Fast singular oscillating limits and global regularity for the 3D primitive equations of geophysics

Mathematical Modelling and Numerical Analysis

Volumn 34, Issue 2, 2000, Pages 201-222

  Babin, Anatoli ^a   Mahalov, Alex ^b   Nicolaenko, Basil ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b ARIZONA STATE UNIVERSITY   (United States)

Author keywords

Fast singular oscillating limits; Primitive equations for geophysical fluid flows; Three dimensional Navier Stokes equations

Indexed keywords

EID: 0034383815     PISSN: 0764583X     EISSN: None     Source Type: Journal    
DOI: 10.1051/m2an:2000138     Document Type: Article

References (49)

1. V.I. Arnold, Small denominators. I. Mappings of the circumference onto itself. Amer. Math. Soc. Transl. Ser. 2. 46 (1965) 213-284.
2. V.I. Arnold and B.A. Khesin, Topological Methods in Hydrodynamics. Appl. Math. Sci. 125 (1997).
3. J. Avrin, A. Babin, A. Mahalov and B. Nicolaenko, On regularity of solutions of 3D Navier-Stokes equations. Appl. Anal. 71 (1999) 197-214.
4. A. Babin, A. Mahalov and B. Nicolaenko, Long-time averaged Euler and Navier-Stokes equations for rotating fluids, In Structure and Dynamics of Nonlinear Waves in Fluids, 1994 IUTAM Conference, K. Kirchgässner and A. Mielke Eds, World Scientific (1995) 145-157.
5. A. Babin, A. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids. Europ. J. Mech. B/Fluids 15, No. 3, (1996) 291-300.
6. A. Babin, A. Mahalov and B. Nicolaenko, Resonances and regularity for Boussinesc equations. Russian J. Math. Phys. 4, No. 4, (1996) 417-428.
7. A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability of rotating shallow-water equations. Proc. Acad. Sci. Paris Ser. 1 324 (1997) 593-598.
8. A. Babin, A. Mahalov and B. Nicolaenko, Global regularity and integrability of 3D Euler and Navier-Stokes equations for uniformly rotating fluids. Asympt. Anal. 15 (1997) 103-150.
9. A. Babin, A. Mahalov and B. Nicolaenko, Global splitting and regularity of rotating shallow-water equations. Eur. J. Mech., B/Fluids 16, No. 1, (1997) 725-754.
10. A. Babin, A. Mahalov and B. Nicolaenko, On the nonlinear baroclinic waves and adjustment of pancake dynamics. Theor. and Comp. Fluid Dynamics 11 (1998) 215-235.
11. A. Babin, A. Mahalov, B. Nicolaenko and Y. Zhou, On the asymptotic regimes and the strongly stratified limit of rotating Boussinesq equations. Theor. and Comp. Fluid Dyn. 9 (1997) 223-251.
12. A. Babin, A. Mahalov and B. Nicolaenko, On the regularity of three-dimensional rotating Euler-Boussinesq equations. Math. Models Methods Appl. Sci., 9, No. 7 (1999) 1089-1121.
13. A. Babin, A. Mahalov and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Lett. Appl. Math. (to appear).
14. A. Babin, A. Mahalov and B. Nicolaenko, Global Regularity of 3D Rotating Navier-Stokes Equations for Resonant Domains. Indiana University Mathematics Journal 48, No. 3, (1999) 1133-1176.
15. A. Babin, A. Mahalov and B. Nicolaenko, Fast singular oscillating limits of stably stratified three-dimensional Euler-Boussinesq equations and ageostrophic wave fronts, to appear in Mathematics of Atmosphere and Ocean Dynamics, Cambridge University Press (1999).
16. A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam (1992).
17. n. Annali della Scuola Normale Superiore di Pisa 4 (1977) 647-687.
18. P. Bartello, Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atm. Sci. 52, No. 24, (1995) 4410-4428.
19. A.J. Bourgeois and J.T. Beale, Validity of the quasigeostrophic model for large-scale flow in the atmosphere and the ocean, SIAM J. Math. Anal. 25, No. 4, (1994) 1023-1068.
20. L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982) 771-831.
