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April  2013, 6(2): 401-422. doi: 10.3934/dcdss.2013.6.401

Asymptotic analysis for the 3D primitive equations in a channel

Makram Hamouda 1, , Chang-Yeol Jung 2, and  Roger Temam 1,

1. 

The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 E. 3rd St., Rawles Hall, Bloomington, IN 47405, United States
2. 

School of Technology Management/ Mechanical, and Advanced Materials Engineering/ Natural Science, Ulsan National Institute of Science and Technology, San 194, Banyeon-ri, Eonyang-eup, Ulju-gun, Ulsan, South Korea

Received  July 2011 Revised  November 2011 Published  November 2012

In this article, we give an asymptotic expansion, with respect to the viscosity which is considered here to be small, of the solutions of the $3D$ linearized Primitive Equations (EPs) in a channel with lateral periodicity. A rigorous convergence result, in some physically relevant space, is proven. This allows, among other consequences, to confirm the natural choice of the non-local boundary conditions for the non-viscous PEs.
Keywords: Primitive equations, boundary layers, singular perturbation analysis..
Mathematics Subject Classification: 35B25, 35C20, 76D05, 76N2.
Citation: Makram Hamouda, Chang-Yeol Jung, Roger Temam. Asymptotic analysis for the 3D primitive equations in a channel. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 401-422. doi: 10.3934/dcdss.2013.6.401
References:
[1]

A. Bousquet, M. Petcu, C.-Y. Shiue, R. Temam and J. Tribbia, Boundary conditions for limited areas models based on the shallow water equations,, to appear., ().   Google Scholar

[2]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,, Ann. of Math. (2), 166 (2007), 245.   Google Scholar

[3]

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics. An Introduction to Rotating Fluids and the Navier-Stokes Equations,", Oxford Lecture Series in Mathematics and its Applications, (2006).   Google Scholar

[4]

G.-M. Gie, M. Hamouda and R. Temam, Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary, Networks and Het- erogeneous Media (NHM),, to appear., ().   Google Scholar

[5]

M. Hamouda, C. Jung and R. Temam, Boundary layers for the 2D linearized primitive equations,, Communications on Pure and Applied Analysis, 8 (2009), 335.   Google Scholar

[6]

M. Hamouda, C. Jung and R. Temam, Boundary layers for the 3D primitive equations in a cube,, in preparation., ().   Google Scholar

[7]

M. Hamouda, C. Jung and R. Temam, Regularity and existence results for the inviscid primitive equations in a channel,, in preparation., ().   Google Scholar

[8]

M. Hamouda and R. Temam, "Some Singular Perturbation Problems Related to the Navier-Sotkes Equations,", Advances in Deterministic and Stochastic Analysis. (Eds. N. M. Chuong et al.), (2007), 197.   Google Scholar

[9]

M. Hamouda and R. Temam, Boundary layers for the Navier-Stokes equation : The case of characteristic boundary,, Georgian Mathematical Journal, 15 (2008), 517.   Google Scholar

[10]

A. Huang, M. Petcu and R. Temam, The one-dimensional supercritical shallow-water equations with topography,, Annals of the University of Bucharest, ().   Google Scholar

[11]

G. M. Koblekov, Existence of a solution n the large for the 3D large-scale ocean dynamics equations,, C. R. Math. Acad. Sci. Paris, 343 (2006), 283.   Google Scholar

[12]

I. Kukavica, R. Temam, V. Vicol and M. Ziane, Existence and uniqueness of solutions for the hydrostatic Euler equations on a bounded domain with analytic data,, C. R. Math. Acad. Sci. Paris, 348 (2010), 639.   Google Scholar

[13]

I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean,, Nonlinearity, 20 (2007), 2739.   Google Scholar

[14]

C. Jung and R. Temam, Numerical approximation of two-dimensional convection-diffusion equations with multiple boundary layers,, International Journal of Numerical Analysis and Modeling, 2 (2005), 367.   Google Scholar

