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A CahnHilliardGurtin model with dynamic boundary conditions
Asymptotic analysis for the 3D primitive equations in a channel
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, Banyeonri, Eonyangeup, Uljugun, Ulsan, South Korea 
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 wellposedness of the threedimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245267. 
[3] 
J.Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics. An Introduction to Rotating Fluids and the NavierStokes 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 NavierStokes equations in a curved domain with a noncharacteristic 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), 335359. 
[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 NavierSotkes Equations," Advances in Deterministic and Stochastic Analysis. (Eds. N. M. Chuong et al.), Springer Verlag, New York, 2007 World Scientific Publishing Co., 197227. ISBN13 9789812705501. 
[9] 
M. Hamouda and R. Temam, Boundary layers for the NavierStokes equation : The case of characteristic boundary, Georgian Mathematical Journal, 15 (2008), 517530. 
[10] 
A. Huang, M. Petcu and R. Temam, The onedimensional supercritical shallowwater equations with topography, Annals of the University of Bucharest, Mathematical Series, to appear. 
[11] 
G. M. Koblekov, Existence of a solution n the large for the 3D largescale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283286. 
[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), 639645. 
[13] 
I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 27392753. 
[14] 
C. Jung and R. Temam, Numerical approximation of twodimensional convectiondiffusion equations with multiple boundary layers, International Journal of Numerical Analysis and Modeling, 2 (2005), 367408. 
[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), 237288. 
[17] 
J. L. Lions, R. Temam and S. Wang, On the equations of the largescale ocean, Nonlinearity, 5 (1992), 10071053. 
[18] 
M. C. Lombardo and M. Sammartino, Zero viscosity limit of the Oseen equations in a channel, SIAM J. Math. Anal., 33 (2001), 390410. (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), 419446. 
[20] 
M. Petcu and R. Temam, An interface problem: The twolayer shallow water equations, to appear. 
[21] 
M. Petcu and R. Temam, The onedimensional 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 NavierStokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921951. 
[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), 12571276. 
[24] 
A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary conditions and wellposedness in the linearized case, J. Math. Pures Appl., 89 (2008), 297319. 
[25] 
MC. 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), 26472660. 
[27] 
R. Temam and X. Wang, Boundary layers associated with incompressible NavierStokes equations: the noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647686. 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 (1997/1998), 227246. 
[29] 
S. Shih and R. B. Kellogg, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal., 18 (1987), 14671511. 
[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), 25992617. 
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. 
[2] 
C. Cao and E. S. Titi, Global wellposedness of the threedimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245267. 
[3] 
J.Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics. An Introduction to Rotating Fluids and the NavierStokes 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 NavierStokes equations in a curved domain with a noncharacteristic 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), 335359. 
[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 NavierSotkes Equations," Advances in Deterministic and Stochastic Analysis. (Eds. N. M. Chuong et al.), Springer Verlag, New York, 2007 World Scientific Publishing Co., 197227. ISBN13 9789812705501. 
[9] 
M. Hamouda and R. Temam, Boundary layers for the NavierStokes equation : The case of characteristic boundary, Georgian Mathematical Journal, 15 (2008), 517530. 
[10] 
A. Huang, M. Petcu and R. Temam, The onedimensional supercritical shallowwater equations with topography, Annals of the University of Bucharest, Mathematical Series, to appear. 
[11] 
G. M. Koblekov, Existence of a solution n the large for the 3D largescale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283286. 
[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), 639645. 
[13] 
I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 27392753. 
[14] 
C. Jung and R. Temam, Numerical approximation of twodimensional convectiondiffusion equations with multiple boundary layers, International Journal of Numerical Analysis and Modeling, 2 (2005), 367408. 
[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), 237288. 
[17] 
J. L. Lions, R. Temam and S. Wang, On the equations of the largescale ocean, Nonlinearity, 5 (1992), 10071053. 
[18] 
M. C. Lombardo and M. Sammartino, Zero viscosity limit of the Oseen equations in a channel, SIAM J. Math. Anal., 33 (2001), 390410. (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), 419446. 
[20] 
M. Petcu and R. Temam, An interface problem: The twolayer shallow water equations, to appear. 
[21] 
M. Petcu and R. Temam, The onedimensional 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 NavierStokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921951. 
[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), 12571276. 
[24] 
A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary conditions and wellposedness in the linearized case, J. Math. Pures Appl., 89 (2008), 297319. 
[25] 
MC. 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), 26472660. 
[27] 
R. Temam and X. Wang, Boundary layers associated with incompressible NavierStokes equations: the noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647686. 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 (1997/1998), 227246. 
[29] 
S. Shih and R. B. Kellogg, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal., 18 (1987), 14671511. 
[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), 25992617. 
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