Recent progresses in boundary layer theory

M. Amar, A note on boundary layer effects in periodic homogenization with Dirichlet boundary conditions, Discrete Contin. Dynam. Systems, 6 (2000), 537-556.doi: 10.3934/dcds.2000.6.537.

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I. Babuška, U. Banerjee and J. E. Osborn, Survey of meshless and generalized finite element methods: A unified approach, Acta Numer., 12 (2003), 1-125.doi: 10.1017/S0962492902000090.

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M. Cannone, M. C. Lombardo and M. Sammartino, Well-posedness of Prandtl equations with non-compatible data, Nonlinearity, 26 (2013), 3077-3100.doi: 10.1088/0951-7715/26/12/3077.

M. Cannone, M. C. Lombardo and M. Sammartino, On the Prandtl boundary layer equations in presence of corner singularities, Acta Appl. Math., 132 (2014), 139-149.doi: 10.1007/s10440-014-9912-1.

T. Chacón-Rebollo, M. Gómez-Mármol and S. Rubino, On the existence and asymptotic stability of solutions for unsteady mixing-layer models, Discrete Contin. Dyn. Syst., 34 (2014), 421-436.doi: 10.3934/dcds.2014.34.421.

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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.

Q. Chen, Z. Qin and R. Temam, Numerical resolution near $t=0$ of nonlinear evolution equations in the presence of corner singularities in space dimension 1, Commun. Comput. Phys., 9 (2011), 568-586.doi: 10.4208/cicp.110909.160310s.

W. Cheng and R. Temam, Numerical approximation of one-dimensional stationary diffusion equations with boundary layers, Dedicated to Professor Roger Peyret on the occasion of his 65th birthday (Marseille, 1999), Comput. & Fluids, 31 (2002), 453-466.doi: 10.1016/S0045-7930(01)00060-3.

W. Cheng, R. Temam and X. Wang, New approximation algorithms for a class of partial differential equations displaying boundary layer behavior, Cathleen Morawetz: A great mathematician, Methods Appl. Anal., 7 (2000), 363-390.

P. G. Ciarlet, An introduction to differential geometry with application to elasticity, With a foreword by Roger Fosdick, J. Elasticity, 78/79 (2005), iv+215pp.doi: 10.1007/s10659-005-4738-8.

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A. J. DeSanti, Perturbed quasilinear Dirichlet problems with isolated turning points, Comm. Partial Differential Equations, 12 (1987), 223-242.doi: 10.1080/03605308708820489.

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Yihong Du, Zongming Guo, and Feng Zhou, Boundary blow-up solutions with interior layers and spikes in a bistable problem, Discrete Contin. Dyn. Syst., 19 (2007), 271-298.doi: 10.3934/dcds.2007.19.271.

Zhuoran Du and Baishun Lai, Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds, Discrete Contin. Dyn. Syst., 33 (2013), 1407-1429.

M. Van Dyke, An Album of Fluid Motion, The Parabolic Press, Stanford, California, 1982.

E. Weinan, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.), 16 (2000), 207-218.doi: 10.1007/s101140000034.

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S.-I. Ei and H. Matsuzawa, The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment, Discrete Contin. Dyn. Syst., 26 (2010), 901-921.doi: 10.3934/dcds.2010.26.901.

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G.-M. Gie, Singular perturbation problems in a general smooth domain, Asymptot. Anal., 62 (2009), 227-249.

G.-M. Gie, Asymptotic expansion of the Stokes solutions at small viscosity: The case of non-compatible initial data, Commun. Math. Sci., 12 (2014), 383-400.doi: 10.4310/CMS.2014.v12.n2.a8.

G.-M. Gie, M. Hamouda, C.-Y. Jung and T. Roger, Singular Perturbations and Boundary Layers, in preparation, Springer-Verlag, 2015.

G.-M. Gie, M. Hamouda and R. Temam, Asymptotic analysis of the Stokes problem on general bounded domains: The case of a characteristic boundary, Appl. Anal., 89 (2010), 49-66.doi: 10.1080/00036810903437796.

G.-M. Gie, M. Hamouda and R. Temam, Boundary layers in smooth curvilinear domains: Parabolic problems, Discrete Contin. Dyn. Syst., 26 (2010), 1213-1240.doi: 10.3934/dcds.2010.26.1213.

G.-M. Gie, M. Hamouda and R. Temam, Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary, Netw. Heterog. Media, 7 (2012), 741-766.doi: 10.3934/nhm.2012.7.741.

G.-M. Gie and C.-Y. Jung, Vorticity layers of the 2D Navier-Stokes equations with a slip type boundary condition, Asymptot. Anal., 84 (2013), 17-33.

G.-M. Gie, C.-Y. Jung and R. Temam, Analysis of mixed elliptic and parabolic boundary layers with corners, Int. J. Differ. Equ., (2013), Art. ID 532987, 13pp.