[15]

J. L. Lions, R. Temam and S. Wang, Models for the coupled atmosphere and ocean. (CAO I,II),, Comput. Mech. Adv., 1 (1993).   Google Scholar

[16]

J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications,, Nonlinearity, 5 (1992), 237.   Google Scholar

[17]

J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean,, Nonlinearity, 5 (1992), 1007.   Google Scholar

[18]

M. C. Lombardo and M. Sammartino, Zero viscosity limit of the Oseen equations in a channel,, SIAM J. Math. Anal., 33 (2001), 390.  doi: 10.1137/S0036141000372015.  Google Scholar

[19]

J. Oliger and A. Sundström, Theoretical and practical aspects of some initial boundary value problems in fluid dynamics,, SIAM J. Appl. Math., 35 (1978), 419.   Google Scholar

[20]

M. Petcu and R. Temam, An interface problem: The two-layer shallow water equations,, to appear., ().   Google Scholar

[21]

M. Petcu and R. Temam, The one-dimensional shallow water equations with transparent boundary conditions,, Mathematical Methods in the Applied Sciences (MMAS), (2011).  doi: 10.1002/mma.1482.  Google Scholar

[22]

J. P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921.   Google Scholar

[23]

A. Rousseau, R. Temam and J. Tribbia, Boundary conditions for the 2D linearized PEs of the ocean in the absence of viscosity,, Discrete Contin. Dyn. Syst., 13 (2005), 1257.   Google Scholar

[24]

A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case,, J. Math. Pures Appl., 89 (2008), 297.   Google Scholar

[25]

M-C. Shiue, J. Laminie, R. Temam and J. Tribbia, Boundary value problems for the shallow water equations with topography,, Journal of Geophysical Research, (2011).  doi: 10.1029/2010JC.  Google Scholar

[26]

R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations,, J. Atmospheric Sci., 60 (2003), 2647.   Google Scholar

[27]

R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case,, J. Differential Equations, 179 (2002), 647.  doi: 10.1006/jdeq.2001.4038.  Google Scholar

[28]

R. Temam and X. Wang, Boundary layers for Oseen's type equation in space dimension three,, Russian J. Math. Phys., 5 (): 227.   Google Scholar

[29]

S. Shih and R. B. Kellogg, Asymptotic analysis of a singular perturbation problem,, SIAM J. Math. Anal., 18 (1987), 1467.   Google Scholar

[30]

T. Warner, R. Peterson and R. Treadon, A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction,, Bull. Amer. Meteor. Soc., (1997), 2599.   Google Scholar

show all references

References:
[1]

A. Bousquet, M. Petcu, C.-Y. Shiue, R. Temam and J. Tribbia, Boundary conditions for limited areas models based on the shallow water equations,, to appear., ().   Google Scholar

[2]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,, Ann. of Math. (2), 166 (2007), 245.   Google Scholar

[3]

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics. An Introduction to Rotating Fluids and the Navier-Stokes Equations,", Oxford Lecture Series in Mathematics and its Applications, (2006).   Google Scholar

[4]

G.-M. Gie, M. Hamouda and R. Temam, Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary, Networks and Het- erogeneous Media (NHM),, to appear., ().   Google Scholar

[5]

M. Hamouda, C. Jung and R. Temam, Boundary layers for the 2D linearized primitive equations,, Communications on Pure and Applied Analysis, 8 (2009), 335.   Google Scholar

[6]

M. Hamouda, C. Jung and R. Temam, Boundary layers for the 3D primitive equations in a cube,, in preparation., ().   Google Scholar

[7]

M. Hamouda, C. Jung and R. Temam, Regularity and existence results for the inviscid primitive equations in a channel,, in preparation., ().   Google Scholar

[8]

