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

Anatoli Babin; Alex Mahalov; Basil Nicolaenko

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2000)

  • Volume: 34, Issue: 2, page 201-222
  • ISSN: 0764-583X

How to cite

top

Babin, Anatoli, Mahalov, Alex, and 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 34.2 (2000): 201-222. <http://eudml.org/doc/193983>.

@article{Babin2000,
author = {Babin, Anatoli, Mahalov, Alex, Nicolaenko, Basil},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {three-dimensional primitive equations; primitive Navier-Stokes equations; infinite time regularity; fast singular oscillating limits; geophysical flows; infinite time intervals; regular solutions; strong stratification; global existence; Littlewood-Paley dyadic decomposition; limit resonance equations; convergence},
language = {eng},
number = {2},
pages = {201-222},
publisher = {Dunod},
title = {Fast singular oscillating limits and global regularity for the 3D primitive equations of geophysics},
url = {http://eudml.org/doc/193983},
volume = {34},
year = {2000},
}

TY - JOUR
AU - Babin, Anatoli
AU - Mahalov, Alex
AU - Nicolaenko, Basil
TI - Fast singular oscillating limits and global regularity for the 3D primitive equations of geophysics
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2000
PB - Dunod
VL - 34
IS - 2
SP - 201
EP - 222
LA - eng
KW - three-dimensional primitive equations; primitive Navier-Stokes equations; infinite time regularity; fast singular oscillating limits; geophysical flows; infinite time intervals; regular solutions; strong stratification; global existence; Littlewood-Paley dyadic decomposition; limit resonance equations; convergence
UR - http://eudml.org/doc/193983
ER -

References

top
  1. [1] V. I. Arnold, Small denominators. I. Mappings of the circumference onto itself. Amer. Math. Soc. Transl. Ser. 2 46 (1965)213-284. Zbl0152.41905
  2. [2] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics. Appl. Math. Sci. 125 (1997). Zbl0902.76001MR1612569
  3. [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. Zbl1022.76010MR1690099
  4. [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. Zbl0872.76097MR1685858
  5. [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. Zbl0882.76096MR1400515
  6. [6] A. Babin, A. Mahalov and B. Nicolaenko, Resonances and regularîty for Boussinesq equations. Russian J. Math. Phys. 4, No. 4, (1996) 417-428. Zbl0955.76521MR1470444
  7. [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. Zbl0883.76014MR1444000
  8. [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. Zbl0890.35109MR1480996
  9. [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. Zbl0889.76007MR1472094
  10. [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. Zbl0957.76092
  11. [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. Zbl0912.76092
  12. [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. Zbl1035.76055MR1710277
  13. [13] A. Babin, A. Mahalov and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Lett. Appl. Math. (to appear). Zbl0932.35160MR1752139
  14. [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. Zbl0932.35160MR1736966
  15. [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. [16] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam (1992). Zbl0778.58002MR1156492
  17. [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. Zbl0366.35022MR454413
  18. [18] P. Bartello, Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atm. Sci. 52, No. 24, (1995)4410-4428. MR1370126
  19. [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. Zbl0811.35097MR1278890
  20. [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. Zbl0509.35067MR673830
  21. [21] J.-Y. Chemin, A propos d'un problème de pénalisation de type antisymétrique. Proc. Parts Acad. Sci. 321 (1995) 861-864. Zbl0842.35082MR1355842
  22. [22] P. Constantin, The Littlewood-Paley spectrum in two-dimensional turbulence, Theor. and Comp. Fluid Dyn. 9, No. 3/4, (1997) 183-191. Zbl0907.76042
  23. [23] P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press (1988). Zbl0687.35071MR972259
  24. [24] A. Craya, Contribution à l'analyse de la turbulence associée à des vitesses moyennes. P.S.T. Ministère de l'Air 345 (1958). Zbl0077.39605MR98536
  25. [25] P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge University Press (1981). Zbl0449.76027MR604359
  26. [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. Zbl0849.35106MR1387463
  27. [27] I. Gallagher, Un résultat de stabilité pour les équations des fluides tournants, C.R. Acad. Sci. Paris, Série I (1997) 183-186. Zbl0878.76081MR1438380
  28. [28] I. Gallagher, Asymptotics of the solutions of hyperbolic equations with a skew-symmetrie perturbation. J. Differential Equations 150 (1998) 363-384. Zbl0921.35095MR1658597
  29. [29] I. Gallagher, Applications of Schochet's methods to parabolic equations. J. Math. Pures Appl. 77 (1998) 989-1054. Zbl1101.35330MR1661025
  30. [30] E. Grenier, Rotating fluids and inertial waves. Proc. Acad. Sci. Paris Ser. 1 321 (1995) 711-714. Zbl0843.35073MR1354711
  31. [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. Zbl0815.35066MR1219817
  32. [32] J. L. Joly, G. Métivier and J. Rauch, Resonant one-dimensional nonlinear geometric optics. J. Funct. Anal. 114 (1993) 106-231. Zbl0851.35023MR1220985
  33. [33] J. L. Joly, G. Métivier and J. Rauch, Coherent nonlinear waves and the Wiener algebra. Ann. Inst. Fourier 44 (1994) 167-196. Zbl0791.35019MR1262884
  34. [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. Zbl0836.35087MR1305424
  35. [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. Zbl0961.76071
  36. [36] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd édition, Gordon and Breach, New York (1969). Zbl0184.52603MR254401
  37. [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. Zbl0846.35106MR1350974
  38. [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. Zbl0992.86001MR1454171
  39. [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. Zbl0708.76044MR1076072
  40. [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. [41] O. Métais and J. R. Herring, Numerical experiments of freely evolving turbulence in stably stratified fluids. J. Fluid Mech. 202 (1989) 117. 
  42. [42] J. Pedlosky, Geophysical Fluid Dynamics, 2nd édition, Springer-Verlag (1987). Zbl0713.76005
  43. [43] H. Poincaré, Sur la précession des corps déformables. Bull. Astronomique 27 (1910) 321. 
  44. [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. Zbl0787.34039MR1179539
  45. [45] S. Schochet, Fast singular limits of hyperbolic PDE's. J. Differential Equations 114 (1994) 476-512. Zbl0838.35071MR1303036
  46. [46] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press (1970). Zbl0207.13501MR290095
  47. [47] S. L. Sobolev, Ob odnoi novoi zadache matematicheskoi fiziki. Izvestiia Akademii Nauk SSSR, Ser. Matematicheskaia 18, No. 1, (1954) 3-50. 
  48. [48] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam (1984). Zbl0568.35002MR769654
  49. [49] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia (1983). Zbl0833.35110MR764933

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.