution to the Birkhoff-Rott system, and we prove consistency of this notion with the classical formulation of the equations. Our main purpose in this paper is to present a sharp criterion for the equivalence of the weak Euler and weak Birkhoff-Rott descriptions of vortex sheet dynamics.References
- G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638
- D. Benedetto and M. Pulvirenti, From vortex layers to vortex sheets, SIAM J. Appl. Math. 52 (1992), no. 4, 1041–1056. MR 1174045, DOI 10.1137/0152061
- Garrett Birkhoff, Helmholtz and Taylor instability, Proc. Sympos. Appl. Math., Vol. XIII, American Mathematical Society, Providence, R.I., 1962, pp. 55–76. MR 0137423
- Russel E. Caflisch and Oscar F. Orellana, Long time existence for a slightly perturbed vortex sheet, Comm. Pure Appl. Math. 39 (1986), no. 6, 807–838. MR 859274, DOI 10.1002/cpa.3160390605
- Russel E. Caflisch and Oscar F. Orellana, Singular solutions and ill-posedness for the evolution of vortex sheets, SIAM J. Math. Anal. 20 (1989), no. 2, 293–307. MR 982661, DOI 10.1137/0520020
- Guy David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 1, 157–189 (French). MR 744071
- Jean-Marc Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc. 4 (1991), no. 3, 553–586 (French). MR 1102579, DOI 10.1090/S0894-0347-1991-1102579-6
- Ronald J. DiPerna and Andrew J. Majda, Concentrations in regularizations for $2$-D incompressible flow, Comm. Pure Appl. Math. 40 (1987), no. 3, 301–345. MR 882068, DOI 10.1002/cpa.3160400304
- Ronald J. DiPerna and Andrew J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), no. 4, 667–689. MR 877643
- Ronald J. DiPerna and Andrew Majda, Reduced Hausdorff dimension and concentration-cancellation for two-dimensional incompressible flow, J. Amer. Math. Soc. 1 (1988), no. 1, 59–95. MR 924702, DOI 10.1090/S0894-0347-1988-0924702-6
- Jean Duchon and Raoul Robert, Global vortex sheet solutions of Euler equations in the plane, J. Differential Equations 73 (1988), no. 2, 215–224. MR 943940, DOI 10.1016/0022-0396(88)90105-2
- Lawrence C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics, vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1034481, DOI 10.1090/cbms/074
- L. C. Evans and S. Müller, Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity, J. Amer. Math. Soc. 7 (1994), no. 1, 199–219. MR 1220787, DOI 10.1090/S0894-0347-1994-1220787-3
- H. Kaden. Aufwicklung einer unstabilen unstetigkeitsfläsche. Ing. Arch., 2:140–168, 1931.
- T. Kambe, Spiral vortex solution of Birkhoff-Rott equation, Phys. D 37 (1989), no. 1-3, 463–473. Advances in fluid turbulence (Los Alamos, NM, 1988). MR 1024399, DOI 10.1016/0167-2789(89)90150-4
- R. Krasny. Desingularization of periodic vortex sheet roll-up. J. Comput. Phys., 65:292–313, 1986.
- Robert Krasny, A study of singularity formation in a vortex sheet by the point-vortex approximation, J. Fluid Mech. 167 (1986), 65–93. MR 851670, DOI 10.1017/S0022112086002732
- R. Krasny. Computation of vortex sheet roll-up in the Trefftz plane. J. Fluid Mech., 184:123–155, 1987.
