May  2016, 36(5): 2521-2583. doi: 10.3934/dcds.2016.36.2521

Recent progresses in boundary layer theory

1. 

Department of Mathematics, University of Louisville, Louisville, KY 40292, United States

2. 

Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 689-798, South Korea

3. 

Department of Mathematics and The Institute, for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405

Received  April 2015 Revised  September 2015 Published  October 2015

In this article, we review recent progresses in boundary layer analysis of some singular perturbation problems. Using the techniques of differential geometry, an asymptotic expansion of reaction-diffusion or heat equations in a domain with curved boundary is constructed and validated in some suitable functional spaces. In addition, we investigate the effect of curvature as well as that of an ill-prepared initial data. Concerning convection-diffusion equations, the asymptotic behavior of their solutions is difficult and delicate to analyze because it largely depends on the characteristics of the corresponding limit problems, which are first order hyperbolic differential equations. Thus, the boundary layer analysis is performed on relatively simpler domains, typically intervals, rectangles, or circles. We consider also the interior transition layers at the turning point characteristics in an interval domain and classical (ordinary), characteristic (parabolic) and corner (elliptic) boundary layers in a rectangular domain using the technique of correctors and the tools of functional analysis. The validity of our asymptotic expansions is also established in suitable spaces.
Citation: Gung-Min Gie, Chang-Yeol Jung, Roger Temam. Recent progresses in boundary layer theory. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2521-2583. doi: 10.3934/dcds.2016.36.2521
References:
[1]

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

[2]

I. Andronov, D. Bouche and F. Molinet, Asymptotic and Hybrid Methods in Electromagnetics,, IEE Electromagnetic Waves Series, (2005). doi: 10.1049/PBEW051E.

[3]

I. Babuška and J. M. Melenk, The partition of unity method,, Internat. J. Numer. Methods Engrg., 40 (1997), 727. doi: 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N.

[4]

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

[5]

C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport,, Ann. Sci. École Norm. Sup. (4), 3 (1970), 185.

[6]

G. K. Batchelor, An Introduction to Fluid Dynamics,, paperback edition, (1999).

[7]

A. E. Berger, H. De Han and R. B. Kellogg, A priori estimates and analysis of a numerical method for a turning point problem,, Math. Comp., 42 (1984), 465. doi: 10.1090/S0025-5718-1984-0736447-2.

[8]

O. Botella, Numerical Solution of Navier-Stokes Singular Problem by a Chebyshev Projection Method,, Ph.D. Thesis, (2012).

[9]

Daniel Bouche and Frédéric Molinet, Méthodes Asymptotiques en Électromagnétisme,, With a preface by Robert Dautray, (1994).

[10]

R. E. Caflisch and M. Sammartino, Existence and singularities for the Prandtl boundary layer equations,, Special issue on the occasion of the 125th anniversary of the birth of Ludwig Prandtl, 80 (2000), 733. doi: 10.1002/1521-4001(200011)80:11/12<733::AID-ZAMM733>3.0.CO;2-L.

[11]

J. R. Cannon, The One-Dimensional Heat Equation,, With a foreword by Felix E. Browder, (1984). doi: 10.1017/CBO9781139086967.

[12]

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

[13]

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. doi: 10.1007/s10440-014-9912-1.

[14]

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. doi: 10.3934/dcds.2014.34.421.

[15]

K. W. Chang and F. A. Howes, Nonlinear Singular Perturbation Phenomena: Theory and Applications,, Applied Mathematical Sciences, (1984). doi: 10.1007/978-1-4612-1114-3.

[16]

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, (2006).

[17]

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. doi: 10.4208/cicp.110909.160310s.

[18]

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, 31 (2002), 453. doi: 10.1016/S0045-7930(01)00060-3.

[19]

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, 7 (2000), 363.

[20]

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

[21]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations,, Trans. Amer. Math. Soc., 277 (1983), 1. doi: 10.1090/S0002-9947-1983-0690039-8.

[22]

A. J. DeSanti, Nonmonotone interior layer theory for some singularly perturbed quasilinear boundary value problems with turning points,, SIAM J. Math. Anal., 18 (1987), 321. doi: 10.1137/0518025.

[23]

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

[24]

B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions,, J. Math. Pures Appl. (9), 78 (1999), 461. doi: 10.1016/S0021-7824(99)00032-X.

[25]

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. doi: 10.3934/dcds.2007.19.271.

[26]

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

[27]

M. Van Dyke, An Album of Fluid Motion,, The Parabolic Press, (1982).

[28]

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

[29]

W. Eckhaus and E. M. de Jager, Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type,, Arch. Rational Mech. Anal., 23 (1966), 26. doi: 10.1007/BF00281135.

[30]

W. Eckhaus, Boundary layers in linear elliptic singular perturbation problems,, SIAM Rev., 14 (1972), 225. doi: 10.1137/1014030.

[31]

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. doi: 10.3934/dcds.2010.26.901.

[32]

N. Flyer and B. Fornberg, Accurate numerical resolution of transients in initial-boundary value problems for the heat equation,, J. Comput. Phys., 184 (2003), 526. doi: 10.1016/S0021-9991(02)00034-7.

