×

zbMATH — the first resource for mathematics

Boundary layers for the subcritical modes of the 3D primitive equations in a cube. (English) Zbl 07046591
Summary: In this article we study the boundary layers for the subcritical modes of the viscous Linearized Primitive Equations (LPEs) in a cube at small viscosity. The boundary layers include the parabolic boundary layers, ordinary boundary layers, and their interaction-corner layers. The boundary layer correctors are determined by a phenomenological study reminiscent of the Prandtl corrector approach and then a rigorous convergence result is proved which a posteriori justifies the phenomenological study.

MSC:
35 Partial differential equations
34 Ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bardos, Claude; Brézis, Haïm, Sur une classe de problèmes d’évolution non linéaires, J. Differential Equations, 6, 345-394, (1969) · Zbl 0176.09003
[2] Bardos, C.; Brézis, D.; Brézis, Perturbations singulières et prolongements maximaux d’opérateurs positifs, Arch. Ration. Mech. Anal., 53, 69-100, (1973/74) · Zbl 0281.47028
[3] Bardos, Claude; Rauch, Jeffrey, Maximal positive boundary value problems as limits of singular perturbation problems, Trans. Amer. Math. Soc., 270, 2, 377-408, (1982) · Zbl 0485.35010
[4] Caloz, Gabriel; Costabel, Martin; Dauge, Monique; Vial, Grégory, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer, Asymptot. Anal., 50, 1-2, 121-173, (2006) · Zbl 1136.35021
[5] Charney, J. G.; Fjörtoft, R.; von Neumann, J., Numerical integration of the barotropic vorticity equation, Tellus, 2, 237-254, (1950)
[6] Van Dyke, M., An Album of Fluid Motion, (1982), The Parabolic Press: The Parabolic Press Stanford, California
[7] Eckhaus, Wiktor, Boundary layers in linear elliptic singular perturbation problems, SIAM Rev., 14, 225-270, (1972) · Zbl 0234.35009
[8] Fei, Mingwen; Han, Daozhi; Wang, Xiaoming, Initial-boundary layer associated with the nonlinear Darcy-Brinkman-Oberbeck-Boussinesq system, Phys. D, 338, 42-56, (2017) · Zbl 1376.76064
[9] Gie, Gung-Min; Jung, Chang-Yeol; Temam, Roger, Recent progresses in boundary layer theory, Discrete Contin. Dyn. Syst., 36, 5, 2521-2583, (2016) · Zbl 1343.35016
[10] Gie, Gung-Min; Hamouda, Makram; Jung, Chang-Yeol; Temam, Roger, Singular Perturbations and Boundary Layers, Applied Mathematics Series, vol. 200, (2018), Springer Nature Switzerland AG · Zbl 1411.35002
[11] Guès, O.; Métivier, G.; Williams, M.; Zumbrun, K., Boundary layer and long time stability for multidimensional viscous shocks, Discrete Contin. Dyn. Syst., 11, 1, 131-160, (2004) · Zbl 1108.35115
[12] Grasman, J., On the Birth of Boundary Layers, Mathematical Centre Tracts, vol. 36, (1971), Mathematisch Centrum: Mathematisch Centrum Amsterdam · Zbl 0228.35012
[13] Grenier, E., Boundary layers, (Handbook of Mathematical Fluid Dynamics, vol. III, (2004), North-Holland: North-Holland Amsterdam), 245-309 · Zbl 1221.76082
[14] Gisclon, Marguerite; Serre, Denis, Étude des conditions aux limites pour un système strictement hyperbolique via l’approximation parabolique, C. R. Acad. Sci. Paris Sér. I Math., 319, 4, 377-382, (1994) · Zbl 0808.35075
[15] Gisclon, Marguerite; Serre, Denis, Conditions aux limites pour un système strictement hyperbolique fournies par le schéma de Godunov, RAIRO Modél. Math. Anal. Numér., 31, 3, 359-380, (1997) · Zbl 0873.65087
[16] Hamouda, Makram; Jung, Chang-Yeol; Temam, Roger, Boundary layers for the 2D linearized primitive equations, Commun. Pure Appl. Anal., 8, 1, 335-359, (2009) · Zbl 1152.35423
