The geometry and analysis of the averaged Euler equations and a new diffeomorphism group

Geometric and Functional Analysis

Volumn 10, Issue 3, 2000, Pages 582-599

  Marsden, Jerrold E ^a   Ratiu, Tudor S ^b   Shkoller, Steve ^c  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b EPFL   (Switzerland)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034407954     PISSN: 1016443X     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00001631     Document Type: Article

References (26)

1. R. ABRAHAM, J.E. MARSDEN, T. RATIU, Manifolds, Tensor Analysis, and Applications, Springer Verlag, New York, 1988.
2. V.I. ARNOLD, Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluids parfaits, Ann. Inst. Grenoble 16 (1966), 319-361.
3. V.I. ARNOLD, B. KHESIN, Topological Methods in Hydrodynamics, Springer Verlag, New York, 1998.
4. G.I. BARENBLATT, A. J. CHORIN, New Perspectives in turbulence: scaling laws, asymptotics, and intermittency, SIAM Rev., 40 (1998), 265-291.
5. G.I. BARENBLATT, A.J. CHORIN, Scaling laws and vanishing viscosity limits in turbulence theory, in "Recent advances in partial differential equations, Venice 1996", Proc. Sympos. Appl. Math. 54 (1998), 1-25.
6. S.Y. CHEN, C. FOIAS, D.D. HOLM, E.J. OLSON, E.S. TITI, S. WYNNE, The Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett. 81 (1999), 5338-5341.
7. A.J. CHORIN, T.J.R. HUGHES, J.E. MARSDEN, M. MCCRACKEN, Product formulas and numerical algorithms, Comm. Pure Appl. Math. 31 (1978), 205-256.
8. D. CIORANESCU, V. GIRAULT, Weak and classical solutions of a family of second grade fluids, Inter. J. Non-Linear Mech. 32 (1997), 317-335.
9. D. CIORANESCU, E.H. OUAZAR, Existence and uniqueness for fluids of second grade, in "Nonlinear Partial Differential Equations" Collège de France Seminar, Pitman 109 (1984), 178-197.
10. J.E. DUNN, R.L. FOSDICK, Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Rat. Mech. Anal. 56 (1974), 191-252.
11. D. EBIN AND J. MARSDEN, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163.
12. C. FOIAS, D.D. HOLM, E.S. TITI, Well-posedness of the averaged Navier-Stokes equations and attractor estimates, in preparation.
13. G.P. GALDI, M. GROBBELAARVANDALSEN, N. SAUER, Existence and uniqueness of classical-solutions of the equations of motion for 2nd-grade fluids, Arch. Rat. Mech. Anal. 124 (1993), 221-237.
14. V. GIRAULT, L.R. SCOTT, Analysis of a two-dimensional grade-two fluid model with a tangential boundary condition, preprint (1999).
15. D.D. HOLM, J.E. MARSDEN, T.S. RATIU, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett. 349 (1998), 4173-4177.
16. D.D. HOLM, J.E. MARSDEN, T.S. RATIU, Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math. 137 (1998), 1-81.
17. J.E. MARSDEN, On product formulas for nonlinear semigroups, J. Funct. Anal. 13 (1973), 51-72.
18. J.E. MARSDEN, D.G. EBIN, A.E. FISCHER, Diffeomorphism groups, hydrodynamics and relativity, Proceedings of the thirteenth biennial seminar of the Canadian Mathematical Congress (J.R. Vanstone, ed.), Montreal (1972), 135-279.
19. J.E. MARSDEN, T.S. RATIU, Introduction to Mechanics and Symmetry, Springer-Verlag, Second Edition, 1999.
20. J.E. MARSDEN, S. SHKOLLER, The anisotropic averaged Euler equations, (2000) E-print, http://xyz.lanl.gov/abs/math.AP/0005033.
21. J.E. MARSDEN, A. WEINSTEIN, Coadjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica D 7 (1983), 305-323.
22. V.G. MAZ'YA, B.A. PLAMENEVSKIǏ, L.I. STUPYALIS, The three-dimensional problem of steady-state motion of a fluid with a free surface, Amer. Math. Soc. Tranl. 123 (1984), 171-268.
23. W. NOLL, C. TRUESDELL, The Nonlinear Field Theories of Mechanics, Springer-Verlag, Berlin, 1965.
24. R.S. RIVLIN, J.L. ERICKSEN, Stress-deformation relations for isotropic materials, J. Rat. Mech. Anal. 4 (1955), 323-425.
25. 1 metric and mean hydrodynamics, J. Funct. Anal. 160 (1998), 337-365.
26. S. SHKOLLER, On averaged incompressible Lagrangian hydrodynamics, (1999), E-print, http://xyz.lanl.gov/abs/math.AP/9908109.