Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T21:12:56.588Z Has data issue: false hasContentIssue false
  • English
  • Français

Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics

Published online by Cambridge University Press:  15 April 2002

Anatoli Babin ,
Alex Mahalov  and
Basil Nicolaenko
Anatoli Babin
Affiliation:
Department of Mathematics, University of California, Irvine, CA, 92697, USA.
Alex Mahalov
Affiliation:
Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA.
Basil Nicolaenko
Affiliation:
Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA.
Get access

Abstract

Fast singular oscillating limits ofthe three-dimensional "primitive" equations ofgeophysical fluid flows are analyzed.We prove existence on infinite time intervals of regular solutions to the3D "primitive" Navier-Stokes equations for strongstratification (large stratification parameter N).This uniform existence is proven forperiodic or stress-free boundary conditionsfor all domain aspect ratios,including the case of three wave resonances which yield nonlinear " $2\frac{1}{2}$ dimensional" limit equations for N → +∞;smoothness assumptions are the same as for localexistence theorems, that is initial data in Hα , α ≥ 3/4.The global existence is proven using techniques ofthe Littlewood-Paley dyadic decomposition.Infinite time regularity for solutions of the3D "primitive" Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit resonantequations and convergence theorems.

Keywords

Fast singular oscillating limitsthree-dimensional Navier-Stokes equationsprimitive equations for geophysical fluid flows.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 2: Special issue for R. Teman's 60th birthday , March 2000 , pp. 201 - 222
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, V.I., Small denominators. I. Mappings of the circumference onto itself. Amer. Math. Soc. Transl. Ser. 2. 46 (1965) 213-284.
V.I. Arnold and B.A. Khesin, Topological Methods in Hydrodynamics. Appl. Math. Sci. 125 (1997).
Avrin, J., Babin, A., Mahalov, A. and Nicolaenko, B., On regularity of solutions of 3D Navier-Stokes equations. Appl. Anal. 71 (1999) 197-214. CrossRef
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.
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.
A. Babin, A. Mahalov and B. Nicolaenko, Resonances and regularity for Boussinesq equations. Russian J. Math. Phys. 4, No. 4, (1996) 417-428.
Babin, A., Mahalov, A. and Nicolaenko, B., Regularity and integrability of rotating shallow-water equations. Proc. Acad. Sci. Paris Ser. 1 324 (1997) 593-598. CrossRef
Babin, A., Mahalov, A. and Nicolaenko, B., Global regularity and integrability of 3D Euler and Navier-Stokes equations for uniformly rotating fluids. Asympt. Anal. 15 (1997) 103-150.
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.
Babin, A., Mahalov, A. and Nicolaenko, B., On the nonlinear baroclinic waves and adjustment of pancake dynamics. Theor. and Comp. Fluid Dynamics 11 (1998) 215-235. CrossRef
Babin, A., Mahalov, A., Nicolaenko, B. and Zhou, Y., On the asymptotic regimes and the strongly stratified limit of rotating Boussinesq equations. Theor. and Comp. Fluid Dyn. 9 (1997) 223-251. CrossRef
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.
A. Babin, A. Mahalov and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Lett. Appl. Math. (to appear).
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.
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).
A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam (1992).
Bardos, C. and Benachour, S., Domaine d'analycité des solutions de l'équation d'Euler dans un ouvert de ${\mathbb {R}}^n$ . Annali della Scuola Normale Superiore di Pisa 4 (1977) 647-687.
P. Bartello, Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atm. Sci. 52, No. 24, (1995) 4410-4428.
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.
Caffarelli, L., Kohn, R. and Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982) 771-831. CrossRef
Chemin, J.-Y., A propos d'un probleme de pénalisation de type antisymétrique. Proc. Paris Acad. Sci. 321 (1995) 861-864.
P. Constantin, The Littlewood-Paley spectrum in two-dimensional turbulence, Theor. and Comp. Fluid Dyn. 9, No. 3/4, (1997) 183-191.
P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press (1988).
A. Craya, Contribution à l'analyse de la turbulence associée à des vitesses moyennes. P.S.T. Ministère de l'Air 345 (1958).
P.G. Drazin and W.H. Reid, Hydrodynamic Stability, Cambridge University Press (1981).
Embid, P.F. and Majda, A.J., Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity, Comm. Partial Diff. Eqs. 21 (1996) 619-658. CrossRef
I. Gallagher, Un résultat de stabilité pour les équations des fluides tournants, C.R. Acad. Sci. Paris, Série I (1997) 183-186.
Gallagher, I., Asymptotics of the solutions of hyperbolic equations with a skew-symmetric perturbation. J. Differential Equations 150 (1998) 363-384. CrossRef
Gallagher, I., Applications of Schochet's methods to parabolic equations. J. Math. Pures Appl. 77 (1998) 989-1054. CrossRef
Grenier, E., Rotating fluids and inertial waves. Proc. Acad Sci. Paris Ser. 1 321 (1995) 711-714.
Joly, J.L., Métivier, G. and Rauch, J., Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves. Duke Math. J. 70 (1993) 373-404. CrossRef
Joly, J.L., Métivier, G. and Rauch, J., Resonant one-dimensional nonlinear geometric optics. J. Funct. Anal. 114 (1993) 106-231. CrossRef
Joly, J.L., Métivier, G. and Rauch, J., Coherent nonlinear waves and the Wiener algebra. Ann. Inst. Fourier 44 (1994) 167-196.
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.
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.
O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd edition, Gordon and Breach, New York (1969).
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.
Lions, J.-L., Temam, R. and Wang, S., A simple global model for the general circulation of the atmosphere. Comm. Pure Appl. Math. 50 (1997) 707-752. 3.0.CO;2-A>CrossRef
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.
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.
Métais, O. and Herring, J.R., Numerical experiments of freely evolving turbulence in stably stratified fluids. J. Fluid Mech. 202 (1989) 117. CrossRef
J. Pedlosky, Geophysical Fluid Dynamics, 2nd edition, Springer-Verlag (1987).
Poincaré, H., Sur la précession des corps déformables. Bull. Astronomique 27 (1910) 321.
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.
Schochet, S., Fast singular limits of hyperbolic PDE's. J. Differential Equations 114 (1994) 476-512. CrossRef
E.M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press (1970).
S.L. Sobolev, Ob odnoi novoi zadache matematicheskoi fiziki. Izvestiia Akademii Nauk SSSR, Ser. Matematicheskaia. 18, No. 1, (1954) 3-50.
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam (1984).
R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia (1983).