Relaxation of Euler equations and hydrodynamic instabilities
Authors:
Jean Duchon and Raoul Robert
Journal:
Quart. Appl. Math. 50 (1992), 235-255
MSC:
Primary 76C05; Secondary 35Q35, 76E99
DOI:
https://doi.org/10.1090/qam/1162274
MathSciNet review:
MR1162274
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Abstract: We present a relaxed version of incompressible Euler equations that permits foliated flows involving two velocities. These relaxed equations allow a two-phase evolution of some vortex sheets as an alternative to discontinuous solutions of Euler equations. In the case of two perfect fluids of different densities superposed one over the other, we show that this relaxation process yields a linearly well-posed two-phase solution.
- V. Arnol′d, Chapitres supplémentaires de la théorie des équations différentielles ordinaires, “Mir”, Moscow, 1984 (French). Translated from the Russian by Djilali Embarek; Reprint of the 1980 edition. MR 898218
- C. Bardos and U. Frisch, Finite-time regularity for bounded and unbounded ideal incompressible fluids using Hölder estimates, Turbulence and Navier-Stokes equations (Proc. Conf., Univ. Paris-Sud, Orsay, 1975) Springer, Berlin, 1976, pp. 1–13. Lecture Notes in Math., Vol. 565. MR 0467034
- Garrett Birkhoff, Helmholtz and Taylor instability, Proc. Sympos. Appl. Math., Vol. XIII, American Mathematical Society, Providence, R.I., 1962, pp. 55–76. MR 0137423
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- 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 https://doi.org/10.1137/0520020
J. R. Chan-Hong, Sur quelques problèmes à frontière libre en hydrologie, Thèse, Université de Lyon, 1987
P. Comte, M. Lesieur, and J. P. Chollet, Simulation numérique d’un jet plan turbulent, C. R. Acad. Sci. Paris Ser. II Méc. Phys. Chim. Sci. Univers. Sci. Terre 305, 1037–1044 (1987)
- Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223–270. MR 775191, DOI https://doi.org/10.1007/BF00752112
- 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 https://doi.org/10.1090/S0894-0347-1988-0924702-6
- Jean Duchon and Raoul Robert, Solutions globales avec nappe tourbillonnaire pour les équations d’Euler dans le plan, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 5, 183–186 (French, with English summary). MR 832068
- Jean Duchon and Raoul Robert, Élargissement diphasique d’une nappe tourbillonnaire en dynamique du fluide parfait incompressible, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 15, 701–704 (French, with English summary). MR 917601
- 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 https://doi.org/10.1016/0022-0396%2888%2990105-2
- J. Duchon and R. Robert, Estimation d’opérateurs intégraux du type de Cauchy dans les échelles d’Ovsjannikov et application, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 1, 83–95 (French). MR 840714
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J. Duchon and R. Robert, to appear
T. Kato, Non stationary flows of viscous and ideal fluids in R$^{3}$, J. Funct. Anal. 9, 296–305 (1972)
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C. Sulem, Thesis, University Paris-Nord, 1983
- 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
- L. C. Young, Lectures on the calculus of variations and optimal control theory, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969. Foreword by Wendell H. Fleming. MR 0259704
J. M. Delort, Existence de nappes de tourbillon pour l’équation d’Euler dans le plan, Prépublication, Université Paris-Sud (1990)
V. Arnold, Chapitres Supplémentaires de la Théorie des Équations Différentielles Ordinaires, Mir, Moscow, 1984
C. Bardos and U. Frisch, Finite time regularity for bounded and unbounded ideal incompressible fluids using Hölder estimates, Lecture Notes in Math. vol. 565, Springer-Verlag, 1975, pp. 1–14
G. Birkhoff, Helmholtz and Taylor instability, Proc. Sympos. Appl. Math., vol. 13, Amer. Math. Soc., Providence, RI, 1962
Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc. 2, 225–255 (1989)
R. Caflisch and O. Orellana, Long time existence for a slightly perturbed vortex sheet, Comm. Pure Appl. Math. 39, 807–838 (1986)
R. Caflisch and O. Orellana, Singular solutions and well posedness for the evolution of vortex sheets, SIAM J. Math. Anal. 20, 293–307 (1989)
J. R. Chan-Hong, Sur quelques problèmes à frontière libre en hydrologie, Thèse, Université de Lyon, 1987
P. Comte, M. Lesieur, and J. P. Chollet, Simulation numérique d’un jet plan turbulent, C. R. Acad. Sci. Paris Ser. II Méc. Phys. Chim. Sci. Univers. Sci. Terre 305, 1037–1044 (1987)
R. Di Perna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88, 223–270 (1985)
R. Di Perna and A. Majda, Reduced Hausdorff dimension and concentration-cancellation for two-dimensional incompressible flow, J. Amer. Math. Soc. 1, 59–95 (1988)
J. Duchon and R. Robert, Solutions globales avec nappe tourbillonnaire pour les équations d’Euler dans le plan, C. R. Acad. Sci. Paris Sér. I Math. 302, 183–186 (1986)
J. Duchon and R. Robert, Elargissement diphasique d’une nappe tourbillonnaire en dynamique du fluide parfait incompressible, C. R. Acad. Sci. Paris Sér. I Math. 305, 701–704 (1987)
J. Duchon and R. Robert, Global vortex sheet solutions of Euler equations in the plane, J. Differential Equations 73, 215–224 (1988)
J. Duchon and R. Robert, Estimation d’opérateurs intégraux du type de Cauchy dans les échelles d’Ovsjannikov et applications, Ann. Inst. Fourier 36, 83–95 (1986)
J. Duchon and R. Robert, Sur quelques problèmes à frontière libre analytique dans le plan, Séminaire Bony-Sjöstrand-Meyer, Exposé no. X, janvier 1985
J. Duchon and R. Robert, to appear
T. Kato, Non stationary flows of viscous and ideal fluids in R$^{3}$, J. Funct. Anal. 9, 296–305 (1972)
R. Krasny, Desingularization of periodic vortex sheet roll-up, J. Comp. Phys. 65, 292–313 (1986)
D. W. McLaughlin, G. C. Papanicolaou, and O. R. Pironneau, Convection of microstructures and related problems, SIAM J. Appl. Math. 45, 780–797 (1985)
D. Serre, Large oscillations in hyperbolic system of conservation laws, Prépublication, Université de St. Etienne, 1987
C. Sulem, Thesis, University Paris-Nord, 1983
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, 485–516 (1981)
L. C. Young, Lectures on the Calculus of Variations, Saunders, Philadelphia, 1969
J. M. Delort, Existence de nappes de tourbillon pour l’équation d’Euler dans le plan, Prépublication, Université Paris-Sud (1990)
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Article copyright:
© Copyright 1992
American Mathematical Society