Recent progresses in boundary layer theory. (English) Zbl 1343.35016
Summary: 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.
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35C20 | Asymptotic expansions of solutions to PDEs |
| 76D10 | Boundary-layer theory, separation and reattachment, higher-order effects |
| 35K05 | Heat equation |
Keywords:
curvilinear coordinates; initial layers; turning points; corner layers; ill-prepared initial data; technique of correctorsReferences:
| [1] | M. Amar, A note on boundary layer effects in periodic homogenization with Dirichlet boundary conditions,, Discrete Contin. Dynam. Systems, 6, 537 (2000) · Zbl 1019.35009 · doi:10.3934/dcds.2000.6.537 |
| [2] | I. Andronov, <em>Asymptotic and Hybrid Methods in Electromagnetics</em>,, IEE Electromagnetic Waves Series (2005) · Zbl 1131.78001 · doi:10.1049/PBEW051E |
| [3] | I. Babuška, The partition of unity method,, Internat. J. Numer. Methods Engrg., 40, 727 (1997) · Zbl 0949.65117 · doi:10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N |
| [4] | I. Babuška, Survey of meshless and generalized finite element methods: A unified approach,, Acta Numer., 12, 1 (2003) · Zbl 1048.65105 · 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, 185 (1970) · Zbl 0202.36903 |
| [6] | G. K. Batchelor, <em>An Introduction to Fluid Dynamics</em>,, paperback edition (1999) · Zbl 0958.76001 |
| [7] | A. E. Berger, A priori estimates and analysis of a numerical method for a turning point problem,, Math. Comp., 42, 465 (1984) · Zbl 0542.34050 · doi:10.1090/S0025-5718-1984-0736447-2 |
| [8] | O. Botella, <em>Numerical Solution of Navier-Stokes Singular Problem by a Chebyshev Projection Method</em>,, Ph.D. Thesis (2012) |
| [9] | Daniel Bouche, <em>Méthodes Asymptotiques en Électromagnétisme</em>,, With a preface by Robert Dautray (1994) · Zbl 0817.35110 |
| [10] | R. E. Caflisch, 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, 733 (2000) · Zbl 0951.76582 · doi:10.1002/1521-4001(200011)80:11/12<733::AID-ZAMM733>3.0.CO;2-L |
| [11] | J. R. Cannon, <em>The One-Dimensional Heat Equation</em>,, With a foreword by Felix E. Browder (1984) · Zbl 0567.35001 · doi:10.1017/CBO9781139086967 |
| [12] | M. Cannone, Well-posedness of Prandtl equations with non-compatible data,, Nonlinearity, 26, 3077 (2013) · Zbl 1396.35047 · doi:10.1088/0951-7715/26/12/3077 |
| [13] | M. Cannone, On the Prandtl boundary layer equations in presence of corner singularities,, Acta Appl. Math., 132, 139 (2014) · Zbl 1305.35112 · doi:10.1007/s10440-014-9912-1 |
| [14] | T. Chacón-Rebollo, On the existence and asymptotic stability of solutions for unsteady mixing-layer models,, Discrete Contin. Dyn. Syst., 34, 421 (2014) · Zbl 1283.35145 · doi:10.3934/dcds.2014.34.421 |
| [15] | K. W. Chang, <em>Nonlinear Singular Perturbation Phenomena: Theory and Applications</em>,, Applied Mathematical Sciences (1984) · Zbl 0559.34013 · doi:10.1007/978-1-4612-1114-3 |
| [16] | J.-Y. Chemin, <em>Mathematical Geophysics. An Introduction to Rotating Fluids and the Navier-Stokes Equations</em>,, Oxford Lecture Series in Mathematics and its Applications (2006) · Zbl 1205.86001 |
| [17] | Q. Chen, Numerical resolution near \(t=0\) of nonlinear evolution equations in the presence of corner singularities in space dimension 1,, Commun. Comput. Phys., 9, 568 (2011) · Zbl 1364.35057 · doi:10.4208/cicp.110909.160310s |
| [18] | W. Cheng, 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, 453 (2002) · Zbl 1004.76049 · doi:10.1016/S0045-7930(01)00060-3 |
| [19] | W. Cheng, New approximation algorithms for a class of partial differential equations displaying boundary layer behavior,, Cathleen Morawetz: A great mathematician, 7, 363 (2000) · Zbl 1001.65106 |
| [20] | P. G. Ciarlet, An introduction to differential geometry with application to elasticity,, With a foreword by Roger Fosdick, 78/79 (2005) · Zbl 1086.74001 · doi:10.1007/s10659-005-4738-8 |
