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October  2016, 36(10): 5681-5707. doi: 10.3934/dcds.2016049

Averaging method applied to the three-dimensional primitive equations

 1 Laboratoire de Mathématiques et applications, Univ. de Poitiers, Teleport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962, Futuroscope Chasseneuil Cedex 2 The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, Indiana 47405 3 Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom

Received  February 2015 Revised  April 2016 Published  July 2016

In this article we study the small Rossby number asymptotics for the three-dimensional primitive equations of the oceans and of the atmosphere. The fast oscillations present in the exact solution are eliminated using an averaging method, the so-called renormalisation group method.
Citation: Madalina Petcu, Roger Temam, Djoko Wirosoetisno. Averaging method applied to the three-dimensional primitive equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5681-5707. doi: 10.3934/dcds.2016049
References:
 [1] P. Bartello, Geostrophic adjustment and inverse cascades in rotating stratified turbulence,, J. Atmos. Sci., 52 (1995), 4410. doi: 10.1175/1520-0469(1995)052<4410:GAAICI>2.0.CO;2. Google Scholar [2] A. Babin, A. Mahalov and B. Nicolaenko, On the regularity of three-dimensional rotating Euler-Boussinesq equations,, Math. Models and Methods in Appl. Sci., 9 (1999), 1089. doi: 10.1142/S021820259900049X. Google Scholar [3] A. Babin, A. Mahalov and B. Nicolaenko, Fast singular oscillating limits and global regularity for the 3d primitive equations of geophysics,, M2AN, 34 (2000), 201. doi: 10.1051/m2an:2000138. Google Scholar [4] A. Babin, A. Mahalov and B. Nicolaenko, 3d Navier-Stokes and Euler equations with initial data characterized by uniformly large velocity,, Indiana Univ. Math. J., 50 (2001), 1. doi: 10.1512/iumj.2001.50.2155. Google Scholar [5] A. Babin, A. Mahalov and B. Nicolaenko, Fast singular oscillating limits of stably-stratified three-dimensional Euler-Boussinesq equations and ageostrophic wave fronts,, Large-scale atmosphere-ocean dynamics I, (2002), 126. doi: 10.1017/CBO9780511549991.005. Google Scholar [6] L. Y. Chen, N. Goldenfeld and Y. Oono, Renormalization group theory for global asymptotics analysis,, Phys. Rev. Lett., 73 (1994), 1311. doi: 10.1103/PhysRevLett.73.1311. Google Scholar [7] J. Y. Chemin, A propos d'un problème de pénalisation de type antisymmétrique,, J. Math. pures et appl., 76 (1997), 739. doi: 10.1016/S0021-7824(97)89967-9. Google Scholar [8] C. Cao and E. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,, Annals of Mathematics, 166 (2007), 245. doi: 10.4007/annals.2007.166.245. Google Scholar [9] P. F. Embid and A. J. Majda, Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity,, Comm. P.D.E., 21 (1996), 619. doi: 10.1080/03605309608821200. Google Scholar [10] I. Gallagher, Existence globale pour des Equations des fluides géostrophiques,, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 623. doi: 10.1016/S0764-4442(97)84772-6. Google Scholar [11] I. Gallagher, Applications of Schochet's methods to parabolic equations,, J. Math. Pures Appl. (9), 77 (1998), 989. doi: 10.1016/S0021-7824(99)80002-6. Google Scholar [12] E. Grenier, Rotating fluids and inertial waves,, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 711. Google Scholar [13] G. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations,, C. R. Math. Acad. Sci. Paris, 343 (2006), 283. doi: 10.1016/j.crma.2006.04.020. Google Scholar [14] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications,, Nonlinearity, 5 (1992), 237. doi: 10.1088/0951-7715/5/2/001. Google Scholar [15] I. Moise and R. Temam, Renormalization Group Method. Application to Navier-Stokes Equation,, Discrete and Continuous Dynamical Systems, 6 (2000), 191. Google Scholar [16] I. Moise, R. Temam and M. Ziane, Asymptotic analysis of the Navier-Stokes equations in thin domains,, Topological Methods in Nonlinear Analysis, 10 (1997), 249. Google Scholar [17] I. Moise and M. Ziane, Renormalization group method. Applications to partial differential equations,, J. Dyn. and Diff. Syst., 13 (2001), 275. doi: 10.1023/A:1016680007953. Google Scholar [18] M. Petcu, On the three dimensional primitive equations,, Adv. Diff. Eq., 11 (2006), 1201. Google Scholar [19] M. Petcu, R. Temam and D. Wirosoetisno, Renormalization group method applied to the primitive equations,, J. Diff. Eq., 208 (2005), 215. doi: 10.1016/j.jde.2003.10.011. Google Scholar [20] M. Petcu and D. Wirosoetisno, Sobolev and Gevrey regularity for the primitive equations in a space dimension 3,, Appl. Anal., 84 (2005), 769. doi: 10.1080/00036810500130745. Google Scholar [21] S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit,, Comm. Math. Phys., 104 (1986), 49. doi: 10.1007/BF01210792. Google Scholar [22] S. Schochet, Asymptotics for symmetric hyperbolic systems with a large parameter,, J. Diff. Eq., 75 (1988), 1. doi: 10.1016/0022-0396(88)90126-X. Google Scholar [23] S. Schochet, Fast singular limit of hyperbolic PDEs,, J. Diff. Eq., 114 (1994), 476. doi: 10.1006/jdeq.1994.1157. Google Scholar [24] R. Temam and D. Wirosoetisno, Averaging of differential equations generating oscillations and an application to control,, Applied Math. and Optimization, 46 (2002), 313. doi: 10.1007/s00245-002-0749-z. Google Scholar [25] R. Temam and D. Wirosoetisno, On the solutions of the renormalized equations at all orders,, Advances in Differential Equations, 8 (2003), 1005. Google Scholar [26] R. Temam and D. Wirosoetisno, Exponential Approximations for the Primitive Equations of the ocean,, in Discrete and Continuous Dynamical Systems-Series B, 7 (2007), 425. Google Scholar [27] R. Temam and D. Wirosoetisno, Stability of the slow manifold in the primitive equations,, SIAM J. Math. Anal., 42 (2010), 427. doi: 10.1137/080733358. Google Scholar [28] R. Temam and D. Wirosoetisno, Slow manifolds and invariant sets of the primitive equations,, J. Atmospheric Sciences, 68 (2011), 675. doi: 10.1175/2010JAS3650.1. Google Scholar [29] R. Temam and M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions,, Advances in Differential Equations, 1 (1996), 499. Google Scholar [30] R. Temam and M. Ziane, Navier-Stokes equations in thin spherical domains,, Contemporary Mathematics, 209 (1997), 281. doi: 10.1090/conm/209/02772. Google Scholar [31] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics,, in Handbook of Mathematical Fluid Dynamics, 3 (2004), 535. Google Scholar [32] M. Ziane, On a certain renormalization group method,, J. Math. Phys., 41 (2000), 3290. doi: 10.1063/1.533307. Google Scholar [33] M. Ziane, Regularity results for the stationary primitive equations of the atmosphere and the ocean,, Nonlinear Analysis, 28 (1997), 289. doi: 10.1016/0362-546X(95)00154-N. Google Scholar [34] M. Ziane, Regularity results for a Stokes type system,, Applicable Analysis, 58 (1995), 263. doi: 10.1080/00036819508840376. Google Scholar