[33]

N. Flyer and B. Fornberg, On the nature of initial-boundary value solutions for dispersive equations,, SIAM J. Appl. Math., 64 (): 546. doi: 10.1137/S0036139902415853.

[34]

S. Garcia, Aperiodic, chaotic lid-driven square cavity flows,, Math. Comput. Simulation, 81 (2011), 1741. doi: 10.1016/j.matcom.2011.01.011.

[35]

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

[36]

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. doi: 10.4310/CMS.2014.v12.n2.a8.

[37]

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

[38]

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. doi: 10.1080/00036810903437796.

[39]

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

[40]

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. doi: 10.3934/nhm.2012.7.741.

[41]

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.

[42]

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

[43]

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. doi: 10.1016/j.jde.2012.06.008.

[44]

G.-M. Gie, J. P. Kelliher, M. C. Lopes Filho, A. L. Mazzucato and H. J. Nussenzveig Lopes, Vanishing viscosity limit of some symmetric flows,, preprint., ().

[45]

J. Grasman, On the Birth of Boundary Layers,, Mathematical Centre Tracts, (1971).

[46]

H. P. Greenspan, The Theory of Rotating Fluids,, Reprint of the 1968 original, (1968).

[47]

Y. Guo and T. Nguyen, A note on Prandtl boundary layers,, Comm. Pure Appl. Math., 64 (2011), 1416. doi: 10.1002/cpa.20377.

[48]

Y. Guo and T. Nguyen, Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate,, , ().

[49]

E. Grenier, Boundary layers,, in Handbook of Mathematical Fluid Dynamics. Vol. III, (2004), 245.

[50]

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

[51]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985).

[52]

P. Grisvard, Singularities in Boundary Value Problems,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], (1992).

[53]

O. Guès, G. Métivier, M. Williams and K. Zumbrun, Boundary layer and long time stability for multidimensional viscous shocks,, Discrete Contin. Dyn. Syst., 11 (2004), 131. doi: 10.3934/dcds.2004.11.131.

[54]

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

[55]

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.

[56]

M. Hamouda and R. Temam, Some singular perturbation problems related to the Navier-Stokes equations,, in Advances in Deterministic and Stochastic Analysis, (2007), 197. doi: 10.1142/9789812770493_0011.

[57]

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

[58]

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

[59]

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. doi: 10.1016/j.jde.2012.02.012.

[60]

H. Han and R. B. Kellogg, Differentiability properties of solutions of the equation $-\epsilon^2\Delta u+ru=f(x,y)$ in a square,, SIAM J. Math. Anal., 21 (1990), 394. doi: 10.1137/0521022.

[61]

H. De Han and R. B. Kellogg, A method of enriched subspaces for the numerical solution of a parabolic singular perturbation problem,, in Computational and Asymptotic Methods for Boundary and Interior Layers (Dublin, (1982), 46.

[62]

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), 293.

[63]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, Reprint of the 1952 edition, (1952).

[64]

P. W. Hemker, A Numerical Study of Stiff Two-Point Boundary Problems,, Mathematisch Centrum, (1977).

[65]

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. doi: 10.1080/00207160.2013.772987.

[66]

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. doi: 10.1007/s00211-013-0585-x.

[67]

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

[68]

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.

[69]

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

[70]

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

[71]

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.

[72]

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. doi: 10.1007/s10915-006-9086-8.

[73]

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

[74]

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

[75]

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

[76]

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

[77]

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

[78]

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. doi: 10.1137/110839515.

[79]

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

[80]

T. Kato, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary,, in Seminar on Nonlinear Partial Differential Equations (Berkeley, (1983), 85. doi: 10.1007/978-1-4612-1110-5_6.

[81]

T. Kato, Remarks on the Euler and Navier-Stokes equations in $R^2$,, in Nonlinear Functional Analysis and its Applications, (1983), 1.

[82]

J. P. Kelliher, On Kato's conditions for vanishing viscosity,, Indiana Univ. Math. J., 56 (2007), 1711. doi: 10.1512/iumj.2007.56.3080.

[83]

J. P. Kelliher, Vanishing viscosity and the accumulation of vorticity on the boundary,, Commun. Math. Sci., 6 (2008), 869. doi: 10.4310/CMS.2008.v6.n4.a4.

[84]

J. P. Kelliher, On the vanishing viscosity limit in a disk,, Math. Ann., 343 (2009), 701. doi: 10.1007/s00208-008-0287-3.

[85]

R. B. Kellogg and M. Stynes, Corner singularities and boundary layers in a simple convection-diffusion problem,, J. Differential Equations, 213 (2005), 81. doi: 10.1016/j.jde.2005.02.011.

[86]

J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods,, Applied Mathematical Sciences, (1996). doi: 10.1007/978-1-4612-3968-0.

[87]

W. Klingenberg, A Course in Differential Geometry,, Translated from the German by David Hoffman, (1978).

[88]

P. A. Lagerstrom, Matched Asymptotic Expansions. Ideas and Techniques,, Applied Mathematical Sciences, (1988). doi: 10.1007/978-1-4757-1990-1.