M. Hamouda and R. Temam, "Some Singular Perturbation Problems Related to the Navier-Sotkes Equations,", Advances in Deterministic and Stochastic Analysis. (Eds. N. M. Chuong et al.), (2007), 197.   Google Scholar

[9]

M. Hamouda and R. Temam, Boundary layers for the Navier-Stokes equation : The case of characteristic boundary,, Georgian Mathematical Journal, 15 (2008), 517.   Google Scholar

[10]

A. Huang, M. Petcu and R. Temam, The one-dimensional supercritical shallow-water equations with topography,, Annals of the University of Bucharest, ().   Google Scholar

[11]

G. M. Koblekov, Existence of a solution n the large for the 3D large-scale ocean dynamics equations,, C. R. Math. Acad. Sci. Paris, 343 (2006), 283.   Google Scholar

[12]

I. Kukavica, R. Temam, V. Vicol and M. Ziane, Existence and uniqueness of solutions for the hydrostatic Euler equations on a bounded domain with analytic data,, C. R. Math. Acad. Sci. Paris, 348 (2010), 639.   Google Scholar

[13]

I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean,, Nonlinearity, 20 (2007), 2739.   Google Scholar

[14]

C. Jung and R. Temam, Numerical approximation of two-dimensional convection-diffusion equations with multiple boundary layers,, International Journal of Numerical Analysis and Modeling, 2 (2005), 367.   Google Scholar

[15]

J. L. Lions, R. Temam and S. Wang, Models for the coupled atmosphere and ocean. (CAO I,II),, Comput. Mech. Adv., 1 (1993).   Google Scholar

[16]

J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications,, Nonlinearity, 5 (1992), 237.   Google Scholar

[17]

J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean,, Nonlinearity, 5 (1992), 1007.   Google Scholar

[18]

M. C. Lombardo and M. Sammartino, Zero viscosity limit of the Oseen equations in a channel,, SIAM J. Math. Anal., 33 (2001), 390.  doi: 10.1137/S0036141000372015.  Google Scholar

[19]

J. Oliger and A. Sundström, Theoretical and practical aspects of some initial boundary value problems in fluid dynamics,, SIAM J. Appl. Math., 35 (1978), 419.   Google Scholar

[20]

M. Petcu and R. Temam, An interface problem: The two-layer shallow water equations,, to appear., ().   Google Scholar

[21]

M. Petcu and R. Temam, The one-dimensional shallow water equations with transparent boundary conditions,, Mathematical Methods in the Applied Sciences (MMAS), (2011).  doi: 10.1002/mma.1482.  Google Scholar

[22]

J. P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921.   Google Scholar

[23]

A. Rousseau, R. Temam and J. Tribbia, Boundary conditions for the 2D linearized PEs of the ocean in the absence of viscosity,, Discrete Contin. Dyn. Syst., 13 (2005), 1257.   Google Scholar

[24]

A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case,, J. Math. Pures Appl., 89 (2008), 297.   Google Scholar

[25]

M-C. Shiue, J. Laminie, R. Temam and J. Tribbia, Boundary value problems for the shallow water equations with topography,, Journal of Geophysical Research, (2011).  doi: 10.1029/2010JC.  Google Scholar

[26]

R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations,, J. Atmospheric Sci., 60 (2003), 2647.   Google Scholar

[27]

R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case,, J. Differential Equations, 179 (2002), 647.  doi: 10.1006/jdeq.2001.4038.  Google Scholar

[28]

R. Temam and X. Wang, Boundary layers for Oseen's type equation in space dimension three,, Russian J. Math. Phys., 5 (): 227.   Google Scholar

[29]

S. Shih and R. B. Kellogg, Asymptotic analysis of a singular perturbation problem,, SIAM J. Math. Anal., 18 (1987), 1467.   Google Scholar

[30]

T. Warner, R. Peterson and R. Treadon, A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction,, Bull. Amer. Meteor. Soc., (1997), 2599.   Google Scholar

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