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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 and Continuous Dynamical Systems, 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), 4410-4428. doi: 10.1175/1520-0469(1995)052<4410:GAAICI>2.0.CO;2.

[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), 1089-1121. doi: 10.1142/S021820259900049X.

[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), 201-222. doi: 10.1051/m2an:2000138.

[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), 1-35. doi: 10.1512/iumj.2001.50.2155.

[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, J. Norbury and I. Roulstone eds., Cambridge University Press, (2002), 126-201. doi: 10.1017/CBO9780511549991.005.

[6]

L. Y. Chen, N. Goldenfeld and Y. Oono, Renormalization group theory for global asymptotics analysis, Phys. Rev. Lett., 73 (1994), 1311-1315. doi: 10.1103/PhysRevLett.73.1311.

[7]

J. Y. Chemin, A propos d'un problème de pénalisation de type antisymmétrique, J. Math. pures et appl., 76 (1997), 739-755. doi: 10.1016/S0021-7824(97)89967-9.

[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), 245-267. doi: 10.4007/annals.2007.166.245.

[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), 619-658. doi: 10.1080/03605309608821200.

[10]

I. Gallagher, Existence globale pour des Equations des fluides géostrophiques, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 623-626. doi: 10.1016/S0764-4442(97)84772-6.

[11]

I. Gallagher, Applications of Schochet's methods to parabolic equations, J. Math. Pures Appl. (9), 77 (1998), 989-1054. doi: 10.1016/S0021-7824(99)80002-6.

[12]

E. Grenier, Rotating fluids and inertial waves, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 711-714.

[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), 283-286. doi: 10.1016/j.crma.2006.04.020.

[14]

J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications, Nonlinearity, 5 (1992), 237-288. doi: 10.1088/0951-7715/5/2/001.

[15]

I. Moise and R. Temam, Renormalization Group Method. Application to Navier-Stokes Equation, Discrete and Continuous Dynamical Systems, 6 (2000), 191-210.

[16]

I. Moise, R. Temam and M. Ziane, Asymptotic analysis of the Navier-Stokes equations in thin domains, Topological Methods in Nonlinear Analysis, volume dedicated to O.A. Ladyzhenskaya, 10 (1997), 249-282.

[17]

I. Moise and M. Ziane, Renormalization group method. Applications to partial differential equations, J. Dyn. and Diff. Syst., 13 (2001), 275-321. doi: 10.1023/A:1016680007953.

[18]

M. Petcu, On the three dimensional primitive equations, Adv. Diff. Eq., 11 (2006), 1201-1226.

[19]

M. Petcu, R. Temam and D. Wirosoetisno, Renormalization group method applied to the primitive equations, J. Diff. Eq., 208 (2005), 215-257. doi: 10.1016/j.jde.2003.10.011.

[20]

M. Petcu and D. Wirosoetisno, Sobolev and Gevrey regularity for the primitive equations in a space dimension 3, Appl. Anal., 84 (2005), 769-788. doi: 10.1080/00036810500130745.

[21]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75. doi: 10.1007/BF01210792.

[22]

S. Schochet, Asymptotics for symmetric hyperbolic systems with a large parameter, J. Diff. Eq., 75 (1988), 1-27. doi: 10.1016/0022-0396(88)90126-X.

[23]

S. Schochet, Fast singular limit of hyperbolic PDEs, J. Diff. Eq., 114 (1994), 476-512. doi: 10.1006/jdeq.1994.1157.

[24]

R. Temam and D. Wirosoetisno, Averaging of differential equations generating oscillations and an application to control, Applied Math. and Optimization, 46 (2002), 313-330. doi: 10.1007/s00245-002-0749-z.

[25]

R. Temam and D. Wirosoetisno, On the solutions of the renormalized equations at all orders, Advances in Differential Equations, 8 (2003), 1005-1024.

[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), 425-440.

[27]

R. Temam and D. Wirosoetisno, Stability of the slow manifold in the primitive equations, SIAM J. Math. Anal., 42 (2010), 427-458. doi: 10.1137/080733358.

[28]

R. Temam and D. Wirosoetisno, Slow manifolds and invariant sets of the primitive equations, J. Atmospheric Sciences, 68 (2011), 675-682. doi: 10.1175/2010JAS3650.1.

[29]

R. Temam and M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Advances in Differential Equations, 1 (1996), 499-546.

[30]

R. Temam and M. Ziane, Navier-Stokes equations in thin spherical domains, Contemporary Mathematics, AMS, 209 (1997), 281-314. doi: 10.1090/conm/209/02772.

[31]

R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, in Handbook of Mathematical Fluid Dynamics, 3, S. Friedlander and D. Serre Editors, Elsevier, (2004), 535-657.

[32]

M. Ziane, On a certain renormalization group method, J. Math. Phys., 41 (2000), 3290-3299. doi: 10.1063/1.533307.

[33]

M. Ziane, Regularity results for the stationary primitive equations of the atmosphere and the ocean, Nonlinear Analysis, Theory, Methods and Applications, 28 (1997), 289-313. doi: 10.1016/0362-546X(95)00154-N.