[89]

N. Levinson, The first boundary value problem for $\varepsilon\Delta u+A(x,y)u_x+B(x,y)u_y+C(x,y)u=D(x,y)$ for small $\varepsilon$,, Ann. of Math. (2), 51 (1950), 428.

[90]

F. Li and K. Nakashima, Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains,, Discrete Contin. Dyn. Syst., 32 (2012), 1391. doi: 10.3934/dcds.2012.32.1391.

[91]

J.-L. Lions, Perturbations Singulières Dans Les Problèmes Aux Limites et en Contrôle Optimal,, Lecture Notes in Mathematics, (1973).

[92]

P.-L. Lions, On the Hamilton-Jacobi-Bellman equations,, Acta Appl. Math., 1 (1983), 17. doi: 10.1007/BF02433840.

[93]

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.

[94]

M. C. Lopes Filho, A. L. Mazzucato and H. J. Nussenzveig Lopes, Vanishing viscosity limit for incompressible flow inside a rotating circle,, Phys. D, 237 (2008), 1324. doi: 10.1016/j.physd.2008.03.009.

[95]

M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussenzveig Lopes and M. Taylor, Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows,, Bull. Braz. Math. Soc. (N.S.), 39 (2008), 471. doi: 10.1007/s00574-008-0001-9.

[96]

M. C. Lopes Filho, Boundary layers and the vanishing viscosity limit for incompressible 2D flow,, in Lectures on the Analysis of Nonlinear Partial Differential Equations. Part 1, (2012), 1.

[97]

T. Ma and S. Wang, Boundary layer separation and structural bifurcation for 2-D incompressible fluid flows. Partial differential equations and applications,, Discrete Contin. Dyn. Syst., 10 (2004), 459. doi: 10.3934/dcds.2004.10.459.

[98]

T. Ma and S. Wang, Bifurcation Theory and Applications,, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, (2005). doi: 10.1142/9789812701152.

[99]

A. Malchiodi, Construction of multidimensional spike-layers,, Discrete Contin. Dyn. Syst., 14 (2006), 187. doi: 10.3934/dcds.2006.14.187.

[100]

N. Masmoudi, The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary,, Arch. Rational Mech. Anal., 142 (1998), 375. doi: 10.1007/s002050050097.

[101]

H. Matsuzawa, On a solution with transition layers for a bistable reaction-diffusion equation with spatially heterogeneous environments,, Discrete Contin. Dyn. Syst., (2009), 516.

[102]

A. Mazzucato, D. Niu and X. Wang, Boundary layer associated with a class of 3D nonlinear plane parallel channel flows,, Indiana Univ. Math. J., 60 (2011), 1113. doi: 10.1512/iumj.2011.60.4479.

[103]

A. Mazzucato and M. Taylor, Vanishing viscosity limits for a class of circular pipe flows,, Comm. Partial Differential Equations, 36 (2011), 328. doi: 10.1080/03605302.2010.505973.

[104]

A. L. Mazzucato, V. Nistor and Q. Qu, A nonconforming generalized finite element method for transmission problems,, SIAM J. Numer. Anal., 51 (2013), 555. doi: 10.1137/100816031.

[105]

A. L. Mazzucato, V. Nistor and Q. Qu, Quasi-optimal rates of convergence for the generalized finite element method in polygonal domains,, J. Comput. Appl. Math., 263 (2014), 466. doi: 10.1016/j.cam.2013.12.026.

[106]

N. Möes, J. Dolbow and T. Belytschko, A finite element method for crack growth without remeshing,, International Journal for Numerical Methods in Engineering, 46 (1999), 131.

[107]

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory,, Applied Mathematics and Mathematical Computation, (1999).

[108]

R. E. O'Malley, Jr., On boundary value problems for a singularly perturbed differential equation with a turning point,, SIAM J. Math. Anal., 1 (1970), 479. doi: 10.1137/0501041.

[109]

R. E. O'Malley, Jr., Introduction to Singular Perturbations,, Applied Mathematics and Mechanics, (1974).

[110]

R. E. O'Malley, Jr., Singular Perturbation Analysis for Ordinary Differential Equations,, Communications of the Mathematical Institute, (1977).

[111]

R. E. O'Malley, Jr., Singular Perturbation Methods for Ordinary Differential Equations,, Applied Mathematical Sciences, (1991). doi: 10.1007/978-1-4612-0977-5.

[112]

R. E. O'Malley, Jr., Singularly perturbed linear two-point boundary value problems,, SIAM Rev., 50 (2008), 459. doi: 10.1137/060662058.

[113]

C. H. Ou and R. Wong, Shooting method for nonlinear singularly perturbed boundary-value problems,, Stud. Appl. Math., 112 (2004), 161. doi: 10.1111/j.0022-2526.2004.01509.x.

[114]

L. Prandtl, Verber flüssigkeiten bei sehr kleiner reibung,, in Verk. III Intem. Math. Kongr. Heidelberg, (1905), 484.

[115]

L. Prandtl, Gesammelte Abhandlungen Zur Angewandten Mechanik, Hydro- und Aerodynamik,, Herausgegeben von Walter Tollmien, (1961).