- Gilles Lebeau, Régularité du problème de Kelvin-Helmholtz pour l’équation d’Euler 2d, ESAIM Control Optim. Calc. Var. 8 (2002), 801–825 (French, with English and French summaries). A tribute to J. L. Lions. MR 1932974, DOI 10.1051/cocv:2002052
- Jian-Guo Liu and Zhou Ping Xin, Convergence of vortex methods for weak solutions to the $2$-D Euler equations with vortex sheet data, Comm. Pure Appl. Math. 48 (1995), no. 6, 611–628. MR 1338471, DOI 10.1002/cpa.3160480603
- Milton C. Lopes Filho, Helena J. Nussenzveig Lopes, and Max O. Souza, On the equation satisfied by a steady Prandtl-Munk vortex sheet, Commun. Math. Sci. 1 (2003), no. 1, 68–73. MR 1979844
- Milton C. Lopes Filho, Helena J. Nussenzveig Lopes, and Eitan Tadmor, Approximate solutions of the incompressible Euler equations with no concentrations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 17 (2000), no. 3, 371–412 (English, with English and French summaries). MR 1771138, DOI 10.1016/S0294-1449(00)00113-X
- M. C. Lopes Filho, H. J. Nussenzveig Lopes, and Zhouping Xin, Existence of vortex sheets with reflection symmetry in two space dimensions, Arch. Ration. Mech. Anal. 158 (2001), no. 3, 235–257. MR 1842346, DOI 10.1007/s002050100145
- Andrew J. Majda, The interaction of nonlinear analysis and modern applied mathematics, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 175–191. MR 1159212
- Andrew J. Majda, Remarks on weak solutions for vortex sheets with a distinguished sign, Indiana Univ. Math. J. 42 (1993), no. 3, 921–939. MR 1254126, DOI 10.1512/iumj.1993.42.42043
- Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
- Carlo Marchioro and Mario Pulvirenti, Mathematical theory of incompressible nonviscous fluids, Applied Mathematical Sciences, vol. 96, Springer-Verlag, New York, 1994. MR 1245492, DOI 10.1007/978-1-4612-4284-0
- Daniel I. Meiron, Gregory R. Baker, and Steven A. Orszag, Analytic structure of vortex sheet dynamics. I. Kelvin-Helmholtz instability, J. Fluid Mech. 114 (1982), 283–298. MR 647268, DOI 10.1017/S0022112082000159
- D. W. Moore, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet, Proc. Roy. Soc. London Ser. A 365 (1979), no. 1720, 105–119. MR 527594, DOI 10.1098/rspa.1979.0009
- M. Munk. Isoperimetrische Aufgaben aus der Theorie des Fluges. Inaug.-dissertation, Göttingen, 1919.
- H. J. Nussenzveig Lopes, A refined estimate of the size of concentration sets for $2$D incompressible inviscid flow, Indiana Univ. Math. J. 46 (1997), no. 1, 165–182. MR 1462801, DOI 10.1512/iumj.1997.46.1334
- Stanley Osher and Ronald Fedkiw, Level set methods and dynamic implicit surfaces, Applied Mathematical Sciences, vol. 153, Springer-Verlag, New York, 2003. MR 1939127, DOI 10.1007/b98879
- L. Prandtl. Uber die Entstehung von Wirbeln in der idealen Flussigkeit, mit Anwendung auf die Tragflugeltheorie und andere Aufgaben, in Von Kármán and Levi-Civita (eds.) Vorträge aus dem Gebiete der Hydro- und Aerodynamik, Springer, Berlin, 1922.
- D. I. Pullin, James D. Buntine, and P. G. Saffman, On the spectrum of a stretched spiral vortex, Phys. Fluids 6 (1994), no. 9, 3010–3027. MR 1290064, DOI 10.1063/1.868127
- Nicholas Rott, Diffraction of a weak shock with vortex generation, J. Fluid Mech. 1 (1956), 111–128 (1 plate). MR 82327, DOI 10.1017/S0022112056000081
- P. G. Saffman, Vortex dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992. MR 1217252
- Steven Schochet, The weak vorticity formulation of the $2$-D Euler equations and concentration-cancellation, Comm. Partial Differential Equations 20 (1995), no. 5-6, 1077–1104. MR 1326916, DOI 10.1080/03605309508821124
- Steven Schochet, The point-vortex method for periodic weak solutions of the 2-D Euler equations, Comm. Pure Appl. Math. 49 (1996), no. 9, 911–965. MR 1399201, DOI 10.1002/(SICI)1097-0312(199609)49:9<911::AID-CPA2>3.0.CO;2-A
- 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 (1981), no. 4, 485–516. MR 628507
- Sijue Wu, Recent progress in mathematical analysis of vortex sheets, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 233–242. MR 1957535
- Yu Xi Zheng, Concentration-cancellation for the velocity fields in two-dimensional incompressible fluid flows, Comm. Math. Phys. 135 (1991), no. 3, 581–594. MR 1091579
Additional Information
- Milton C. Lopes Filho
- Affiliation: Departamento de Matemática, IMECC-UNICAMP, Cx. Postal 6065, Campinas SP 13081-970, Brazil
- Email: mlopes@ime.unicamp.br
- Helena J. Nussenzveig Lopes
- Affiliation: Departamento de Matemática, IMECC-UNICAMP, Cx. Postal 6065, Campinas SP 13081-970, Brazil
- Email: hlopes@ime.unicamp.br
- Steven Schochet
- Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978 Israel
- Email: schochet@post.tau.ac.il
- Received by editor(s): March 4, 2005
- Published electronically: April 11, 2007
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 4125-4142
- MSC (2000): Primary 76B03; Secondary 35Q35, 76B47
- DOI: https://doi.org/10.1090/S0002-9947-07-04309-7
- MathSciNet review: 2309179