[34]

M. Ziane, Regularity results for a Stokes type system, Applicable Analysis, 58 (1995), 263-293. doi: 10.1080/00036819508840376.

show all references

References:
[1]

P. Bartello, Geostrophic adjustment and inverse cascades in rotating stratified turbulence, J. Atmos. Sci., 52 (1995), 4410-4428. doi: 10.1175/1520-0469(1995)052<4410:GAAICI>2.0.CO;2.

[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), 1089-1121. doi: 10.1142/S021820259900049X.

[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), 201-222. doi: 10.1051/m2an:2000138.

[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), 1-35. doi: 10.1512/iumj.2001.50.2155.

[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, J. Norbury and I. Roulstone eds., Cambridge University Press, (2002), 126-201. doi: 10.1017/CBO9780511549991.005.

[6]

L. Y. Chen, N. Goldenfeld and Y. Oono, Renormalization group theory for global asymptotics analysis, Phys. Rev. Lett., 73 (1994), 1311-1315. doi: 10.1103/PhysRevLett.73.1311.

[7]

J. Y. Chemin, A propos d'un problème de pénalisation de type antisymmétrique, J. Math. pures et appl., 76 (1997), 739-755. doi: 10.1016/S0021-7824(97)89967-9.

[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), 245-267. doi: 10.4007/annals.2007.166.245.

[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), 619-658. doi: 10.1080/03605309608821200.

[10]

I. Gallagher, Existence globale pour des Equations des fluides géostrophiques, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 623-626. doi: 10.1016/S0764-4442(97)84772-6.

[11]

I. Gallagher, Applications of Schochet's methods to parabolic equations, J. Math. Pures Appl. (9), 77 (1998), 989-1054. doi: 10.1016/S0021-7824(99)80002-6.

[12]

E. Grenier, Rotating fluids and inertial waves, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 711-714.

[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), 283-286. doi: 10.1016/j.crma.2006.04.020.

[14]

J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications, Nonlinearity, 5 (1992), 237-288. doi: 10.1088/0951-7715/5/2/001.

[15]

I. Moise and R. Temam, Renormalization Group Method. Application to Navier-Stokes Equation, Discrete and Continuous Dynamical Systems, 6 (2000), 191-210.

[16]

I. Moise, R. Temam and M. Ziane, Asymptotic analysis of the Navier-Stokes equations in thin domains, Topological Methods in Nonlinear Analysis, volume dedicated to O.A. Ladyzhenskaya, 10 (1997), 249-282.

[17]

I. Moise and M. Ziane, Renormalization group method. Applications to partial differential equations, J. Dyn. and Diff. Syst., 13 (2001), 275-321. doi: 10.1023/A:1016680007953.

[18]

M. Petcu, On the three dimensional primitive equations, Adv. Diff. Eq., 11 (2006), 1201-1226.

[19]

M. Petcu, R. Temam and D. Wirosoetisno, Renormalization group method applied to the primitive equations, J. Diff. Eq., 208 (2005), 215-257. doi: 10.1016/j.jde.2003.10.011.

[20]

M. Petcu and D. Wirosoetisno, Sobolev and Gevrey regularity for the primitive equations in a space dimension 3, Appl. Anal., 84 (2005), 769-788. doi: 10.1080/00036810500130745.

[21]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75. doi: 10.1007/BF01210792.

[22]

S. Schochet, Asymptotics for symmetric hyperbolic systems with a large parameter, J. Diff. Eq., 75 (1988), 1-27. doi: 10.1016/0022-0396(88)90126-X.

[23]

S. Schochet, Fast singular limit of hyperbolic PDEs, J. Diff. Eq., 114 (1994), 476-512. doi: 10.1006/jdeq.1994.1157.

[24]

R. Temam and D. Wirosoetisno, Averaging of differential equations generating oscillations and an application to control, Applied Math. and Optimization, 46 (2002), 313-330. doi: 10.1007/s00245-002-0749-z.

[25]

R. Temam and D. Wirosoetisno, On the solutions of the renormalized equations at all orders, Advances in Differential Equations, 8 (2003), 1005-1024.

[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), 425-440.

[27]

R. Temam and D. Wirosoetisno, Stability of the slow manifold in the primitive equations, SIAM J. Math. Anal., 42 (2010), 427-458. doi: 10.1137/080733358.

[28]

R. Temam and D. Wirosoetisno, Slow manifolds and invariant sets of the primitive equations, J. Atmospheric Sciences, 68 (2011), 675-682. doi: 10.1175/2010JAS3650.1.

[29]

R. Temam and M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Advances in Differential Equations, 1 (1996), 499-546.

[30]

R. Temam and M. Ziane, Navier-Stokes equations in thin spherical domains, Contemporary Mathematics, AMS, 209 (1997), 281-314. doi: 10.1090/conm/209/02772.

[31]

R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, in Handbook of Mathematical Fluid Dynamics, 3, S. Friedlander and D. Serre Editors, Elsevier, (2004), 535-657.

[32]

M. Ziane, On a certain renormalization group method, J. Math. Phys., 41 (2000), 3290-3299. doi: 10.1063/1.533307.

[33]

M. Ziane, Regularity results for the stationary primitive equations of the atmosphere and the ocean, Nonlinear Analysis, Theory, Methods and Applications, 28 (1997), 289-313. doi: 10.1016/0362-546X(95)00154-N.

[34]

M. Ziane, Regularity results for a Stokes type system, Applicable Analysis, 58 (1995), 263-293. doi: 10.1080/00036819508840376.

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