Euler equations for incompressible ideal fluids

Russian Mathematical Surveys

Volumn 62, Issue 3, 2007, Pages 409-451

  Bardos, C ^a   Titi, E S ^b,c  

^a UNIVERSITÉ PARIS DESCARTES   (France)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c WEIZMANN INSTITUTE OF SCIENCE   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 35748978223     PISSN: 00360279     EISSN: 14684829     Source Type: Journal    
DOI: 10.1070/RM2007v062n03ABEH004410     Document Type: Conference Paper

References (79)

1. S. Montgomery-Smith, "Finite time blow up for a Navier-Stokes like equation", Proc. Amer. Math. Soc. 129:10 (2001), 3025-3029.
2. H. Bellout, S. Benachour, and E. S. Titi, "Finite-time singularity versus global regularity for hyper-viscous Hamilton-Jacobi-like equations", Nonlinearity 16:6 (2003), 1967-1989.
3. T. Kato, "Remarks on zero viscosity limit for non stationary Navier-Stokes flows with boundary", Seminar on nonlinear partial differential equations (Berkeley, 1983) (S.S. Chern, ed.), Math. Sci. Res. Inst. Publ., vol. 2, Springer, New York 1984, pp. 85-98.
4. G. Lebeau, "Régularité du problème de Kelvin-Helmholtz pour l'équation d'Euler 2d", ESAIM Control Optim. Cale. Var. 8 (2002), 801-825.
5. S. Wu, "Recent progress in mathematical analysis of vortex sheets", Proceedings of the international congress of mathematicians, vol. III. Invited lectures (Beijing, China, 2002), Higher Ed. Press, Beijing 2002, pp. 233-242.
6. S. Wu, "Mathematical analysis of vortex sheets", Comm. Pure Appl. Math. 59:8 (2006), 1065-1206.
7. A. J. Majda and A. L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts Appl. Math., vol. 27, Cambridge Univ. Press, Cambridge 2002.
8. C. Marchioro and M. Pulvirenti, Mathematical theory of incompressible nonviscous fluids, Appl. Math. Sei., vol. 96, Springer-Verlag, New York 1994.
9. L. Lichtenstein, "Über einige Existenzprobleme der Hydrodynamik homogener, unzusammendrückbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsätze", Math. Z. 23:1 (1925), 89-154;
10. "Über einige Existenzprobleme der Hydrodynamik. Zweite Abhandlung. Nichthomogene, unzusammendrückbare, reibungslose Flüssigkeiten", Math. Z. 26:1 (1927), 196-323;
11. "Über einige Existenzprobleme der Hydrodynamik. III. Permanente Bewegungen einer homogenen, inkompressiblen, zähen Flüssigkeit", Math. Z. 28 (1928), 387-415;
12. "Über einige Existenzprobleme der Hydrodynamik. IV. Stetigkeitssätze. Eine Begründung der Helmholtz-KirchhofTschen Theorie geradliniger Wirbelfäden", Math. Z. 32:1 (1930), 608-640.
13. n", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4:4 (1977), 647-687.
14. J. T. Beale, T. Kato, and A. Majda, "Remarks on the breakdown of smooth solutions for the 3-D Euler equations", Comm. Math. Phys. 94:1 (1984), 61-66.
15. A. Ferrari, "On the blow-up of solutions of the 3-D Euler equations in a bounded domain", Comm. Math. Phys. 155:2 (1993), 277-294.
16. ∞ and BMO", Kyushu J. Math. 57:2 (2003), 303-324.
17. Y. Meyer, Wavelets and operators, vol. I, Cambridge Stud. Adv. Math., vol. 37, Cambridge Univ. Press, Cambridge 1992.
18. G. Ponce, "Remarks on a paper "Remarks on the breakdown of smooth solutions for the 3-D Euler equations" by J. T. Beale, T. Kato and A. Majda", Comm. Math. Phys. 98:3 (1985), 349-353.
19. P. Constantin, C. Fefferman, and A. Majda, "Geometric constraints on potential singular solutions for the 3-D Euler equations", Comm. Partial Differential Equations 21:3-4 (1996), 559-571.
20. P. Constantin, "Euler equations, Navier-Stokes equations and turbulence", Mathematical foundation of turbulent viscous flows (Martina Franca, Italy, 2003) (M. Cannone and T. Miyakawa, eds.), Lecture Notes in Math., vol. 1871, Springer, Berlin 2006, pp. 1-43.
21. Russian source