| [21] | M. G. Crandall, Viscosity solutions of Hamilton-Jacobi equations,, Trans. Amer. Math. Soc., 277, 1 (1983) · Zbl 0599.35024 · 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, 321 (1987) · Zbl 0622.34063 · doi:10.1137/0518025 |
| [23] | A. J. DeSanti, Perturbed quasilinear Dirichlet problems with isolated turning points,, Comm. Partial Differential Equations, 12, 223 (1987) · Zbl 0628.35039 · doi:10.1080/03605308708820489 |
| [24] | B. Desjardins, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions,, J. Math. Pures Appl. (9), 78, 461 (1999) · Zbl 0992.35067 · doi:10.1016/S0021-7824(99)00032-X |
| [25] | Yihong Du, Boundary blow-up solutions with interior layers and spikes in a bistable problem,, Discrete Contin. Dyn. Syst., 19, 271 (2007) · Zbl 1142.35036 · doi:10.3934/dcds.2007.19.271 |
| [26] | Zhuoran Du, Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds,, Discrete Contin. Dyn. Syst., 33, 1407 (2013) · Zbl 1266.58007 |
| [27] | M. Van Dyke, <em>An Album of Fluid Motion</em>,, 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, 207 (2000) · Zbl 0961.35101 · doi:10.1007/s101140000034 |
| [29] | W. Eckhaus, Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type,, Arch. Rational Mech. Anal., 23, 26 (1966) · Zbl 0151.15101 · doi:10.1007/BF00281135 |
| [30] | W. Eckhaus, Boundary layers in linear elliptic singular perturbation problems,, SIAM Rev., 14, 225 (1972) · Zbl 0234.35009 · doi:10.1137/1014030 |
| [31] | S.-I. Ei, The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment,, Discrete Contin. Dyn. Syst., 26, 901 (2010) · Zbl 1191.35033 · doi:10.3934/dcds.2010.26.901 |
| [32] | N. Flyer, Accurate numerical resolution of transients in initial-boundary value problems for the heat equation,, J. Comput. Phys., 184, 526 (2003) · Zbl 1017.65079 · doi:10.1016/S0021-9991(02)00034-7 |
| [33] | N. Flyer, On the nature of initial-boundary value solutions for dispersive equations,, SIAM J. Appl. Math., 64, 546 · Zbl 1053.35017 · doi:10.1137/S0036139902415853 |
| [34] | S. Garcia, Aperiodic, chaotic lid-driven square cavity flows,, Math. Comput. Simulation, 81, 1741 (2011) · Zbl 1419.76162 · doi:10.1016/j.matcom.2011.01.011 |
| [35] | G.-M. Gie, Singular perturbation problems in a general smooth domain,, Asymptot. Anal., 62, 227 (2009) · Zbl 1163.35375 |
| [36] | G.-M. Gie, Asymptotic expansion of the Stokes solutions at small viscosity: The case of non-compatible initial data,, Commun. Math. Sci., 12, 383 (2014) · Zbl 1302.35100 · doi:10.4310/CMS.2014.v12.n2.a8 |
| [37] | G.-M. Gie, <em>Singular Perturbations and Boundary Layers</em>,, in preparation (2015) |
| [38] | G.-M. Gie, Asymptotic analysis of the Stokes problem on general bounded domains: The case of a characteristic boundary,, Appl. Anal., 89, 49 (2010) · Zbl 1191.35034 · doi:10.1080/00036810903437796 |
| [39] | G.-M. Gie, Boundary layers in smooth curvilinear domains: Parabolic problems,, Discrete Contin. Dyn. Syst., 26, 1213 (2010) · Zbl 1191.35035 · doi:10.3934/dcds.2010.26.1213 |
| [40] | G.-M. Gie, Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary,, Netw. Heterog. Media, 7, 741 (2012) · Zbl 1270.35046 · doi:10.3934/nhm.2012.7.741 |
| [41] | G.-M. Gie, Vorticity layers of the 2D Navier-Stokes equations with a slip type boundary condition,, Asymptot. Anal., 84, 17 (2013) · Zbl 1280.35090 |
| [42] | G.-M. Gie, Analysis of mixed elliptic and parabolic boundary layers with corners,, Int. J. Differ. Equ. (2013) · Zbl 1272.35094 |
| [43] | G.-M. Gie, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions,, J. Differential Equations, 253, 1862 (2012) · Zbl 1248.35144 · doi:10.1016/j.jde.2012.06.008 |
| [44] | G.-M. Gie, Vanishing viscosity limit of some symmetric flows,, preprint. |
| [45] | J. Grasman, <em>On the Birth of Boundary Layers</em>,, Mathematical Centre Tracts (1971) · Zbl 0228.35012 |
| [46] | H. P. Greenspan, <em>The Theory of Rotating Fluids</em>,, Reprint of the 1968 original (1968) · Zbl 0182.28103 |
| [47] | Y. Guo, A note on Prandtl boundary layers,, Comm. Pure Appl. Math., 64, 1416 (2011) · Zbl 1232.35126 · doi:10.1002/cpa.20377 |