[116]

J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921. doi: 10.1016/j.anihpc.2006.06.008.

[117]

W. H. Reid, Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow,, Studies in Appl. Math., 53 (1974), 91. doi: 10.1002/sapm197453291.

[118]

W. H. Reid, Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory,, Studies in Appl. Math., 53 (1974), 217. doi: 10.1002/sapm1974533217.

[119]

H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems,, Springer Series in Computational Mathematics, (1996). doi: 10.1007/978-3-662-03206-0.

[120]

L. Ruan and C. Zhu, Boundary layer for nonlinear evolution equations with damping and diffusion,, Discrete Contin. Dyn. Syst., 32 (2012), 331. doi: 10.3934/dcds.2012.32.331.

[121]

H. Schlichting, Boundary Layer Theory,, Translated by J. Kestin, (1955).

[122]

S.-D. Shih and R. B. Kellogg, Asymptotic analysis of a singular perturbation problem,, SIAM J. Math. Anal., 18 (1987), 1467. doi: 10.1137/0518107.

[123]

E. Simonnet, M. Ghil, K. Ide, R. Temam and S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation, I. Steady-state solution,, J. Phys. Oceanogr., 33 (2003), 712. doi: 10.1175/1520-0485(2003)33<712:LVISMO>2.0.CO;2.

[124]

S. Smale, Smooth solutions of the heat and wave equations,, Comment. Math. Helv., 55 (1980), 1. doi: 10.1007/BF02566671.

[125]

D. R. Smith, Singular-Perturbation Theory. An Introduction with Applications,, Cambridge University Press, (1985).

[126]

M. Stynes, Steady-state convection-diffusion problems,, Acta Numer., 14 (2005), 445. doi: 10.1017/S0962492904000261.

[127]

G. Fu Sun and M. Stynes, Finite element methods on piecewise equidistant meshes for interior turning point problems,, Numer. Algorithms, 8 (1994), 111. doi: 10.1007/BF02145699.

[128]

R. Temam, Behaviour at time $t=0$ of the solutions of semilinear evolution equations,, J. Differential Equations, 43 (1982), 73. doi: 10.1016/0022-0396(82)90075-4.

[129]

R. Temam and X. Wang, Remarks on the Prandtl equation for a permeable wall,, Special issue on the occasion of the 125th anniversary of the birth of Ludwig Prandtl, 80 (2000), 835. doi: 10.1002/1521-4001(200011)80:11/12<835::AID-ZAMM835>3.0.CO;2-9.

[130]

R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: The noncharacteristic boundary case,, J. Differential Equations, 179 (2002), 647. doi: 10.1006/jdeq.2001.4038.

[131]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Reprint of the 1984 edition, (1984).

[132]

R. Temam and X. M. Wang, Asymptotic analysis of the linearized Navier-Stokes equations in a channel,, Differential Integral Equations, 8 (1995), 1591.

[133]

R. Temam and X. Wang, Asymptotic analysis of Oseen type equations in a channel at small viscosity,, Indiana Univ. Math. J., 45 (1996), 863. doi: 10.1512/iumj.1996.45.1290.

[134]

R. Temam and X. Wang, Asymptotic analysis of the linearized Navier-Stokes equations in a general $2$D domain,, Asymptot. Anal., 14 (1997), 293.

[135]

R. Temam and X. Wang, Boundary layers for Oseen's type equation in space dimension three,, Russian J. Math. Phys., 5 (1997), 227.

[136]

N. M. Temme, Analytical methods for an elliptic singular perturbation problem in a circle,, J. Comput. Appl. Math., 207 (2007), 301. doi: 10.1016/j.cam.2006.10.049.

[137]

M. Urano, K. Nakashima and Y. Yamada, Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity,, Discrete Contin. Dyn. Syst., (2005), 868.

[138]

F. Verhulst, Methods and Applications of Singular Perturbations. Boundary Layers and Multiple Timescale Dynamics,, Texts in Applied Mathematics, (2005). doi: 10.1007/0-387-28313-7.

[139]

M. I. Višik and L. A. Ljusternik, Regular degeneration and boundary layer for linear differential equations with small parameter,, Amer. Math. Soc. Transl. (2), 20 (1962), 239.

[140]

M. I. Višik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter,, Uspehi Mat. Nauk (N.S.), 12 (1957), 3.

[141]

T. von Kármán, Progress in the statistical theory of turbulence,, J. Marine Research, 7 (1948), 252.

[142]

L. Wang and J. Wei, Solutions with interior bubble and boundary layer for an elliptic problem,, Discrete Contin. Dyn. Syst., 21 (2008), 333. doi: 10.3934/dcds.2008.21.333.

[143]

L. Wang and C. Zhao, Solutions with clustered bubbles and a boundary layer of an elliptic problem,, Discrete Contin. Dyn. Syst., 34 (2014), 2333.

[144]

W. Wasow, Linear Turning Point Theory,, Applied Mathematical Sciences, (1985). doi: 10.1007/978-1-4612-1090-0.

[145]

R. Wong and H. Yang, On a boundary-layer problem,, Stud. Appl. Math., 108 (2002), 369. doi: 10.1111/1467-9590.01430.

[146]

R. Wong and H. Yang, On an internal boundary layer problem,, J. Comput. Appl. Math., 144 (2002), 301. doi: 10.1016/S0377-0427(01)00569-6.