22. English transl., V. I. Yudovich, "Non-stationary flows of an ideal incompressible fluid", USSR Comput. Math. and Math. Phys. 3 (1963), 1407-1456.
23. J.-M. Delort, "Existence de nappes de tourbillon en dimension deux", J. Amer. Math. Soc. 4:3 (1991), 553-586.
24. V. Scheffer, "An inviscid flow with compact support in space-time", J. Geom. Anal. 3:4 (1993), 343-401.
25. A. Shnirelman, "On the nonuniqueness of weak solution of the Euler equation", Comm. Pure Appl. Math. 50:12 (1997), 1261-1286.
26. R. Caflisch and O. Orellana, "Singular solutions and ill-posedness for the evolution of vortex sheets", SIAM J. Math. Anal. 20:2 (1989), 293-307.
27. K. Ohkitani and J. D. Gibbon, "Numerical study of singularity formation in a class of Euler and Navier-Stokes flows", Phys. Fluids 12:12 (2000), 3181-3194.
28. T. Hou and C. Li, Dynamic stability of the 3D axi-symmetric Navier-Stokes equations with swirl, arXiv: math/0608295, 2006.
29. J. Leray, "Etude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l'hydrodynamique", J. Math. Pures Appl. (9) 12 (1933), 1-82.
30. J. Leray, "Essai sur les mouvements plans d'une liquide visqueux que limitent des parois", J. Math. Pures Appl. (9) 13 (1934), 331-418.
31. J. Leray, "Sur le mouvement d'un liquide visqueux emplissant l'espace", Acta Math. 63:1 (1934), 193-248.
32. E. Hopf, "Ein allgemeiner Endlichkeitssatz der Hydrodynamik", Math. Ann. 117:1 (1941), 764-775.
33. E. Hopf, "Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen", Math. Nachr. 4 (1951), 213-231.
34. E. Hopf, "On nonlinear partial differential equations", Lecture series of the symposium on partial differential equations (Berkeley, 1955), Univ. Kansas Press, Lawrence 1957, pp. 1-29.
35. YMH 58:2 (2003), 251-286.
36. English transl., O. A. Ladyzhenskaya, "Sixth problem of the millennium: Navier-Stokes equations, existence and smoothness", Russian Math. Surveys 58:2 (2003), 251-286.
37. L. Caffarelli, R. Kohn, and L. Nirenberg, "Partial regularity of suitable weak solutions of the Navier-Stokes equations", Comm. Pure Appl. Math. 35:6 (1982), 771-831.
38. B. E. Launder and D. B. Spalding, Lectures in mathematical models of turbulence, Academic Press, London-New York 1972.
39. B. Mohammadi and O. Pironneau, Analysis of the k-epsilon turbulence model, RAM Res. Appl. Math., Masson, Paris; Wiley, Chichester 1994.
40. S. B. Pope, Turbulent flows, Cambridge Univ. Press, Cambridge 2000.
41. J. S. Smagorinsky, "General circulation experiments with the primitive equations", Monthly Weather Rev. 91:3 (1963), 99-164.
42. L. Onsager, "Statistical hydrodynamics", Nuovo Cimento 6:2 (suppl.) (1949), 279-287.
43. G. L. Eyink, "Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer", Phys. D 78:3-4 (1994), 222-240.
44. P. Constantin, W. E, and E. S. Titi, "Onsager's conjecture on the energy conservation for solutions of Euler's equation", Comm. Math. Phys. 165:2 (1994), 207-209.
45. P.-L. Lions, Mathematical topics in fluid mechanics, vol. 1. Incompressible models, Oxford Lecture Ser. Math. Appl., vol. 3, Clarendon Press, Oxford 1996.
46. P. Constantin and C. Foias, Navier-Stokes equations, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL 1988.
47. G. David, "Courbes corde-arc et espaces de Hardy généralisés", Ann. Inst. Fourier (Grenoble) 32:3 (1982), 227-239.
48. C. Bardos, F. Golse, and D. Levermore, "Fluid dynamic limits of kinetic equations. I. Formal derivations", J. Statist. Phys. 63:1-2 (1991), 323-344.
49. Z. R. DiPerna and P.-L. Lions, "On the Cauchy problem for Boltzmann equations: global existence and weak stability", Ann. of Math. (2) 130:2 (1989), 321-366.
50. F. Golse and L. Saint-Raymond, "The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels", Invent. Math. 155:1 (2004), 81-161.
51. L. Saint-Raymond, "Convergence of solutions to the Boltzmann equation in the incompressible Euler limit", Arch. Ration. Mech. Anal. 166:1 (2003), 47-80.
