Some Mathematical Problems in Geophysical Fluid Dynamics

Handbook of Numerical Analysis

Volumn 14, Issue , 2009, Pages 577-750

  Petcu, Madalina ^a   Temam, Roger M ^b   Ziane, Mohammed ^c  

^a UNIVERSITÉ DE POITIERS   (France)

^b INDIANA UNIVERSITY   (United States)

^c UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70350313482     PISSN: 15708659     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1570-8659(08)00212-3     Document Type: Chapter

References (70)

1. Agmon S., Douglis A., and Nirenberg L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12 (1959) 209-230
2. Agmon S., Douglis A., and Nirenberg L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math. 17 (1964) 35-92
3. Azérad P., and Guillén F. Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics. SIAM J. Math. Anal. 33 (2001) 847-859
4. Bardos C., and Tartar L. Sur l'unicité rétrograde des équations paraboliques et quelques questions voisines. Arch. Rational Mech. Anal. 50 (1973) 10-25
5. Bartello P. Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atmos. Sci. 52 24 (1995) 4410-4428
6. Bewley T., Temam R., and Ziane M. A general framework for robust control in fluid mechanics. Phys. D. 138 (2000) 360-392
7. Bewley T., Temam R., and Ziane M. Existence and uniqueness of optimal control to the Navier-Stokes equations. C. R. Acad. Sci. Paris Sér. I. 330 (2000) 1007-1011
8. Bresch D., Kazhikhov A., and Lemoine J. On the two-dimensional hydrostatic Navier-Stokes equations. SIAM J. Math. Anal. 36 1 (2004) 796-814
9. Constantin P., Foias C., and Temam R. Attractors representing turbulent flows. Mem. Am. Math. Soc. 53 (1985)
10. Constantin P., and Foias C. Navier-Stokes Equations (1988), Univ. Chicago Press, Chicago, IL
11. Chepyzhov V.V., and Vishik M.I. Attractors for Equations of Mathematical Physics (American Mathematical Society Colloquium Publications, vol. 49) (2002), American Mathematical Society, Providence, RI
12. Cao C., and Titi E.S. Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, arXiv: Math. AP/0503028 2 16 Nov 2005 and. Ann. of Math. 166 (2007) 245-267
13. Ewald B., and Temam R. Maximum principles for the primitive equations of the atmosphere. Discrete Contin. Dyn. Syst. 7 2 (2001) 343-362
14. Foias C., and Temam R. Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Funct. Anal. 87 (1989) 359-369
15. Ghidaglia J.M. Régularité des solutions de certains problèmes aux limites linéaires liés aux équations d'Euler (in French) [Regularity of the solutions of certain linear boundary value problems associated with the Euler equations]. Commun. Partial Differ. Equ. 9 13 (1984) 1265-1298
16. Gill A.F. Atmosphere-Ocean Dynamics (1982), Academic Press, New York
17. Grisvard P. Elliptic Problems in Non-Smooth Domains (Monographs Studies in Mathematics, vol. 24) (1985), Pitman, London
18. Guillén-González F., Masmoudi N., and Rodríguez-Bellido M.A. Anisotropic estimates and strong solutions of the primitive equations. Differ. Integral Equ. 14 11 (2001) 1381-1408
19. Haltiner G., and Williams R. Numerical Weather Prediction and Dynamic Meteorology. second ed. (1980), Wiley, New York
20. Hale J.K. Attractors and dynamics in partial differential equations. From Finite to Infinite Dimensional Dynamical Systems (Cambridge, 1995), of NATO Sci. Ser. II Math. Phys. Chem. vol. 19 (2001), Kluwer Acad. Publ., Dordrecht 85-112
21. Haney R. Surface thermal boundary condition for ocean circulation models. J. Phys. Oceanogr. 1 (1971) 241-248