[147]

R. Wong and H. Yang, On the Ackerberg-O'Malley resonance,, Stud. Appl. Math., 110 (2003), 157. doi: 10.1111/1467-9590.00235.

[148]

R. Wong and Y. Zhao, A singularly perturbed boundary-value problem arising in phase transitions,, European J. Appl. Math., 17 (2006), 705. doi: 10.1017/S095679250600670X.

[149]

L. Zhang, Ph.D. Thesis, Indiana University,, in preparation, (2015).

show all references

References:
[1]

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

[2]

I. Andronov, D. Bouche and F. Molinet, Asymptotic and Hybrid Methods in Electromagnetics,, IEE Electromagnetic Waves Series, (2005). doi: 10.1049/PBEW051E.

[3]

I. Babuška and J. M. Melenk, The partition of unity method,, Internat. J. Numer. Methods Engrg., 40 (1997), 727. doi: 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N.

[4]

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

[5]

C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport,, Ann. Sci. École Norm. Sup. (4), 3 (1970), 185.

[6]

G. K. Batchelor, An Introduction to Fluid Dynamics,, paperback edition, (1999).

[7]

A. E. Berger, H. De Han and R. B. Kellogg, A priori estimates and analysis of a numerical method for a turning point problem,, Math. Comp., 42 (1984), 465. doi: 10.1090/S0025-5718-1984-0736447-2.

[8]

O. Botella, Numerical Solution of Navier-Stokes Singular Problem by a Chebyshev Projection Method,, Ph.D. Thesis, (2012).

[9]

Daniel Bouche and Frédéric Molinet, Méthodes Asymptotiques en Électromagnétisme,, With a preface by Robert Dautray, (1994).

[10]

R. E. Caflisch and M. Sammartino, Existence and singularities for the Prandtl boundary layer equations,, Special issue on the occasion of the 125th anniversary of the birth of Ludwig Prandtl, 80 (2000), 733. doi: 10.1002/1521-4001(200011)80:11/12<733::AID-ZAMM733>3.0.CO;2-L.

[11]

J. R. Cannon, The One-Dimensional Heat Equation,, With a foreword by Felix E. Browder, (1984). doi: 10.1017/CBO9781139086967.

[12]

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

[13]

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. doi: 10.1007/s10440-014-9912-1.

[14]

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. doi: 10.3934/dcds.2014.34.421.

[15]

K. W. Chang and F. A. Howes, Nonlinear Singular Perturbation Phenomena: Theory and Applications,, Applied Mathematical Sciences, (1984). doi: 10.1007/978-1-4612-1114-3.

[16]

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, (2006).

[17]

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. doi: 10.4208/cicp.110909.160310s.

[18]

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, 31 (2002), 453. doi: 10.1016/S0045-7930(01)00060-3.

[19]

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, 7 (2000), 363.

[20]

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

[21]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations,, Trans. Amer. Math. Soc., 277 (1983), 1. doi: 10.1090/S0002-9947-1983-0690039-8.

[22]

A. J. DeSanti, Nonmonotone interior layer theory for some singularly perturbed quasilinear boundary value problems with turning points,, SIAM J. Math. Anal., 18 (1987), 321. doi: 10.1137/0518025.

[23]

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

[24]

B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions,, J. Math. Pures Appl. (9), 78 (1999), 461. doi: 10.1016/S0021-7824(99)00032-X.

[25]

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. doi: 10.3934/dcds.2007.19.271.

[26]

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

[27]

M. Van Dyke, An Album of Fluid Motion,, The Parabolic Press, (1982).

[28]

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

[29]

W. Eckhaus and E. M. de Jager, Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type,, Arch. Rational Mech. Anal., 23 (1966), 26. doi: 10.1007/BF00281135.

[30]

W. Eckhaus, Boundary layers in linear elliptic singular perturbation problems,, SIAM Rev., 14 (1972), 225. doi: 10.1137/1014030.

[31]

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. doi: 10.3934/dcds.2010.26.901.

[32]

N. Flyer and B. Fornberg, Accurate numerical resolution of transients in initial-boundary value problems for the heat equation,, J. Comput. Phys., 184 (2003), 526. doi: 10.1016/S0021-9991(02)00034-7.

[33]

N. Flyer and B. Fornberg, On the nature of initial-boundary value solutions for dispersive equations,, SIAM J. Appl. Math., 64 (): 546. doi: 10.1137/S0036139902415853.

[34]

S. Garcia, Aperiodic, chaotic lid-driven square cavity flows,, Math. Comput. Simulation, 81 (2011), 1741. doi: 10.1016/j.matcom.2011.01.011.

[35]

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

[36]

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. doi: 10.4310/CMS.2014.v12.n2.a8.

[37]

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

[38]

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. doi: 10.1080/00036810903437796.

[39]

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

[40]

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. doi: 10.3934/nhm.2012.7.741.

[41]

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.