52. C. Cheverry, "Propagation of oscillations in real vanishing viscosity limits", Comm. Math. Phys. 247:3 (2004), 655-695.
53. C. Bardos, "Existence et unicité de la solution de l'équation d'Euler en dimension deux", J. Math. Anal. Appl. 40:3 (1972), 769-790.
54. P. Constantin and J. Wu, "The inviscid limit for non-smooth vorticity", Indiana Univ. Math. J. 45:1 (1996), 67-81.
55. M. C. Lopes Filho, H. J. Nussenzveig Lopes, and Z. Xin, "Existence of vortex sheets with reflection symmetry in two space dimensions", Arch. Ration. Mech. Anal. 158:3 (2001), 235-257.
56. P. Gérard, P. Markowich, N. Mauser, and F. Poupaud, "Homogenization limits and Wigner transforms", Comm. Pure Appl. Math. 50:4 (1997), 323-379.
57. Russian source
58. English transl., M. Vishik and A. Fursikov, Mathematical problems of statistical hydromechanics, Math. Appl. (Soviet Ser.), vol. 9, Kluwer, Dordrecht 1988.
59. C. Foias, O. Manely, R. Rosa, and R. Temam, Navier-Stokes equations and turbulence, Encyclopedia Math. Appl., vol. 83, Cambridge Univ. Press, Cambridge 2001.
60. C. Foias, "What do the Navier-Stokes equations tell us about turbulence?", Harmonic analysis and nonlinear differential equations (Riverside, CA, USA, 1995), Contemp. Math., vol. 208, Providence, RI, Amer. Math. Soc. 1997, pp. 151-180.
61. U. Frisch, Turbulence. The legacy of A. N. Kolmogorov, Cambridge Univ. Press, Cambridge 1995.
62. E. Grenier, "On the nonlinear instability of Euler and Prandtl equations", Comm. Pure Appl. Math. 53:9 (2000), 1067-1109.
63. IIMM 30:5 (1966),801-821
64. English transl., O. A. Oleinik, "On the mathematical theory of boundary layer for an unsteady flow of incompressible fluid", J. Appl. Math. Mech. 30:5 (1966), 951-974.
65. W. E and B. Engquist, "Blowup of solutions to the unsteady Prandtl's equation", Comm. Pure Appl. Math. 50:12 (1997), 1287-1293.
66. W. E, "Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation", Acta Math. Sin. (Engl. Ser.) 16:2 (2000), 207-218.
67. K. Asano, "Zero-viscosity limit of the incompressible Navier-Stokes equations. II", Mathematical analysis of fluid and plasma dynamics, I (Kyoto, 1986), Sūrikaisekikenkyūsho Kōkyūroku, vol. 656, Kyoto Univ., Kyoto 1988, pp. 105-128.
68. R. Caflisch and M. Sammartino, "Navier-Stokes equations on a circular domain: construction of the solution and zero viscosity limit", C. R. Acad. Sci. Paris. Sér. I Math. 324:8 (1997), 861-866.
69. M. Cannone, M. C. Lombardo, and M. Sammartino, "Existence and uniqueness for Prandtl equations", C. R. Acad. Sci. Paris. Sér. I Math. 332:3 (2001), 277-282.
70. 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 (1981), 485-516.
71. J. Duchon and R. Robert, "Global vortex sheet solutions of Euler equations in the plane", J. Differential Equations 73:2 (1988), 215-224.
72. 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 (1979), 105-119.
73. D. Meiron, G. Baker, and S. Orszag, "Analytic structure of vortex sheet dynamics. I. Kelvin-Helmholtz instability", J. Fluid Mech. 114 (1982), 283-298.
74. V. Kamotski and G. Lebeau, "On 2D Ray leigh-Taylor instabilities", Asymptot. Anal. 42:1-2 (2005), 1-27.
75. R. Krasny, "Computation of vortex sheet roll-up in Trefftz plane", J. Fluid Mech. 184 (1987), 123-155.
76. S. Wu, "Well-posedness in Sobolev spaces of the full water wave problem in 2-D", Invent. Math. 130:1 (1997), 39-72.
77. 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 (2003), 68-73.
78. M. C. Lopes Filho, H. J. Nussenzveig Lopes, and S. Schochet, A criterion for the equivalence of the Birkhoff-Rott and Euler description of vortex sheet evolution, arXiv: math/0502215, 2005.
79. C. M. Dafermos, Hyperbolic conservation laws in continuum physics, 1st edn., Grundlehren Math. Wiss., vol. 325, Springer-Verlag, Berlin 2000.