| [48] | Y. Guo, Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate,, <a href= · Zbl 1403.35204 |
| [49] | E. Grenier, Boundary layers,, in Handbook of Mathematical Fluid Dynamics. Vol. III, 245 (2004) · Zbl 1221.76082 |
| [50] | E. Grenier, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems,, J. Differential Equations, 143, 110 (1998) · Zbl 0896.35078 · doi:10.1006/jdeq.1997.3364 |
| [51] | P. Grisvard, <em>Elliptic Problems in Nonsmooth Domains</em>,, Monographs and Studies in Mathematics (1985) · Zbl 0695.35060 |
| [52] | P. Grisvard, <em>Singularities in Boundary Value Problems</em>,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] (1992) · Zbl 0766.35001 |
| [53] | O. Guès, Boundary layer and long time stability for multidimensional viscous shocks,, Discrete Contin. Dyn. Syst., 11, 131 (2004) · Zbl 1108.35115 · doi:10.3934/dcds.2004.11.131 |
| [54] | M. Hamouda, Boundary layers for the 2D linearized primitive equations,, Commun. Pure Appl. Anal., 8, 335 (2009) · Zbl 1152.35423 · doi:10.3934/cpaa.2009.8.335 |
| [55] | M. Hamouda, Asymptotic analysis for the 3D primitive equations in a channel,, Discrete Contin. Dyn. Syst. Ser. S, 6, 401 (2013) · Zbl 1262.35078 |
| [56] | M. Hamouda, Some singular perturbation problems related to the Navier-Stokes equations,, in Advances in Deterministic and Stochastic Analysis, 197 (2007) · Zbl 1180.35409 · doi:10.1142/9789812770493_0011 |
| [57] | M. Hamouda, Boundary layers for the Navier-Stokes equations. The case of a characteristic boundary,, Georgian Math. J., 15, 517 (2008) · Zbl 1157.35333 |
| [58] | M. Hamouda, Very weak solutions of the Stokes problem in a convex polygon,, to appear (2015) |
| [59] | D. Han, Boundary layer for a class of nonlinear pipe flow,, J. Differential Equations, 252, 6387 (2012) · Zbl 1246.35159 · doi:10.1016/j.jde.2012.02.012 |
| [60] | H. Han, Differentiability properties of solutions of the equation \(-\epsilon^2\Delta u+ru=f(x,y)\) in a square,, SIAM J. Math. Anal., 21, 394 (1990) · Zbl 0732.35020 · doi:10.1137/0521022 |
| [61] | H. De Han, 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, 46 (1982) · Zbl 0511.65083 |
| [62] | H. D. Han, The use of enriched subspaces for singular perturbation problems,, in Proceedings of the China-France Symposium on Finite Element Methods (Beijing, 293 (1982) · Zbl 0611.65054 |
| [63] | G. H. Hardy, <em>Inequalities</em>,, Reprint of the 1952 edition (1952) |
| [64] | P. W. Hemker, <em>A Numerical Study of Stiff Two-Point Boundary Problems</em>,, Mathematisch Centrum (1977) · Zbl 0426.65043 |
| [65] | Y. Hong, Singularly perturbed reaction-diffusion equations in a circle with numerical applications,, Int. J. Comput. Math., 90, 2308 (2013) · Zbl 1284.65168 · doi:10.1080/00207160.2013.772987 |
| [66] | Y. Hong, On the numerical approximations of stiff convection-diffusion equations in a circle,, Numer. Math., 127, 291 (2014) · Zbl 1295.65112 · 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, 41 (2008) · Zbl 1155.35307 |
| [68] | C.-Y. Jung, Semi-analytical numerical methods for convection-dominated problems with turning points,, Int. J. Numer. Anal. Model., 10, 314 (2013) · Zbl 1267.65090 |
| [69] | C.-Y. Jung, Singular perturbation analysis on a homogeneous ocean circulation model,, Anal. Appl. (Singap.), 9, 275 (2011) · Zbl 1229.76091 · doi:10.1142/S0219530511001832 |
| [70] | C.-Y. Jung, Boundary layer theory for convection-diffusion equations in a circle,, Russian Math. Surveys, 69, 435 (2014) · Zbl 1307.35025 |
| [71] | C.-Y. Jung, Numerical approximation of two-dimensional convection-diffusion equations with multiple boundary layers,, Int. J. Numer. Anal. Model., 2, 367 (2005) · Zbl 1103.65115 |
| [72] | C.-Y. Jung, On parabolic boundary layers for convection-diffusion equations in a channel: analysis and numerical applications,, J. Sci. Comput., 28, 361 (2006) · Zbl 1158.76422 · doi:10.1007/s10915-006-9086-8 |