[42]

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

[43]

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. doi: 10.1016/j.jde.2012.06.008.

[44]

G.-M. Gie, J. P. Kelliher, M. C. Lopes Filho, A. L. Mazzucato and H. J. Nussenzveig Lopes, Vanishing viscosity limit of some symmetric flows,, preprint., ().

[45]

J. Grasman, On the Birth of Boundary Layers,, Mathematical Centre Tracts, (1971).

[46]

H. P. Greenspan, The Theory of Rotating Fluids,, Reprint of the 1968 original, (1968).

[47]

Y. Guo and T. Nguyen, A note on Prandtl boundary layers,, Comm. Pure Appl. Math., 64 (2011), 1416. doi: 10.1002/cpa.20377.

[48]

Y. Guo and T. Nguyen, Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate,, , ().

[49]

E. Grenier, Boundary layers,, in Handbook of Mathematical Fluid Dynamics. Vol. III, (2004), 245.

[50]

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

[51]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985).

[52]

P. Grisvard, Singularities in Boundary Value Problems,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], (1992).

[53]

O. Guès, G. Métivier, M. Williams and K. Zumbrun, Boundary layer and long time stability for multidimensional viscous shocks,, Discrete Contin. Dyn. Syst., 11 (2004), 131. doi: 10.3934/dcds.2004.11.131.

[54]

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

[55]

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.

[56]

M. Hamouda and R. Temam, Some singular perturbation problems related to the Navier-Stokes equations,, in Advances in Deterministic and Stochastic Analysis, (2007), 197. doi: 10.1142/9789812770493_0011.

[57]

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

[58]

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

[59]

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. doi: 10.1016/j.jde.2012.02.012.

[60]

H. Han and R. B. Kellogg, Differentiability properties of solutions of the equation $-\epsilon^2\Delta u+ru=f(x,y)$ in a square,, SIAM J. Math. Anal., 21 (1990), 394. doi: 10.1137/0521022.

[61]

H. De Han and R. B. Kellogg, A method of enriched subspaces for the numerical solution of a parabolic singular perturbation problem,, in Computational and Asymptotic Methods for Boundary and Interior Layers (Dublin, (1982), 46.

[62]

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), 293.

[63]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, Reprint of the 1952 edition, (1952).

[64]

P. W. Hemker, A Numerical Study of Stiff Two-Point Boundary Problems,, Mathematisch Centrum, (1977).

[65]

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. doi: 10.1080/00207160.2013.772987.

[66]

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. doi: 10.1007/s00211-013-0585-x.

[67]

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

[68]

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.

[69]

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

[70]

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

[71]

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.

[72]

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. doi: 10.1007/s10915-006-9086-8.

[73]

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

[74]

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

[75]

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

[76]

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

[77]

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

[78]

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. doi: 10.1137/110839515.

[79]

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

[80]

T. Kato, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary,, in Seminar on Nonlinear Partial Differential Equations (Berkeley, (1983), 85. doi: 10.1007/978-1-4612-1110-5_6.

[81]

T. Kato, Remarks on the Euler and Navier-Stokes equations in $R^2$,, in Nonlinear Functional Analysis and its Applications, (1983), 1.

[82]

J. P. Kelliher, On Kato's conditions for vanishing viscosity,, Indiana Univ. Math. J., 56 (2007), 1711. doi: 10.1512/iumj.2007.56.3080.

[83]

J. P. Kelliher, Vanishing viscosity and the accumulation of vorticity on the boundary,, Commun. Math. Sci., 6 (2008), 869. doi: 10.4310/CMS.2008.v6.n4.a4.

[84]

J. P. Kelliher, On the vanishing viscosity limit in a disk,, Math. Ann., 343 (2009), 701. doi: 10.1007/s00208-008-0287-3.

[85]

R. B. Kellogg and M. Stynes, Corner singularities and boundary layers in a simple convection-diffusion problem,, J. Differential Equations, 213 (2005), 81. doi: 10.1016/j.jde.2005.02.011.

[86]

J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods,, Applied Mathematical Sciences, (1996). doi: 10.1007/978-1-4612-3968-0.

[87]

W. Klingenberg, A Course in Differential Geometry,, Translated from the German by David Hoffman, (1978).

[88]

P. A. Lagerstrom, Matched Asymptotic Expansions. Ideas and Techniques,, Applied Mathematical Sciences, (1988). doi: 10.1007/978-1-4757-1990-1.

[89]

N. Levinson, The first boundary value problem for $\varepsilon\Delta u+A(x,y)u_x+B(x,y)u_y+C(x,y)u=D(x,y)$ for small $\varepsilon$,, Ann. of Math. (2), 51 (1950), 428.

[90]

F. Li and K. Nakashima, Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains,, Discrete Contin. Dyn. Syst., 32 (2012), 1391. doi: 10.3934/dcds.2012.32.1391.

[91]

J.-L. Lions, Perturbations Singulières Dans Les Problèmes Aux Limites et en Contrôle Optimal,, Lecture Notes in Mathematics, (1973).

[92]

P.-L. Lions, On the Hamilton-Jacobi-Bellman equations,, Acta Appl. Math., 1 (1983), 17. doi: 10.1007/BF02433840.

