Mathematical analysis of vortex sheets

Communications on Pure and Applied Mathematics

Volumn 59, Issue 8, 2006, Pages 1065-1206

  Wu, Sijue ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33745626744     PISSN: 00103640     EISSN: 00103640     Source Type: Journal    
DOI: 10.1002/cpa.20110     Document Type: Article

References (52)

1. Bergh, J.; Löfström, J, Interpolation spaces: an introduction. Grundlehren der Mathematischen Wissenschaften, 223. Springer, Berlin-New York, 1976.
2. Birkhoff, G. Helmholtz and Taylor instability. Proc. Sympos. Appl. Math., Vol. XIII, 55-76. American Mathematical Society, Providence, R.I., 1962.
3. Caflisch, R. E.; Orellana, O. F. Long time existence for a slightly perturbed vortex sheet. Comm. Pure Appl. Math. 39 (1986), no. 6, 807-838.
4. Caflisch, R, E.; Orellana, O. F. Singular solutions and ill-posedness for the evolution of vortex sheets. SIAM J. Math. Anal. 20 (1989), no. 2, 293-307.
5. Calderón, A.-P. Cauchy integrals on Lipschitz curves and related operators. Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 4, 1324-1327.
6. Chazarain, J.; Piriou, A. Introduction to the theory of linear partial differential equations. Studies in Mathematics and Its Applications, 14. North-Holland, Amsterdam-New York, 1982.
7. Christ, M. Lectures on singular integral operators. CBMS Regional Conference Series in Mathematics, 77. American Mathematical Society, Providence, R.I., 1990.
8. Coifman, R. R. Topics in nonlinear analysis. Lecture notes by M. Murray. Yale University, New Haven, Conn., 1982.
9. Coifman, R. R.; Fefferman, C. Weighted norm inequalities for maximal functions and singular integrals. Studio Math. 51 (1974), 241-250.
10. 2 boundedness of Cauchy integrals on Lipschitz curves. J. Amer. Math. Soc. 2 (1989), no. 3, 553-564.
11. 2 pour les courbes lipschitziennes. Ann. of Math. (2) 116 (1982), no. 2, 361-387.
12. Coifman, R.; Meyer, Y. Lavrentiev's curves and conformai mappings. Report no. 5. Institut Mittag-Leffler, Djursholm, Sweden, 1983.
13. Coifman, R. R.; Meyer, Y. Nonlinear harmonic analysis, operator theory and P.D.E. Beijing lectures in harmonic analysis (Beijing, 1984), 3-45. Annals of Mathematics Studies, 112. Princeton University Press, Princeton, N.J., 1986.
14. Coifman, R. R.; Rochberg, R.; Weiss, G. Factorization theorems for Hardy spaces in several variables. Ann. of Math. (2) 103 (1976), no. 3, 611-635.
15. David, G. Thése de troisiéme cycle. Mathématique. Doctoral dissertation, Université de Paris XI, 1981.
16. David, G. Courbes corde-arc et espaces de Hardy généralisés. Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, xi, 227-239.
17. David, G. Opérateurs intégraux singuliers sur certaines courbes du plan complexe. Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 1, 157-189.
18. David, G.; Journé, J.-L. A boundedness criterion for generalized Calderón-Zygmund operators. Ann. of Math. (2) 120 (1984), no. 2, 371-397.
19. David, G.; Journé, J.-L.; Semmes, S. Opérateurs de Calderón-Zygmund, fonctions paraaccrétives et interpolation. Rev. Mat. Iberoamericana 1 (1985), no. 4, 1-56.
20. Delort, J.-M. Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc. 4 (1991), no. 3, 553-586.
21. DiPerna, R. J.; Majda, A. J. Concentrations in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. 40 (1987), no. 3, 301-345.
22. Duchon, J.; Robert, R. Global vortex sheet solutions of Euler equations in the plane. J. Differential Equations 73 (1988), no. 2, 215-224.
23. Ebin, D. G. Ill-posedness of the Rayleigh-Taylor and Heimholte problems for incompressible fluids. Comm. Partial Differential Equations 13 (1988), no. 10, 1265-1295.
24. Evans, L. C. Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, R.I., 1998.
