Asymptotic analysis for the 3D primitive equations in a channel

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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

Abstract
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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 and 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., ().
[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), 245-267.
[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, 32. The Clarendon Press, Oxford University Press, Oxford, 2006. xii+250 pp.
[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., ().
[5]	M. Hamouda, C. Jung and R. Temam, Boundary layers for the 2D linearized primitive equations, Communications on Pure and Applied Analysis, 8 (2009), 335-359.
[6]	M. Hamouda, C. Jung and R. Temam, Boundary layers for the 3D primitive equations in a cube,, in preparation., ().
[7]	M. Hamouda, C. Jung and R. Temam, Regularity and existence results for the inviscid primitive equations in a channel,, in preparation., ().
[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.), Springer Verlag, New York, 2007 World Scientific Publishing Co., 197-227. ISBN-13 978-981-270-550-1.
[9]	M. Hamouda and R. Temam, Boundary layers for the Navier-Stokes equation : The case of characteristic boundary, Georgian Mathematical Journal, 15 (2008), 517-530.
[10]	A. Huang, M. Petcu and R. Temam, The one-dimensional supercritical shallow-water equations with topography,, Annals of the University of Bucharest, ().
[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), 283-286.
[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), 639-645.
[13]	I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753.
[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), 367-408.
[15]	J. L. Lions, R. Temam and S. Wang, Models for the coupled atmosphere and ocean. (CAO I,II), Comput. Mech. Adv., 1 (1993), 120 pp.
[16]	J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.
[17]	J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053.
[18]	M. C. Lombardo and M. Sammartino, Zero viscosity limit of the Oseen equations in a channel, SIAM J. Math. Anal., 33 (2001), 390-410. (electronic). doi: 10.1137/S0036141000372015.
[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), 419-446.
[20]	M. Petcu and R. Temam, An interface problem: The two-layer shallow water equations,, to appear., ().
[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.
[22]	J. P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921-951.
[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), 1257-1276.
[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), 297-319.
[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, Oceans, 116, CO2015, 006315 (2011). doi: 10.1029/2010JC.
[26]	R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci., 60 (2003), 2647-2660.
[27]	R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647-686. doi: 10.1006/jdeq.2001.4038.
[28]	R. Temam and X. Wang, Boundary layers for Oseen's type equation in space dimension three,, Russian J. Math. Phys., 5 (): 227.
[29]	S. Shih and R. B. Kellogg, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal., 18 (1987), 1467-1511.
[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), 2599-2617.

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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., ().
[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), 245-267.
[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, 32. The Clarendon Press, Oxford University Press, Oxford, 2006. xii+250 pp.
[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., ().
[5]	M. Hamouda, C. Jung and R. Temam, Boundary layers for the 2D linearized primitive equations, Communications on Pure and Applied Analysis, 8 (2009), 335-359.
[6]	M. Hamouda, C. Jung and R. Temam, Boundary layers for the 3D primitive equations in a cube,, in preparation., ().
[7]	M. Hamouda, C. Jung and R. Temam, Regularity and existence results for the inviscid primitive equations in a channel,, in preparation., ().
[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.), Springer Verlag, New York, 2007 World Scientific Publishing Co., 197-227. ISBN-13 978-981-270-550-1.
[9]	M. Hamouda and R. Temam, Boundary layers for the Navier-Stokes equation : The case of characteristic boundary, Georgian Mathematical Journal, 15 (2008), 517-530.
[10]	A. Huang, M. Petcu and R. Temam, The one-dimensional supercritical shallow-water equations with topography,, Annals of the University of Bucharest, ().
[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), 283-286.
[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), 639-645.
[13]	I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753.
[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), 367-408.
[15]	J. L. Lions, R. Temam and S. Wang, Models for the coupled atmosphere and ocean. (CAO I,II), Comput. Mech. Adv., 1 (1993), 120 pp.
[16]	J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.
[17]	J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053.
[18]	M. C. Lombardo and M. Sammartino, Zero viscosity limit of the Oseen equations in a channel, SIAM J. Math. Anal., 33 (2001), 390-410. (electronic). doi: 10.1137/S0036141000372015.
[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), 419-446.
[20]	M. Petcu and R. Temam, An interface problem: The two-layer shallow water equations,, to appear., ().
[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.
[22]	J. P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921-951.
[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), 1257-1276.
[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), 297-319.
[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, Oceans, 116, CO2015, 006315 (2011). doi: 10.1029/2010JC.
[26]	R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci., 60 (2003), 2647-2660.
[27]	R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647-686. doi: 10.1006/jdeq.2001.4038.
[28]	R. Temam and X. Wang, Boundary layers for Oseen's type equation in space dimension three,, Russian J. Math. Phys., 5 (): 227.
[29]	S. Shih and R. B. Kellogg, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal., 18 (1987), 1467-1511.
[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), 2599-2617.

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