[93]

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.

[94]

M. C. Lopes Filho, A. L. Mazzucato and H. J. Nussenzveig Lopes, Vanishing viscosity limit for incompressible flow inside a rotating circle,, Phys. D, 237 (2008), 1324. doi: 10.1016/j.physd.2008.03.009.

[95]

M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussenzveig Lopes and M. Taylor, Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows,, Bull. Braz. Math. Soc. (N.S.), 39 (2008), 471. doi: 10.1007/s00574-008-0001-9.

[96]

M. C. Lopes Filho, Boundary layers and the vanishing viscosity limit for incompressible 2D flow,, in Lectures on the Analysis of Nonlinear Partial Differential Equations. Part 1, (2012), 1.

[97]

T. Ma and S. Wang, Boundary layer separation and structural bifurcation for 2-D incompressible fluid flows. Partial differential equations and applications,, Discrete Contin. Dyn. Syst., 10 (2004), 459. doi: 10.3934/dcds.2004.10.459.

[98]

T. Ma and S. Wang, Bifurcation Theory and Applications,, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, (2005). doi: 10.1142/9789812701152.

[99]

A. Malchiodi, Construction of multidimensional spike-layers,, Discrete Contin. Dyn. Syst., 14 (2006), 187. doi: 10.3934/dcds.2006.14.187.

[100]

N. Masmoudi, The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary,, Arch. Rational Mech. Anal., 142 (1998), 375. doi: 10.1007/s002050050097.

[101]

H. Matsuzawa, On a solution with transition layers for a bistable reaction-diffusion equation with spatially heterogeneous environments,, Discrete Contin. Dyn. Syst., (2009), 516.

[102]

A. Mazzucato, D. Niu and X. Wang, Boundary layer associated with a class of 3D nonlinear plane parallel channel flows,, Indiana Univ. Math. J., 60 (2011), 1113. doi: 10.1512/iumj.2011.60.4479.

[103]

A. Mazzucato and M. Taylor, Vanishing viscosity limits for a class of circular pipe flows,, Comm. Partial Differential Equations, 36 (2011), 328. doi: 10.1080/03605302.2010.505973.

[104]

A. L. Mazzucato, V. Nistor and Q. Qu, A nonconforming generalized finite element method for transmission problems,, SIAM J. Numer. Anal., 51 (2013), 555. doi: 10.1137/100816031.

[105]

A. L. Mazzucato, V. Nistor and Q. Qu, Quasi-optimal rates of convergence for the generalized finite element method in polygonal domains,, J. Comput. Appl. Math., 263 (2014), 466. doi: 10.1016/j.cam.2013.12.026.

[106]

N. Möes, J. Dolbow and T. Belytschko, A finite element method for crack growth without remeshing,, International Journal for Numerical Methods in Engineering, 46 (1999), 131.

[107]

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory,, Applied Mathematics and Mathematical Computation, (1999).

[108]

R. E. O'Malley, Jr., On boundary value problems for a singularly perturbed differential equation with a turning point,, SIAM J. Math. Anal., 1 (1970), 479. doi: 10.1137/0501041.

[109]

R. E. O'Malley, Jr., Introduction to Singular Perturbations,, Applied Mathematics and Mechanics, (1974).

[110]

R. E. O'Malley, Jr., Singular Perturbation Analysis for Ordinary Differential Equations,, Communications of the Mathematical Institute, (1977).

[111]

R. E. O'Malley, Jr., Singular Perturbation Methods for Ordinary Differential Equations,, Applied Mathematical Sciences, (1991). doi: 10.1007/978-1-4612-0977-5.

[112]

R. E. O'Malley, Jr., Singularly perturbed linear two-point boundary value problems,, SIAM Rev., 50 (2008), 459. doi: 10.1137/060662058.

[113]

C. H. Ou and R. Wong, Shooting method for nonlinear singularly perturbed boundary-value problems,, Stud. Appl. Math., 112 (2004), 161. doi: 10.1111/j.0022-2526.2004.01509.x.

[114]

L. Prandtl, Verber flüssigkeiten bei sehr kleiner reibung,, in Verk. III Intem. Math. Kongr. Heidelberg, (1905), 484.

[115]

L. Prandtl, Gesammelte Abhandlungen Zur Angewandten Mechanik, Hydro- und Aerodynamik,, Herausgegeben von Walter Tollmien, (1961).

[116]

J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921. doi: 10.1016/j.anihpc.2006.06.008.

[117]

W. H. Reid, Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow,, Studies in Appl. Math., 53 (1974), 91. doi: 10.1002/sapm197453291.

[118]

W. H. Reid, Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory,, Studies in Appl. Math., 53 (1974), 217. doi: 10.1002/sapm1974533217.

[119]

H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems,, Springer Series in Computational Mathematics, (1996). doi: 10.1007/978-3-662-03206-0.

[120]

L. Ruan and C. Zhu, Boundary layer for nonlinear evolution equations with damping and diffusion,, Discrete Contin. Dyn. Syst., 32 (2012), 331. doi: 10.3934/dcds.2012.32.331.