22. Hu C., Temam R., and Ziane M. Regularity results for linear elliptic problems related to the primitive equations. Chin. Ann. Math. Ser. B. 23 2 (2002) 277-292
23. Hu C., Temam R., and Ziane M. The primitive equations of the large scale ocean under the small depth hypothesis. Discrete Cont. Dyn. Syst. 9 1 (2003) 97-131
24. Ju N. The global attractor for the solutions to the 3D viscous primitive equations. Discrete Contin. Dyn. Syst. 17 1 (2007) 159-179
25. Kobelkov G.M. Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations. C. R. Math. Acad. Sci. Paris. 343 4 (2006) 283-286
26. Kozlov V.A., Mazya V.G., and Rossmann J. Math. Surveys Monogr. Elliptic Boundary Value Problems in Domains with Point Singularities vol. 52 (1997), Amer. Math. Soc., Providence, RI
27. Kukavica I., and Ziane M. On the regularity of the primitive equations of the ocean. C. R. Math. Acad. Sci. Paris. 345 5 (2007) 257-260
28. Kukavica I., and Ziane M. On the regularity of the primitive equations. Nonlinearity. 20 (2007) 2739-2753
29. Laprise R. The Euler equation of motion with hydrostatic pressure as independent coordinate. Mon. Wea. Rev. 120 1 (1992) 197-207
30. Lions J.L. Quelques Méthodes De Résolution Des Problèmes Aux Limites Non Linéaires (1969), Dunod, Paris re-edited (2002).
31. Lions J.L., and Magenes E. Nonhomogeneous Boundary Value Problems and Applications vol. 1 (1972), Springer-Verlag, New York
32. Lions J.L., and Malgrange B. Sur l'unicité rétrograde dans les problèmes mixtes paraboliques. Math. Scand. 8 (1960) 277-286
33. Lions J.L., Temam R., and Wang S. New formulations of the primitive equations of the atmosphere and applications. Nonlinearity 5 (1992) 237-288
34. Lions J.L., Temam R., and Wang S. On the equations of the large-scale ocean. Nonlinearity 5 (1992) 1007-1053
35. Lions J.L., Temam R., and Wang S. Models of the coupled atmosphere and ocean (CAO I). Comput. Mech. Adv. 1 (1993) 5-54
36. Lions J.L., Temam R., and Wang S. Numerical analysis of the coupled models of atmosphere and ocean (CAO II). Comput. Mech. Adv. 1 (1993) 55-119
37. Lions J.L., Temam R., and Wang S. Mathematical study of the coupled models of atmosphere and ocean (CAO III). J. Math. Pures Appl. 74 (1995) 105-163
38. Ma T., and Wang S. Geometric theory of incompressible flows with applications to fluid dynam-ics. Mathematical Surveys and Monographs, 119 (2005), American Mathematical Society, Providence, RI x+234
39. Ma T., and Wang S. Bifurcation theory and applications. World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises vol. 53 (2005), World Scientific Publishing Co. Pvt. Ltd., Hackensack, NJ xiv+375
40. Mazya V.G., and Rossmann J. On the behaviour of solutions to the Dirichlet problem for second order elliptic equations near edges and polyhedral vertices with critical angles. Z. Anal. Anwendungen 13 1 (1994) 19-47
41. Moise I., Simonnet E., Temam R., and Ziane M. Numerical simulation of differential systems displaying rapidly oscillating solutions. J. Eng. Math. 34 (1998) 201-214
42. Moise I., Temam R., and Ziane M. Asymptotic analysis of the Navier-Stokes equations in thin domains (volume dedicated to O.A. Ladyzhenskaya). Topol. Methods Nonlinear Anal. 10 2 (1997) 249-282
43. Orenga P. Un théorème d'existence de solutions d'un problème de shallow water (in French) [A theorem on the existence of solutions of a shallow-water problem]. Arch. Ration. Mech. Anal. 130 2 (1995) 183-204