[17] Hamouda, Makram; Jung, Chang-Yeol; Temam, Roger, Asymptotic analysis for the 3D primitive equations in a channel, Discrete Contin. Dyn. Syst. Ser., 6, 2, 401-422, (2013) · Zbl 1262.35078
[18] Makram Hamouda, Daozhi Han, Chang-Yeol Jung, Roger Temam, Well-posedness of the 3D linearized inviscid primitive equations in a cube, in preparation. · Zbl 1382.35306
[19] Hamouda, Makram; Han, Daozhi; Jung, Chang-Yeol; Temam, Roger, Boundary layers for the 3D primitive equations in a cube: the zero-mode, J. Appl. Anal. Comput., 8, 3, 873-889, (2018)
[20] Hamouda, Makram; Jung, Chang-Yeol; Temam, Roger, Boundary layers for the 3D primitive equations in a cube: the supercritical modes, Nonlinear Anal., 132, 288-317, (2016) · Zbl 1382.35306
[21] Hamouda, Makram; Jung, Chang-Yeol; Temam, Roger, Existence and regularity results for the inviscid primitive equations with lateral periodicity, Appl. Math. Optim., 73, 3, 501-522, (2016) · Zbl 1351.35138
[22] Han, D.; Wang, X., Initial-boundary layer associated with the nonlinear Darcy-Brinkman system, J. Differential Equations, 256, 2, 609-639, (2014) · Zbl 1331.35278
[23] Han, H.; Kellogg, R. B., Differentiability properties of solutions of the equation \(- \epsilon^2 \operatorname{\Delta} u + r u = f(x, y)\) in a square, SIAM J. Math. Anal., 21, 2, 394-408, (1990) · Zbl 0732.35020
[24] Han, Daozhi; Mazzucato, Anna L.; Niu, Dongjuan; Wang, Xiaoming, Boundary layer for a class of nonlinear pipe flow, J. Differential Equations, 252, 12, 6387-6413, (2012) · Zbl 1246.35159
[25] Halpern, Laurence; Rauch, Jeffrey, Hyperbolic boundary value problems with trihedral corners, Discrete Contin. Dyn. Syst., 36, 8, 4403-4450, (2016) · Zbl 1336.35217
[26] Huang, Aimin; Temam, Roger, The linearized 2D inviscid shallow water equations in a rectangle: boundary conditions and well-posedness, Arch. Ration. Mech. Anal., 211, 3, 1027-1063, (2014) · Zbl 1293.35247
[27] Huang, Aimin; Temam, Roger, The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction, Commun. Pure Appl. Anal., 13, 5, 2005-2038, (2014) · Zbl 1310.76015
[28] Huang, Aimin; Petcu, Madalina; Temam, Roger, The nonlinear 2D supercritical inviscid shallow water equations in a rectangle, Asymptot. Anal., 93, 3, 187-218, (2015) · Zbl 1326.35275
[29] Jung, Chang-Yeol; Temam, Roger, Numerical approximation of two-dimensional convection-diffusion equations with multiple boundary layers, Int. J. Numer. Anal. Model., 2, 4, 367-408, (2005) · Zbl 1103.65115
[30] Jung, Chang-Yeol; Temam, Roger, Singular perturbations and boundary layer theory for convection-diffusion equations in a circle: the generic noncompatible case, SIAM J. Math. Anal., 44, 6, 4274-4296, (2012) · Zbl 1261.35013
[31] Kellogg, R. Bruce; Stynes, Martin, Corner singularities and boundary layers in a simple convection-diffusion problem, J. Differential Equations, 213, 1, 81-120, (2005) · Zbl 1159.35309
[32] Levinson, Norman, The first boundary value problem for \(\varepsilon \operatorname{\Delta} u + A(x, y) u_x + B(x, y) u_y + C(x, y) u = D(x, y)\) for small ε, Ann. of Math. (2), 51, 428-445, (1950) · Zbl 0036.06801
[33] Lions, J.-L., Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics, vol. 323, (1973), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0268.49001
[34] Lions, J. L.; Temam, R.; Wang, S., New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5, 2, 237-288, (1992) · Zbl 0746.76019
[35] Lions, J. L.; Temam, R.; Wang, S., On the equations of the large-scale ocean, Nonlinearity, 5, 5, 1007-1053, (1992) · Zbl 0766.35039
[36] Masmoudi, Nader, The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary, Arch. Ration. Mech. Anal., 142, 4, 375-394, (1998) · Zbl 0915.76017