[121]

H. Schlichting, Boundary Layer Theory,, Translated by J. Kestin, (1955).

[122]

S.-D. Shih and R. B. Kellogg, Asymptotic analysis of a singular perturbation problem,, SIAM J. Math. Anal., 18 (1987), 1467. doi: 10.1137/0518107.

[123]

E. Simonnet, M. Ghil, K. Ide, R. Temam and S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation, I. Steady-state solution,, J. Phys. Oceanogr., 33 (2003), 712. doi: 10.1175/1520-0485(2003)33<712:LVISMO>2.0.CO;2.

[124]

S. Smale, Smooth solutions of the heat and wave equations,, Comment. Math. Helv., 55 (1980), 1. doi: 10.1007/BF02566671.

[125]

D. R. Smith, Singular-Perturbation Theory. An Introduction with Applications,, Cambridge University Press, (1985).

[126]

M. Stynes, Steady-state convection-diffusion problems,, Acta Numer., 14 (2005), 445. doi: 10.1017/S0962492904000261.

[127]

G. Fu Sun and M. Stynes, Finite element methods on piecewise equidistant meshes for interior turning point problems,, Numer. Algorithms, 8 (1994), 111. doi: 10.1007/BF02145699.

[128]

R. Temam, Behaviour at time $t=0$ of the solutions of semilinear evolution equations,, J. Differential Equations, 43 (1982), 73. doi: 10.1016/0022-0396(82)90075-4.

[129]

R. Temam and X. Wang, Remarks on the Prandtl equation for a permeable wall,, Special issue on the occasion of the 125th anniversary of the birth of Ludwig Prandtl, 80 (2000), 835. doi: 10.1002/1521-4001(200011)80:11/12<835::AID-ZAMM835>3.0.CO;2-9.

[130]

R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: The noncharacteristic boundary case,, J. Differential Equations, 179 (2002), 647. doi: 10.1006/jdeq.2001.4038.

[131]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Reprint of the 1984 edition, (1984).

[132]

R. Temam and X. M. Wang, Asymptotic analysis of the linearized Navier-Stokes equations in a channel,, Differential Integral Equations, 8 (1995), 1591.

[133]

R. Temam and X. Wang, Asymptotic analysis of Oseen type equations in a channel at small viscosity,, Indiana Univ. Math. J., 45 (1996), 863. doi: 10.1512/iumj.1996.45.1290.

[134]

R. Temam and X. Wang, Asymptotic analysis of the linearized Navier-Stokes equations in a general $2$D domain,, Asymptot. Anal., 14 (1997), 293.

[135]

R. Temam and X. Wang, Boundary layers for Oseen's type equation in space dimension three,, Russian J. Math. Phys., 5 (1997), 227.

[136]

N. M. Temme, Analytical methods for an elliptic singular perturbation problem in a circle,, J. Comput. Appl. Math., 207 (2007), 301. doi: 10.1016/j.cam.2006.10.049.

[137]

M. Urano, K. Nakashima and Y. Yamada, Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity,, Discrete Contin. Dyn. Syst., (2005), 868.

[138]

F. Verhulst, Methods and Applications of Singular Perturbations. Boundary Layers and Multiple Timescale Dynamics,, Texts in Applied Mathematics, (2005). doi: 10.1007/0-387-28313-7.

[139]

M. I. Višik and L. A. Ljusternik, Regular degeneration and boundary layer for linear differential equations with small parameter,, Amer. Math. Soc. Transl. (2), 20 (1962), 239.

[140]

M. I. Višik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter,, Uspehi Mat. Nauk (N.S.), 12 (1957), 3.

[141]

T. von Kármán, Progress in the statistical theory of turbulence,, J. Marine Research, 7 (1948), 252.

[142]

L. Wang and J. Wei, Solutions with interior bubble and boundary layer for an elliptic problem,, Discrete Contin. Dyn. Syst., 21 (2008), 333. doi: 10.3934/dcds.2008.21.333.

[143]

L. Wang and C. Zhao, Solutions with clustered bubbles and a boundary layer of an elliptic problem,, Discrete Contin. Dyn. Syst., 34 (2014), 2333.

[144]

W. Wasow, Linear Turning Point Theory,, Applied Mathematical Sciences, (1985). doi: 10.1007/978-1-4612-1090-0.

[145]

R. Wong and H. Yang, On a boundary-layer problem,, Stud. Appl. Math., 108 (2002), 369. doi: 10.1111/1467-9590.01430.

[146]

R. Wong and H. Yang, On an internal boundary layer problem,, J. Comput. Appl. Math., 144 (2002), 301. doi: 10.1016/S0377-0427(01)00569-6.

[147]

R. Wong and H. Yang, On the Ackerberg-O'Malley resonance,, Stud. Appl. Math., 110 (2003), 157. doi: 10.1111/1467-9590.00235.

[148]

R. Wong and Y. Zhao, A singularly perturbed boundary-value problem arising in phase transitions,, European J. Appl. Math., 17 (2006), 705. doi: 10.1017/S095679250600670X.

[149]

L. Zhang, Ph.D. Thesis, Indiana University,, in preparation, (2015).

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