G.-M. Gie and J. P. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions, J. Differential Equations, 253 (2012), 1862-1892.doi: 10.1016/j.jde.2012.06.008.

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E. Grenier, Boundary layers, in Handbook of Mathematical Fluid Dynamics. Vol. III, North-Holland, Amsterdam, 2004, 245-309.

E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems, J. Differential Equations, 143 (1998), 110-146.doi: 10.1006/jdeq.1997.3364.

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M. Hamouda, C.-Y. Jung and R. Temam, Boundary layers for the 2D linearized primitive equations, Commun. Pure Appl. Anal., 8 (2009), 335-359.doi: 10.3934/cpaa.2009.8.335.

M. Hamouda, C.-Y. Jung and R. Temam, Asymptotic analysis for the 3D primitive equations in a channel, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 401-422.

M. Hamouda and R. Temam, Some singular perturbation problems related to the Navier-Stokes equations, in Advances in Deterministic and Stochastic Analysis, World Sci. Publ., Hackensack, NJ, 2007, 197-227.doi: 10.1142/9789812770493_0011.

M. Hamouda and R. Temam, Boundary layers for the Navier-Stokes equations. The case of a characteristic boundary, Georgian Math. J., 15 (2008), 517-530.

M. Hamouda, R. Temam and L. Zhang, Very weak solutions of the Stokes problem in a convex polygon, to appear, 2015.

D. Han, A. L. Mazzucato, D. Niu and X. Wang, Boundary layer for a class of nonlinear pipe flow, J. Differential Equations, 252 (2012), 6387-6413.doi: 10.1016/j.jde.2012.02.012.

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H. D. Han and R. B. Kellogg, The use of enriched subspaces for singular perturbation problems, in Proceedings of the China-France Symposium on Finite Element Methods (Beijing, 1982), Sci. Press Beijing, Beijing, 1983, 293-305.

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Y. Hong, C.-Y. Jung and J. Laminie, Singularly perturbed reaction-diffusion equations in a circle with numerical applications, Int. J. Comput. Math., 90 (2013), 2308-2325.doi: 10.1080/00207160.2013.772987.

Y. Hong, C.-Y. Jung and R. Temam, On the numerical approximations of stiff convection-diffusion equations in a circle, Numer. Math., 127 (2014), 291-313.doi: 10.1007/s00211-013-0585-x.

C.-Y. Jung, Finite elements scheme in enriched subspaces for singularly perturbed reaction-diffusion problems on a square domain, Asymptot. Anal., 57 (2008), 41-69.

C.-Y. Jung and T. B. Nguyen, Semi-analytical numerical methods for convection-dominated problems with turning points, Int. J. Numer. Anal. Model., 10 (2013), 314-332.

C.-Y. Jung, M. Petcu and R. Temam, Singular perturbation analysis on a homogeneous ocean circulation model, Anal. Appl. (Singap.), 9 (2011), 275-313.doi: 10.1142/S0219530511001832.

C.-Y. Jung and R. Temam, Boundary layer theory for convection-diffusion equations in a circle, Russian Math. Surveys, 69 (2014), 435-480.

C.-Y. Jung and R. Temam, Numerical approximation of two-dimensional convection-diffusion equations with multiple boundary layers, Int. J. Numer. Anal. Model., 2 (2005), 367-408.

C.-Y. Jung and R. Temam, On parabolic boundary layers for convection-diffusion equations in a channel: analysis and numerical applications, J. Sci. Comput., 28 (2006), 361-410.doi: 10.1007/s10915-006-9086-8.

C.-Y. Jung and R. Temam, Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point, J. Math. Phys., 48 (2007), 065301, 27pp.doi: 10.1063/1.2347899.

C.-Y. Jung and R. Temam, Finite volume approximation of one-dimensional stiff convection-diffusion equations, J. Sci. Comput., 41 (2009), 384-410.doi: 10.1007/s10915-009-9304-2.

C.-Y. Jung and R. Temam, Interaction of boundary layers and corner singularities, Discrete Contin. Dyn. Syst., 23 (2009), 315-339.doi: 10.3934/dcds.2009.23.315.

C.-Y. Jung and R. Temam, Finite volume approximation of two-dimensional stiff problems, Int. J. Numer. Anal. Model., 7 (2010), 462-476.

C.-Y. Jung and R. Temam, Convection-diffusion equations in a circle: The compatible case, J. Math. Pures Appl. (9), 96 (2011), 88-107.doi: 10.1016/j.matpur.2011.03.006.

C.-Y. Jung and R. Temam, Singular perturbations and boundary layer theory for convection-diffusion equations in a circle: The generic noncompatible case, SIAM J. Math. Anal., 44 (2012), 4274-4296.doi: 10.1137/110839515.

C.-Y. Jung and R. Temam, Singularly perturbed problems with a turning point: The non-compatible case, Anal. Appl. (Singap.), 12 (2014), 293-321.doi: 10.1142/S0219530513500279.

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