Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations

Communications on Pure and Applied Mathematics

Volumn 52, Issue 4, 1999, Pages 411-452

  Brenier, Yann ^a  

^a ECOLE NORMALE SUPÉRIEURE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033483717     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-0312(199904)52:4<411::AID-CPA1>3.0.CO;2-3     Document Type: Article

References (32)

1. Arnold, V. Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits. (French) Ann. Inst. Fourier (Grenoble) 16 (1966), fasc. 1, 319-361.
2. Arnol'd, V. I.; Khesin, B. A. Topological methods in hydrodynamics. Applied Mathematical Sciences, 125. Springer-Verlag, New York, 1998.
3. Barles, G. Solutions de viscosité des équations d'Hamilton-Jacobi. (French) [Viscosity solutions of Hamilton-Jacobi equations] Mathématiques & Applications [Mathematics & Applications], 17. Springer-Verlag, Paris, 1994.
4. Brenier, Y. The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Amer. Math. Soc. 2 (1989), no. 2, 225-255.
5. Brenier, Y. The dual least action problem for an ideal, incompressible fluid. Arch. Rational Mech. Anal. 122 (1993), no. 4, 323-351.
6. Brenier, Y. On the motion of an ideal incompressible fluid. Partial differential equations of elliptic type (Cortona, 1992), 123-148, Sympos. Math., XXXV. Cambridge Univ. Press, Cambridge, 1994.
7. Brenier, Y. On the motion of an ideal incompressible fluid. Partial differential equations of elliptic type (Cortona, 1992), 123-148, Sympos. Math., XXXV. Cambridge Univ. Press, Cambridge, 1994.
8. Brenier, Y. Quelques lois de conservations issues de modèles cinétiques. (French) [Some conservation laws arising in kinetic models] Journées "Équations aux Dérivées Partielles" (Saint-Jean-de-Monts, 1995), Exp. No. I, École Polytechnique, Palaiseau, 1995.
9. Brenier, Y. A homogenized model for vortex sheets. Arch. Rational Mech. Anal. 138 (1997), no. 4, 319-353.
10. Brézis, H. Analyse fonctionnelle. (French) [Functional analysis] Théorie et applications. [Theory and applications] Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree] Masson, Paris, 1983.
11. Caflisch, R.; Papanicolaou, G. C. Dynamic theory of suspensions with Brownian effects. SIAM J. Appl. Math. 43 (1983), no. 4, 885-906.
12. Chemin, J.-Y. Fluides parfaits incompressibles. (French) [Incompressible perfect fluids] Astérisque No. 230, (1995).
13. Chorin, A. J. Vorticity and Turbulence. Applied Mathematical Sciences, 103. Springer-Verlag, New York, 1994.
14. Dacorogna, B.; Moser, J. On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 1, 1-26.
15. DiPerna, R. J.; Majda, A. J. Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108 (1987), 667-689.
16. Duchon, J.; Robert, R. Relaxation of Euler equations and hydrodynamic instabilities. Quart. Appl. Math. 50 (1992), 235-255.
17. Ebin, D. G.; Marsden, J. Groups of diffeomorphisms and the notion of an incompressible fluid. Ann. of Math. (2) 92 (1970), 102-163.
18. Grenier, E. Oscillations in quasineutral plasmas. Comm. Partial Differential Equations 21 (1996), 363-394.
19. Grenier, E. Pseudo-differential energy estimates of singular perturbations. Comm. Pure Appl. Math. 50 (1997), 821-865.
20. Lions, P.-L. Mathematical topics in fluid mechanics, Vol. I. Incompressible models. Oxford Lecture Series in Mathematics and Its Applications, 3, Oxford University Press, New York, 1996.
21. Majda, A. J. The interaction of nonlinear analysis and modern applied mathematics. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 175-191. Math. Soc. Japan, Tokyo, 1991.
22. Majda, A.; Zheng, Y. X. Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data. Comm. Pure Appl. Math. 47 (1994), 1365-1401.
23. Marchioro, C.; Pulvirenti, M. Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences, 96, Springer-Verlag, New York, 1994.
24. Roesch, M. Contribution à la description géométrique des fluides parfaits incompressibles. Thèse, Université Paris 6, 1995.
25. Royden, H. L. Real analysis. MacMillan, New York, 1963.
26. Shelukhin, V. Existence theorem in the variational problem for compressible inviscid fluids. Manuscripta Math. 61 (1988), 495-509.
27. Shnirelman, A. I. The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid. Mat. Sb. (N. S.) 128(170) (1985), 82-109, 144.
28. Shnirelman, A. I. Attainable diffeomorphisms. Geom. Funct. Anal. 3 (1993), 279-294.
29. Shnirelman, A. I. Generalized fluid flows, their approximation and applications. Geom. Funct. Anal. 4 (1994), 586-620.
30. Tartar, L. The compensated compactness method applied to systems of conservation laws. Systems of nonlinear partial differential equations (Oxford, 1982), 263-285. NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 111. Reidel, Dordrecht-Boston, 1983.
31. Tartar, L. The compensated compactness method applied to systems of conservation laws. Systems of nonlinear partial differential equations (Oxford, 1982), 263-285. NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 111. Reidel, Dordrecht-Boston, 1983.
32. Young, L. C. Lectures on the calculus of variations and optimal control theory. Chelsea, New York, 1980.