Fast singular oscillating limits and global regularity for the 3D primitive equations of geophysics
Babin, Anatoli ; Mahalov, Alex ; Nicolaenko, Basil
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 34 (2000), p. 201-222 / Harvested from Numdam
Publié le : 2000-01-01
@article{M2AN_2000__34_2_201_0,
     author = {Babin, Anatoli and Mahalov, Alex and Nicolaenko, Basil},
     title = {Fast singular oscillating limits and global regularity for the 3D primitive equations of geophysics},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {34},
     year = {2000},
     pages = {201-222},
     mrnumber = {1765657},
     zbl = {0962.76020},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2000__34_2_201_0}
}
Babin, Anatoli; Mahalov, Alex; Nicolaenko, Basil. Fast singular oscillating limits and global regularity for the 3D primitive equations of geophysics. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 34 (2000) pp. 201-222. http://gdmltest.u-ga.fr/item/M2AN_2000__34_2_201_0/

[1] V. I. Arnold, Small denominators. I. Mappings of the circumference onto itself. Amer. Math. Soc. Transl. Ser. 2 46 (1965)213-284. | Zbl 0152.41905

[2] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics. Appl. Math. Sci. 125 (1997). | MR 1612569 | Zbl 0902.76001

[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. | MR 1690099 | Zbl 1022.76010

[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. | MR 1685858 | Zbl 0872.76097

[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. | MR 1400515 | Zbl 0882.76096

[6] A. Babin, A. Mahalov and B. Nicolaenko, Resonances and regularîty for Boussinesq equations. Russian J. Math. Phys. 4, No. 4, (1996) 417-428. | MR 1470444 | Zbl 0955.76521

[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. | MR 1444000 | Zbl 0883.76014

[8] A. Babin, A. Mahalov and B. Nicolaenko, Global regularity and ïntegrability of 3D Euler and Navier-Stokes equations for uniformly rotating fluids. Asympt. Anal. 15 (1997) 103-150. | MR 1480996 | Zbl 0890.35109

[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. | MR 1472094 | Zbl 0889.76007

[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. | Zbl 0957.76092

[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. | Zbl 0912.76092

[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. | MR 1710277 | Zbl 1035.76055

[13] A. Babin, A. Mahalov and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Lett. Appl. Math. (to appear). | MR 1752139

[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. | MR 1736966 | Zbl 0932.35160

[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). | MR 1156492 | Zbl 0778.58002

[17] C. Bardos and S. Benachour, Domaine d'analycité des solutions de l'équation d'Euler dans un ouvert de Rn. Annali della Scuola Normale Superiore di Pisa 4 (1977) 647-687. | Numdam | MR 454413 | Zbl 0366.35022

[18] P. Bartello, Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atm. Sci. 52, No. 24, (1995)4410-4428. | MR 1370126

[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. | MR 1278890 | Zbl 0811.35097

[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. | MR 673830 | Zbl 0509.35067

[21] J.-Y. Chemin, A propos d'un problème de pénalisation de type antisymétrique. Proc. Parts Acad. Sci. 321 (1995) 861-864. | MR 1355842 | Zbl 0842.35082

[22] P. Constantin, The Littlewood-Paley spectrum in two-dimensional turbulence, Theor. and Comp. Fluid Dyn. 9, No. 3/4, (1997) 183-191. | Zbl 0907.76042

[23] P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press (1988). | MR 972259 | Zbl 0687.35071

[24] A. Craya, Contribution à l'analyse de la turbulence associée à des vitesses moyennes. P.S.T. Ministère de l'Air 345 (1958). | MR 98536

[25] P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge University Press (1981). | MR 604359 | Zbl 0449.76027

[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. | MR 1387463 | Zbl 0849.35106

[27] I. Gallagher, Un résultat de stabilité pour les équations des fluides tournants, C.R. Acad. Sci. Paris, Série I (1997) 183-186. | MR 1438380 | Zbl 0878.76081

[28] I. Gallagher, Asymptotics of the solutions of hyperbolic equations with a skew-symmetrie perturbation. J. Differential Equations 150 (1998) 363-384. | MR 1658597 | Zbl 0921.35095

[29] I. Gallagher, Applications of Schochet's methods to parabolic equations. J. Math. Pures Appl. 77 (1998) 989-1054. | MR 1661025 | Zbl 1101.35330

[30] E. Grenier, Rotating fluids and inertial waves. Proc. Acad. Sci. Paris Ser. 1 321 (1995) 711-714. | MR 1354711 | Zbl 0843.35073

[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. | MR 1219817 | Zbl 0815.35066

[32] J. L. Joly, G. Métivier and J. Rauch, Resonant one-dimensional nonlinear geometric optics. J. Funct. Anal. 114 (1993) 106-231. | MR 1220985 | Zbl 0851.35023

[33] J. L. Joly, G. Métivier and J. Rauch, Coherent nonlinear waves and the Wiener algebra. Ann. Inst. Fourier 44 (1994) 167-196. | Numdam | MR 1262884 | Zbl 0791.35019

[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. | Numdam | MR 1305424 | Zbl 0836.35087

[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. | Zbl 0961.76071

[36] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd édition, Gordon and Breach, New York (1969). | MR 254401 | Zbl 0184.52603

[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. | MR 1350974 | Zbl 0846.35106

[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. | MR 1454171 | Zbl 0992.86001

[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. | MR 1076072 | Zbl 0708.76044

[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 édition, Springer-Verlag (1987). | Zbl 0713.76005

[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. | MR 1179539 | Zbl 0787.34039

[45] S. Schochet, Fast singular limits of hyperbolic PDE's. J. Differential Equations 114 (1994) 476-512. | MR 1303036 | Zbl 0838.35071

[46] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press (1970). | MR 290095 | Zbl 0207.13501

[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). | MR 769654 | Zbl 0568.35002

[49] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia (1983). | MR 764933 | Zbl 0833.35110