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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., (). Google Scholar 
[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), 245. Google Scholar 
[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, (2006). Google Scholar 
[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., (). 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 NavierSotkes 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 NavierStokes equation : The case of characteristic boundary,, Georgian Mathematical Journal, 15 (2008), 517. Google Scholar 
[10] 
A. Huang, M. Petcu and R. Temam, The onedimensional supercritical shallowwater 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 largescale 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 twodimensional convectiondiffusion 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 largescale 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 twolayer shallow water equations,, to appear., (). Google Scholar 
[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. Google Scholar 
[22] 
J. P. Raymond, Stokes and NavierStokes 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 wellposedness in the linearized case,, J. Math. Pures Appl., 89 (2008), 297. Google Scholar 
[25] 
MC. 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 NavierStokes 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 wellposedness of the threedimensional 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 NavierStokes 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 NavierStokes equations in a curved domain with a noncharacteristic 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 NavierSotkes 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 NavierStokes equation : The case of characteristic boundary,, Georgian Mathematical Journal, 15 (2008), 517. Google Scholar 
[10] 
A. Huang, M. Petcu and R. Temam, The onedimensional supercritical shallowwater 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 largescale 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 twodimensional convectiondiffusion 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 largescale 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 twolayer shallow water equations,, to appear., (). Google Scholar 
[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. Google Scholar 
[22] 
J. P. Raymond, Stokes and NavierStokes 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 wellposedness in the linearized case,, J. Math. Pures Appl., 89 (2008), 297. Google Scholar 
[25] 
MC. 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 NavierStokes 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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