Lecture notes : Mathematical study of singular perturbation problems Applications to large-scale oceanography
Journées équations aux dérivées partielles (2010), article no. 1, 49 p.
DOI : 10.5802/jedp.58
Saint-Raymond, Laure 1

1 Université Paris VI and DMA, Ecole normale supérieure, 45 rue d’Ulm, 75230 Paris cedex 05, France
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Saint-Raymond, Laure. Lecture notes : Mathematical study of singular perturbation problems Applications to large-scale oceanography. Journées équations aux dérivées partielles (2010), article  no. 1, 49 p. doi : 10.5802/jedp.58. http://archive.numdam.org/articles/10.5802/jedp.58/

[1.1] D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equation and convergence to the quasi-geostrophic model. Commun. Math. Phys. 238 (2003), pages 211-223. | MR | Zbl

[1.2] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Basics of Mathematical Geophysics, to appear in Oxford University Press, 2006. | MR

[1.3] F. C. Fuglister, Gulf Stream ‘60, Progress in Oceanography I, Pergamon Press, 1963.

[1.4] H. Fujita and T. Kato, On the Navier–Stokes initial value problem I, Archive for Rational Mechanics and Analysis, 16 (1964) pages 269–315. | MR | Zbl

[1.5] I. Gallagher & L. Saint-Raymond, On the influence of the Earth’s rotation on geophysical flows, Handbook of Mathematical Fluid Dynamics, S. Friedlander and D. Serre Editors Vol 4, Chapter 5, 201-329, 2007.

[1.6] Gerbeau, J.-F.; Perthame, B. Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001), no. 1, 89–102. | MR | Zbl

[1.7] A. E. Gill, Atmosphere-Ocean Dynamics, International Geophysics Series, Vol. 30, 1982.

[1.8] A. E. Gill and M. S. Longuet-Higgins, Resonant interactions between planetary waves, Proc. Roy. Soc. London, A 299 (1967), pages 120–140.

[1.9] J. Leray, Essai sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Matematica, 63 (1933), pages 193-248. | MR

[1.10] J. Leray, Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique. Journal de Mathématiques Pures et Appliquées 12 (1933), pages 1–82. | Numdam | Zbl

[1.11] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. II, Compressible Models, Oxford Science Publications, 1997. | Zbl

[1.12] A. Majda, Introduction to PDEs and waves for the atmosphere and ocean. Courant Lecture Notes in Mathematics, 9. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. | MR

[1.13] A. Mellet and A. Vasseur, On the isentropic compressible Navier-Stokes equation, Preprint, 2005.

[1.14] E. Palmén & C.W. Newton, Atmospheric Circulation Systems, Academic Press, 1969.

[1.15] J. Pedlosky, Geophysical fluid dynamics, Springer, 1979. | Zbl

[1.16] R. Temam and M. Ziane, Some mathematical Problems in Geophysical Fluid Dynamics, Handbook of Mathematical Fluid Dynamics, vol. III, eds S. Friedlander and D. Serre, 535–657, 2004. | MR

[2.1] A. Babin, A. Mahalov, and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier–Stokes equations for uniformly rotating fluids, European Journal of Mechanics, 15 (1996), pages 291–300. | MR | Zbl

[2.2] A. Babin, A. Mahalov, and B. Nicolaenko, Resonances and regularity for Boussinesq equations, Russian Journal of Mathematical Physics, 4 (1996), pages 417-428. | MR | Zbl

[2.3] A. Babin, A. Mahalov, and B. Nicolaenko, Global regularity of 3D rotating Navier–Stokes equations for resonant domains, Indiana University Mathematics Journal, 48 (1999), pages 1133–1176. | MR | Zbl

[2.4] F. Charve, Global well-posedness and asymptotics for a geophysical fluid system, Communications in Partial Differential Equations, 29 (2004), pages 1919-1940. | MR | Zbl

[2.5] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Basics of Mathematical Geophysics, to appear in Oxford University Press, 2006. | MR

[2.6] I. Gallagher, Applications of Schochet’s methods to parabolic equations, Journal de Mathématiques Pures et Appliquées, 77 (1998), pages 989-1054. | MR | Zbl

[2.7] I. Gallagher & L. Saint-Raymond, On the influence of the Earth’s rotation on geophysical flows, Handbook of Mathematical Fluid Dynamics, S. Friedlander and D. Serre Editors Vol 4, Chapter 5, 201-329, 2007.

[2.8] H.P. Greenspan, The theory of rotating fluids, Cambridge monographs on mechanics and applied mathematics, 1969. | Zbl

[2.9] E. Grenier, Oscillatory perturbations of the Navier–Stokes equations. Journal de Mathématiques Pures et Appliquées, 76 (1997), pages 477-498. | MR | Zbl

[2.10] S. Schochet, Fast singular limits of hyperbolic PDEs. Journal of Differential Equations 114 (1994), pages 476-512. | MR | Zbl

[2.11] W. Thomson (Lord Kelvin), On gravitational oscillations of rotating water. Proc. Roy. Soc. Edinburgh 10, 1879, pages 92-100. | MR

[3.1] D. Bresch, B. Desjardins and D. Gérard-Varet, Rotating fluids in a cylinder, Discrete and Contininous Dynamical Systems. 11 (2004), no. 1, pages 47–82. | MR | Zbl

[3.2] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Basics of Mathematical Geophysics, to appear in Oxford University Press, 2006. | MR

[3.3] A.-L. Dalibard, L. Saint-Raymond, Mathematical study of rotating fluids with resonant surface stress. J. Differential Equations 246 (2009), 2304–2354. | MR | Zbl

[3.4] A.-L. Dalibard, L. Saint-Raymond, Mathematical study of the beta-plane model for rotating fluids in a thin layer, J. Math. Pures Appl. (2010).