44. Pedlosky J. Geophysical Fluid Dynamics. second ed. (1987), Springer-Verlag, New York
45. Petcu M. On the three dimensional primitive equations. Adv. Differ. Equ. 11 (2006) 1201-1226
46. Petcu M. On the backward uniqueness of the primitive equations. J. Math. Pure Appl. 87 (2007) 275-289
47. Petcu M. Gevrey class regularity for the primitive equations in space dimension 2. Asymp. Anal. 39 (2004) 1-13
48. Petcu M., Temam R., and Wirosoetisno D. Existence and Regularity for the primitive equations in space dimension 2. Comm. Pure Appl. Anal. 3 1 (2004) 115-131
49. Petcu M., Temam R., and Wirosoetisno D. Renormalization group method applied to the primitive equation. J. Differ. Equ. 208 1 (2004) 215-257
50. Petcu M., and Wirosoetisno D. Sobolev and Gevrey regularity for the primitive equations in a space dimension 3. Appl. Anal. 84 (2005) 769-788
51. Schlichting H. Boundary Layer Theory. seventh ed. (1979), McGraw-Hill, New York
52. Simonnet, E., Dijkstra, H.A., Ghil, M. (2008). Bifurcation analysis of ocean, atmosphere and climate models. In this volume.
53. Smagorinsky J. General circulation experiments with the primitive equations. I. The basic experiment. Mon. Wea. Rev. 91 (1963) 98-164
54. Smolarkiewicz P.K., Margolin L.G., and Wyszogrodzki A.A. A class of nonhydrostatic global models. J. Atmos. Sci. 58 4 (2001) 349-364
55. Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. second ed. (1997), Springer-Verlag, New York
56. Temam R. Navier-Stokes Equations, Studies in Mathematics Application (1977), North-Holland, Amsterdam [reedited, AMS-Chelsea Ser., Amer. Math. Soc. (Providence, RI) (2001)]
57. Temam R., and Wirosoetisno D. Exponential Approximations for the Primitive Equations of the ocean. Discrete and Continuous Dynamical Sytems-Series B 7 2 (2007) 425-440
58. Temam, R., Wirosoetisno, D. (2008). Stability of the slow manifold in the primitive equations arXiv:0808.2878v1.
59. Temam R., and Ziane M. Navier-Stokes equations in three-dimensional thin domains with various boundary conditions. Adv. Differ. Equ. 1 4 (1996) 499-546
60. Temam R., and Ziane M. Navier-Stokes equations in thin spherical domains. Contemp. Math. 209 (1997) 281-314
61. Temam R., and Ziane M. Some mathematical problems in geophysical fluid dynamics. In: Friedlander S., and Serre D. (Eds). Handbook of Mathematical Fluid Dynamics 3 (2004), Elsevier
62. Trenberth K.E. Climate System Modeling (1992), Cambridge Univ. Press, Cambridge
63. von Neumann J. Collected works. In: Taub A.H. (Ed). Theory of Games, Astrophysics, Hydrodynamics and Meteorology Vol. VI (1963), A Pergamon Press Book The Macmillan Co., New York 1963 x+538
64. Wang S. On the 2D model of large-scale atmospheric motion: well-posedness and attractors. Nonlinear Anal. 18 1 (1992) 17-60
65. Wang S. Attractors for the 3D baroclinic quasi-geostrophic equations of largescale atmosphere. J. Math. Anal. Appl. 165 1 (1992) 266-283
66. Washington W.M., and Parkinson C.L. An Introduction to Three Dimensional Climate Modeling (1986), Oxford Univ. Press, Oxford
67. Zeng Q.C. Mathematical and Physical Foundations of Numerical Weather Prediction (1979), Science Press, Beijing, China
68. Ziane M. Regularity results for Stokes type systems. Appl. Anal. 58 (1995) 263-292
69. Ziane M. Regularity results for stationary primitive equations of atmosphere and the ocean. Nonlinear Anal. 28 (1997) 289-313
70. Ziane, M. (2000). unpublished manuscript.