25. Journé, J.-L. Calderón-Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón. Lecture Notes in Mathematics, 994. Springer, Berlin, 1983.
26. Kambe, T. Spiral vortex solution of Birkhoff-Rott equation. Advances in fluid turbulence (Los Alamos, NM, 1988). Phys. D 37 (1989), no. 1-3, 463-473.
27. Krasny, R. Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65 (1986), 292-313.
28. Krasny, R. A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167 (1986), 65-93.
29. Krasny, R. Computing vortex sheet motion. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 1573-1583. Mathematical Society of Japan, Tokyo, 1991.
30. Ladyzhenskaya, O. A. Solution "in the large" of the nonstationary boundary value problem for the Navier-Stokes System with two space variables. Comm. Pure Appl, Math. 12 (1959), 427-433.
31. Lebeau, G. Régularité du problème de Kelvin-Helmholtz pour l'équation d'Euler 2d. Séminaire: Équations aux Dérivées Partielles, 2000-2001, Exp. No. II, 12 pp. Séminaire: Équations aux Dérivées Partielles. École Polytechnique, Palaiseau, France, 2001.
32. Liu, J.-G.; Xin, Z. P. 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.
33. Lopes, M. C.; Lowengrub, J.; Nussenzveig Lopes, H. J.; Zheng, Y. X. Numerical evidence for nonuniqueness evolution for the 2D incompressible Euler equations: a vortex sheet example. Preprint, 2000.
34. Lopes, M. C.; Nussenzveig Lopes, H. J.; Schochet, S. A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution. arXiv: math.AP/0502215. Preprint, 2005.
35. Marchioro, C.; Pulvirenti, M. Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences, 96. Springer, New York, 1994.
36. Meiron, D. I.; Baker, G. R.; Orszag, S. A. Analytic structure of vortex sheet dynamics. I. Kelvin-Helmholtz instability. J. Fluid Mech. 114 (1982), 283-298.
37. Moore, D. W. 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.
38. Moore, D. W. Numerical and analytical aspects of Helmholtz instability. Theoretical and applied mechanics (Lyngby, 1984), 263-274. North-Holland, Amsterdam, 1985.
39. Pullin, D. I. The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88 (1978), no. 3, 401-430.
40. Pullin, D. I. On similarity flows containing two-branched vortex sheets. Mathematical aspects of vortex dynamics (Leesburg, VA, 1988), 97-106. SIAM, Philadelphia, 1989.
41. Pullin, D. I.; Phillips, W. R. C. On a generalization of Kaden's problem. J. Fluid Mech. 104 (1981), 45-53.
42. Rott, N. Diffraction of a weak shock with vortex generation. J. Fluid Mech. 1 (1956), 111-128.
43. Saffman, P. G. Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York, 1992.
44. Scheffer, V. An inviscid flow with compact support in space-time. J. Geom. Anal. 3 (1993), no. 4, 343-401.
45. Shnirelman, A. On the nonuniqueness of weak solution of the Euler equation. Comm. Pure Appl. Math. 50 (1997), no. 12, 1261-1286.
46. Stein, E. M. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, 30. Princeton University Press, Princeton, N.J., 1970.
47. Sulem, C.; Sulem, P.-L.; Bardos, C.; Frisch, U. Finite time analyticity for the two- and three-dimensional Kelvin-Helmholtz instability. Comm. Math. Phys. 80 (1981), no. 4, 485-516.
48. Van Dyke, M. An album of fluid motion. Parabolic Press, Stanford, Calif., 1982.
49. Vishik, M. Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 6, 769-812.
50. Wu, S. Recent progress in mathematical analysis of vortex sheets. Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 233-242. Higher Education Press, Beijing, 2002.
51. [51 ] Yudovich, V. Non-stationary flow of an ideal incompressible liquid. Comp. Math. Math. Phys. 3 (1963), 1407-1457.
52. Yudovich, V. I. Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett. 2 (1995), no. 1, 27-38.