[3.5] A.-L. Dalibard, L. Saint-Raymond, About degenerate Northern boundary layers, in preparation.

[3.6] B. Desjardins and E. Grenier, On the Homogeneous Model of Wind-Driven Ocean Circulation, SIAM Journal on Applied Mathematics 60 (1999), pages 43–60. | MR | Zbl

[3.7] D. Gérard-Varet, Highly rotating fluids in rough domains, Journal de Mathématiques Pures et Appliquées 82 (2003), pages 1453–1498. | MR | Zbl

[3.8] I. Gallagher & L. Saint-Raymond, On the influence of the Earth’s rotation on geophysical flows, Handbook of Mathematical Fluid Dynamics, S. Friedlander and D. Serre Editors Vol 4, Chapter 5, 201-329, 2007.

[3.9] E. Grenier and N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial data, Communications in Partial Differential Equations 22 (1997), no. 5-6, pages 953–975. | MR | Zbl

[3.10] N. Masmoudi, Ekman layers of rotating fluids: the case of general initial data, Communications in Pure and Applied Mathematics, 53, (2000), pages 432–483. | MR | Zbl

[3.11] N. Masmoudi, F. Rousset. Stability of oscillating boundary layers in rotating fluids. Ann. Sci. Éc. Norm. Supér. 41 (2008), 955–1002. | Numdam | MR | Zbl

[3.12] F. Rousset. Asymptotic behavior of geophysical fluids in highly rotating balls. Z. Angew. Math. Phys. 58 (2007), 53–67. | MR | Zbl

[3.13] Saint-Raymond, Weak compactness methods for singular penalization problems with boundary layers. SIAM J. Math. Anal. 41 (2009), 153–177. | MR

[4.1] C. Cheverry, I. Gallagher, T. Paul & L. Saint-Raymond, Trapping Rossby waves, C. R. Math. Acad. Sci. Paris 347 (2009), 879–884. | MR | Zbl

[4.2] C. Cheverry, I. Gallagher, T. Paul & L. Saint-Raymond. Semiclassical and spectral analysis of oceanic waves. Submitted (2010).

[4.3] M. Dimassi & S. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit; Cambridge University Press, London Mathematical Society Lecture Note Series 268, 1999. | MR | Zbl

[4.4] A. Dutrifoy & A. J. Majda, Fast Wave Averaging for the Equatorial Shallow Water Equations, Comm. PDE, 32 (2007), 1617 –1642. | MR | Zbl

[4.5] A. Dutrifoy, A. J. Majda & S. Schochet, A Simple Justification of the Singular Limit for Equatorial Shallow-Water Dynamics, in Communications on Pure and Applied Math. LXI (2008) 0002-0012. | Zbl

[4.6] I. Gallagher and L. Saint-Raymond, Weak convergence results for inhomogeneous rotating fluid equations, Journal d’Analyse Mathématique, 99 (2006), 1-34. | MR | Zbl

[4.7] I. Gallagher & L. Saint-Raymond, Mathematical study of the betaplane model: equatorial waves and convergence results. Mém. Soc. Math. Fr. (N.S.). 107 (2006), v+116 pp. | Numdam | MR | Zbl

[4.8] I. Gallagher & L. Saint-Raymond, On the influence of the Earth’s rotation on geophysical flows, Handbook of Mathematical Fluid Dynamics, S. Friedlander and D. Serre Editors Vol 4, Chapter 5, 201-329, 2007.

[4.9] A. E. Gill, Atmosphere-Ocean Dynamics, International Geophysics Series, Vol. 30, 1982.

[4.10] A. E. Gill & M. S. Longuet-Higgins, Resonant interactions between planetary waves, Proc. Roy. Soc. London, A 299 (1967), 120–140.

[4.11] J.-L. Joly, G. Métivier & J. Rauch, Generic Rigorous Asymptotic Expansions for Weakly Nonlinear Multidimensional Oscillatory Waves, Duke Mathematical Journal (1993), 70, 373-404. | MR | Zbl

[4.12] J.-L. Joly, G. Métivier & J. Rauch, Coherent nonlinear waves and the Wiener algebra, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 1, 167–196. | Numdam | MR | Zbl

[4.13] M. Majdoub, M. Paicu. Uniform local existence for inhomogeneous rotating fluid equations. J. Dynam. Differential Equations 21 (2009), 21–44. | MR | Zbl

[4.14] A. Martinez, An introduction to semiclassical and microlocal analysis, Springer (2002) | MR | Zbl

[4.15] J. Pedlosky, Geophysical fluid dynamics, Springer (1979). | Zbl

[4.16] J. Pedlosky, Ocean Circulation Theory, Springer (1996).

[4.17] S. Vũ Ngoc, Systèmes intégrables semi-classiques : du local au global, Panoramas et Synthèses 22, 2006. | MR | Zbl

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