[37] Mazzucato, Anna; Niu, Dongjuan; Wang, Xiaoming, Boundary layer associated with a class of 3D nonlinear plane parallel channel flows, Indiana Univ. Math. J., 60, 4, 1113-1136, (2011) · Zbl 1426.76125
[38] Nguyen, Toan T.; Sueur, Franck, Boundary-layer interactions in the plane-parallel incompressible flows, Nonlinearity, 25, 12, 3327-3342, (2012) · Zbl 1263.35176
[39] O’Malley, Robert E., Singular Perturbation Methods for Ordinary Differential Equations, Appl. Math. Sci., vol. 89, (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0743.34059
[40] O’Malley, Robert E., Singularly perturbed linear two-point boundary value problems, SIAM Rev., 50, 3, 459-482, (2008) · Zbl 1362.34092
[41] Oliger, Joseph; Sundström, Arne, Theoretical and practical aspects of some initial boundary value problems in fluid dynamics, SIAM J. Appl. Math., 35, 3, 419-446, (1978) · Zbl 0397.35067
[42] Oleinik, O. A.; Samokhin, V. N., Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation, vol. 15, (1999), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL · Zbl 0928.76002
[43] Pedlosky, Joseph, Geophysical Fluid Dynamics, (1987), Springer-Verlag: Springer-Verlag New York · Zbl 0713.76005
[44] Tollmien, Walter; Schlichting, Hermann; Görtler, Henry, Ludwig Prandtl Gesammelte Abhandlungen zur angewandten Mechanik, Hydro- und Aerodynamik, (Riegels, F. W., (1961), Springer-Verlag: Springer-Verlag Berlin)
[45] Petcu, M.; Temam, R.; Ziane, M., Some mathematical problems in geophysical fluid dynamics, (Handbook of Numerical Analysis. Vol. XIV. Special Volume: Computational Methods for the Atmosphere and the Oceans. Handbook of Numerical Analysis. Vol. XIV. Special Volume: Computational Methods for the Atmosphere and the Oceans, Handb. Numer. Anal., vol. 14, (2009), Elsevier/North-Holland: Elsevier/North-Holland Amsterdam), 577-750
[46] Rauch, Jeffrey B.; Massey, Frank J., Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc., 189, 303-318, (1974) · Zbl 0282.35014
[47] Roos, H.-G.; Stynes, M.; Tobiska, L., Numerical Methods for Singularly Perturbed Differential EquationsConvection-Diffusion and Flow Problems, Springer Series in Computational Mathematics, vol. 24, (1996), Springer-Verlag: Springer-Verlag Berlin
[48] Shih, Shagi-Di; Kellogg, R. Bruce, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal., 18, 5, 1467-1511, (1987) · Zbl 0642.35006
[49] Stynes, Martin, Steady-state convection-diffusion problems, Acta Numer., 14, 445-508, (2005) · Zbl 1115.65108
[50] Temam, Roger; Tribbia, Joseph, Open boundary conditions for the primitive and Boussinesq equations, J. Atmos. Sci., 60, 21, 2647-2660, (2003)
[51] Tordeux, Sébastien; Vial, Grégory; Dauge, Monique, Matching and multiscale expansions for a model singular perturbation problem, C. R. Math. Acad. Sci. Paris, 343, 10, 637-642, (2006) · Zbl 1109.35013
[52] Verhulst, Ferdinand, Methods and Applications of Singular PerturbationsBoundary Layers and Multiple Timescale Dynamics, Texts in Applied Mathematics, vol. 50, (2005), Springer: Springer New York · Zbl 1148.35006
[53] Vial, Grégory, Efficiency of approximate boundary conditions for corner domains coated with thin layers, C. R. Math. Acad. Sci. Paris, 340, 3, 215-220, (2005) · Zbl 1061.65126
[54] Višik, M. I.; Ljusternik, L. A., Regular degeneration and boundary layer for linear differential equations with small parameter, Amer. Math. Soc. Transl., 2, 20, 239-364, (1962) · Zbl 0122.32402
[55] Warner, T. T.; Peterson, R. A.; Treadon, R. E., A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction, Bull. Am. Meteorol. Soc., 78, 11, 2599-2617, (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.