21. J.-Y. Chemin, A propos d'un probleme de pénalisation de type antisymétrique. Proc. Paris Acad. Sci. 321 (1995) 861-864.
22. P. Constantin, The Littlewood-Paley spectrum in two-dimensional turbulence, Theor. and Comp. Fluid Dyn. 9, No. 3/4, (1997) 183-191.
23. P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press (1988).
24. A. Craya, Contribution à l'analyse de la turbulence associée à des vitesses moyennes. P.S. T. Ministère de l'Air 345 (1958).
25. P.G. Drazin and W.H. Reid, Hydrodynamic Stability, Cambridge University Press (1981).
26. P.F. Embid and A.J. Majda, Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity, Comm. Partial Diff. Eqs. 21 (1996) 619-658.
27. I. Gallagher, Un résultat de stabilité pour les équations des fluides tournants, C.R. Acad. Sci. Paris, Série I (1997) 183-186.
28. I. Gallagher, Asymptotics of the solutions of hyperbolic equations with a skew-symmetric perturbation. J. Differential Equations 150 (1998) 363-384.
29. I. Gallagher, Applications of Schochet's methods to parabolic equations. J. Math. Pures Appl. 77 (1998) 989-1054.
30. E. Grenier, Rotating fluids and inertial waves. Proc. Acad Sci. Paris Ser. 1 321 (1995) 711-714.
31. J.L. Joly, G. Métivier and J. Rauch, Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves. Duke Math. J. 70 (1993) 373-404.
32. J.L. Joly, G. Métivier and J. Rauch, Resonant one-dimensional nonlinear geometric optics. J. Funct. Anal. 114 (1993) 106-231.
33. J.L. Joly, G. Métivier and J. Rauch, Coherent nonlinear waves and the Wiener algebra. Ann. Inst. Fourier 44 (1994) 167-196.
34. J.L. Joly, G. Métivier and J. Rauch, Coherent and focusing multidimensional nonlinear geometric optics. Ann. Scient. E. N. S. Paris 4 (1995) 28, 51-113.
35. D.A. Jones, A. Mahalov and B. Nicolaenko, A numerical study of an operator splitting method for rotating flows with large ageostrophic initial data. Theor. and Comp. Fluid Dyn. 13, No. 2, (1998) 143-159.
36. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd edition, Gordon and Breach, New York (1969).
37. J.-L. Lions, R. Temam and S. Wang, Geostrophic asymptotics of the primitive equations of the atmosphere. Topological Methods in Nonlinear Analysis 4 (1994) 253-287, special issue dedicated to J. Leray.
38. J.-L. Lions, R. Temam and S. Wang, A simple global model for the general circulation of the atmosphere. Comm. Pure Appl. Math. 50 (1997) 707-752.
39. A. Mahalov, S. Leibovich and E.S. Titi, Invariant helical subspaces for the Navier-Stokes Equations. Arch. for Rational Mech. and Anal. 112, No. 3, (1990) 193-222.
40. A. Mahalov and P.S. Marcus, Long-time averaged rotating shallow-water equations, Proc. of the First Asian Computational Fluid Dynamics Conference, W.H. Hui, Y.-K. Kwok and J.R. Chasnov Eds, vol. 3, Hong Kong University of Science and Technology (1995) 1227-1230.
41. O. Métais and J.R. Herring, Numerical experiments of freely evolving turbulence in stably stratified fluids. J. Fluid Mech. 202 (1989) 117.
42. J. Pedlosky, Geophysical Fluid Dynamics, 2nd edition, Springer-Verlag (1987).
43. H. Poincaré, Sur la précession des corps déformables. Bull. Astronomique 27 (1910) 321.
44. G. Raugel and G. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6, No. 3, (1993) 503-568.
45. S. Schochet, Fast singular limits of hyperbolic PDE's. J. Differential Equations 114 (1994) 476-512.
46. E.M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press (1970).
47. S.L. Sobolev, Ob odnoi novoi zadache matematicheskoi fiziki. Izvestiia Akademii Nauk SSSR, Ser. Matematicheskaia. 18, No. 1, (1954) 3-50.
48. R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam (1984).
49. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia (1983).