show all references

References:
 [1] P. Bartello, Geostrophic adjustment and inverse cascades in rotating stratified turbulence,, J. Atmos. Sci., 52 (1995), 4410. doi: 10.1175/1520-0469(1995)052<4410:GAAICI>2.0.CO;2. Google Scholar [2] A. Babin, A. Mahalov and B. Nicolaenko, On the regularity of three-dimensional rotating Euler-Boussinesq equations,, Math. Models and Methods in Appl. Sci., 9 (1999), 1089. doi: 10.1142/S021820259900049X. Google Scholar [3] A. Babin, A. Mahalov and B. Nicolaenko, Fast singular oscillating limits and global regularity for the 3d primitive equations of geophysics,, M2AN, 34 (2000), 201. doi: 10.1051/m2an:2000138. Google Scholar [4] A. Babin, A. Mahalov and B. Nicolaenko, 3d Navier-Stokes and Euler equations with initial data characterized by uniformly large velocity,, Indiana Univ. Math. J., 50 (2001), 1. doi: 10.1512/iumj.2001.50.2155. Google Scholar [5] A. Babin, A. Mahalov and B. Nicolaenko, Fast singular oscillating limits of stably-stratified three-dimensional Euler-Boussinesq equations and ageostrophic wave fronts,, Large-scale atmosphere-ocean dynamics I, (2002), 126. doi: 10.1017/CBO9780511549991.005. Google Scholar [6] L. Y. Chen, N. Goldenfeld and Y. Oono, Renormalization group theory for global asymptotics analysis,, Phys. Rev. Lett., 73 (1994), 1311. doi: 10.1103/PhysRevLett.73.1311. Google Scholar [7] J. Y. Chemin, A propos d'un problème de pénalisation de type antisymmétrique,, J. Math. pures et appl., 76 (1997), 739. doi: 10.1016/S0021-7824(97)89967-9. Google Scholar [8] C. Cao and E. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,, Annals of Mathematics, 166 (2007), 245. doi: 10.4007/annals.2007.166.245. Google Scholar [9] P. F. Embid and A. J. Majda, Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity,, Comm. P.D.E., 21 (1996), 619. doi: 10.1080/03605309608821200. Google Scholar [10] I. Gallagher, Existence globale pour des Equations des fluides géostrophiques,, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 623. doi: 10.1016/S0764-4442(97)84772-6. Google Scholar [11] I. Gallagher, Applications of Schochet's methods to parabolic equations,, J. Math. Pures Appl. (9), 77 (1998), 989. doi: 10.1016/S0021-7824(99)80002-6. Google Scholar [12] E. Grenier, Rotating fluids and inertial waves,, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 711. Google Scholar [13] G. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations,, C. R. Math. Acad. Sci. Paris, 343 (2006), 283. doi: 10.1016/j.crma.2006.04.020. Google Scholar [14] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications,, Nonlinearity, 5 (1992), 237. doi: 10.1088/0951-7715/5/2/001. Google Scholar [15] I. Moise and R. Temam, Renormalization Group Method. Application to Navier-Stokes Equation,, Discrete and Continuous Dynamical Systems, 6 (2000), 191. Google Scholar [16] I. Moise, R. Temam and M. Ziane, Asymptotic analysis of the Navier-Stokes equations in thin domains,, Topological Methods in Nonlinear Analysis, 10 (1997), 249. Google Scholar [17] I. Moise and M. Ziane, Renormalization group method. Applications to partial differential equations,, J. Dyn. and Diff. Syst., 13 (2001), 275. doi: 10.1023/A:1016680007953. Google Scholar [18] M. Petcu, On the three dimensional primitive equations,, Adv. Diff. Eq., 11 (2006), 1201. Google Scholar [19] M. Petcu, R. Temam and D. Wirosoetisno, Renormalization group method applied to the primitive equations,, J. Diff. Eq., 208 (2005), 215. doi: 10.1016/j.jde.2003.10.011. Google Scholar [20] M. Petcu and D. Wirosoetisno, Sobolev and Gevrey regularity for the primitive equations in a space dimension 3,, Appl. Anal., 84 (2005), 769. doi: 10.1080/00036810500130745. Google Scholar [21] S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit,, Comm. Math. Phys., 104 (1986), 49. doi: 10.1007/BF01210792. Google Scholar [22] S. Schochet, Asymptotics for symmetric hyperbolic systems with a large parameter,, J. Diff. Eq., 75 (1988), 1. doi: 10.1016/0022-0396(88)90126-X. Google Scholar [23] S. Schochet, Fast singular limit of hyperbolic PDEs,, J. Diff. Eq., 114 (1994), 476. doi: 10.1006/jdeq.1994.1157. Google Scholar [24] R. Temam and D. Wirosoetisno, Averaging of differential equations generating oscillations and an application to control,, Applied Math. and Optimization, 46 (2002), 313. doi: 10.1007/s00245-002-0749-z. Google Scholar [25] R. Temam and D. Wirosoetisno, On the solutions of the renormalized equations at all orders,, Advances in Differential Equations, 8 (2003), 1005. Google Scholar [26] R. Temam and D. Wirosoetisno, Exponential Approximations for the Primitive Equations of the ocean,, in Discrete and Continuous Dynamical Systems-Series B, 7 (2007), 425. Google Scholar [27] R. Temam and D. Wirosoetisno, Stability of the slow manifold in the primitive equations,, SIAM J. Math. Anal., 42 (2010), 427. doi: 10.1137/080733358. Google Scholar [28] R. Temam and D. Wirosoetisno, Slow manifolds and invariant sets of the primitive equations,, J. Atmospheric Sciences, 68 (2011), 675. doi: 10.1175/2010JAS3650.1. Google Scholar [29] R. Temam and M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions,, Advances in Differential Equations, 1 (1996), 499. Google Scholar [30] R. Temam and M. Ziane, Navier-Stokes equations in thin spherical domains,, Contemporary Mathematics, 209 (1997), 281. doi: 10.1090/conm/209/02772. Google Scholar [31] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics,, in Handbook of Mathematical Fluid Dynamics, 3 (2004), 535. Google Scholar [32] M. Ziane, On a certain renormalization group method,, J. Math. Phys., 41 (2000), 3290. doi: 10.1063/1.533307. Google Scholar [33] M. Ziane, Regularity results for the stationary primitive equations of the atmosphere and the ocean,, Nonlinear Analysis, 28 (1997), 289. doi: 10.1016/0362-546X(95)00154-N. Google Scholar [34] M. Ziane, Regularity results for a Stokes type system,, Applicable Analysis, 58 (1995), 263. doi: 10.1080/00036819508840376. Google Scholar
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