| [73] | C.-Y. Jung, Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point,, J. Math. Phys., 48 (2007) · Zbl 1144.81363 · doi:10.1063/1.2347899 |
| [74] | C.-Y. Jung, Finite volume approximation of one-dimensional stiff convection-diffusion equations,, J. Sci. Comput., 41, 384 (2009) · Zbl 1203.65159 · doi:10.1007/s10915-009-9304-2 |
| [75] | C.-Y. Jung, Interaction of boundary layers and corner singularities,, Discrete Contin. Dyn. Syst., 23, 315 (2009) · Zbl 1161.65086 · doi:10.3934/dcds.2009.23.315 |
| [76] | C.-Y. Jung, Finite volume approximation of two-dimensional stiff problems,, Int. J. Numer. Anal. Model., 7, 462 (2010) · Zbl 1194.65127 |
| [77] | C.-Y. Jung, Convection-diffusion equations in a circle: The compatible case,, J. Math. Pures Appl. (9), 96, 88 (2011) · Zbl 1228.35021 · doi:10.1016/j.matpur.2011.03.006 |
| [78] | C.-Y. Jung, Singular perturbations and boundary layer theory for convection-diffusion equations in a circle: The generic noncompatible case,, SIAM J. Math. Anal., 44, 4274 (2012) · Zbl 1261.35013 · doi:10.1137/110839515 |
| [79] | C.-Y. Jung, Singularly perturbed problems with a turning point: The non-compatible case,, Anal. Appl. (Singap.), 12, 293 (2014) · Zbl 1296.34138 · 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, 85 (1983) · Zbl 0559.35067 · 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, 1 (1983) · Zbl 0598.35093 |
| [82] | J. P. Kelliher, On Kato’s conditions for vanishing viscosity,, Indiana Univ. Math. J., 56, 1711 (2007) · Zbl 1125.76014 · 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, 869 (2008) · Zbl 1161.76012 · doi:10.4310/CMS.2008.v6.n4.a4 |
| [84] | J. P. Kelliher, On the vanishing viscosity limit in a disk,, Math. Ann., 343, 701 (2009) · Zbl 1155.76020 · doi:10.1007/s00208-008-0287-3 |
| [85] | R. B. Kellogg, Corner singularities and boundary layers in a simple convection-diffusion problem,, J. Differential Equations, 213, 81 (2005) · Zbl 1159.35309 · doi:10.1016/j.jde.2005.02.011 |
| [86] | J. Kevorkian, <em>Multiple Scale and Singular Perturbation Methods</em>,, Applied Mathematical Sciences (1996) · Zbl 0846.34001 · doi:10.1007/978-1-4612-3968-0 |
| [87] | W. Klingenberg, <em>A Course in Differential Geometry</em>,, Translated from the German by David Hoffman (1978) · Zbl 0366.53001 |
| [88] | P. A. Lagerstrom, <em>Matched Asymptotic Expansions. Ideas and Techniques</em>,, Applied Mathematical Sciences (1988) · Zbl 0666.34064 · 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, 428 (1950) · Zbl 0036.06801 |
| [90] | F. Li, Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains,, Discrete Contin. Dyn. Syst., 32, 1391 (2012) · Zbl 1250.35085 · doi:10.3934/dcds.2012.32.1391 |
| [91] | J.-L. Lions, <em>Perturbations Singulières Dans Les Problèmes Aux Limites et en Contrôle Optimal</em>,, Lecture Notes in Mathematics (1973) · Zbl 0268.49001 |
| [92] | P.-L. Lions, On the Hamilton-Jacobi-Bellman equations,, Acta Appl. Math., 1, 17 (1983) · Zbl 0594.93069 · doi:10.1007/BF02433840 |
| [93] | M. C. Lombardo, Zero viscosity limit of the Oseen equations in a channel,, SIAM J. Math. Anal., 33, 390 (2001) · Zbl 0988.35130 · doi:10.1137/S0036141000372015 |
| [94] | M. C. Lopes Filho, Vanishing viscosity limit for incompressible flow inside a rotating circle,, Phys. D, 237, 1324 (2008) · Zbl 1143.76416 · doi:10.1016/j.physd.2008.03.009 |
| [95] | M. C. Lopes Filho, Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows,, Bull. Braz. Math. Soc. (N.S.), 39, 471 (2008) · Zbl 1178.35288 · 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, 1 (2012) · Zbl 1291.35189 |
| [97] | T. Ma, Boundary layer separation and structural bifurcation for 2-D incompressible fluid flows. Partial differential equations and applications,, Discrete Contin. Dyn. Syst., 10, 459 (2004) · Zbl 1050.76017 · doi:10.3934/dcds.2004.10.459 |
| [98] | T. Ma, <em>Bifurcation Theory and Applications</em>,, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises (2005) · Zbl 1085.35001 · doi:10.1142/9789812701152 |
| [99] | A. Malchiodi, Construction of multidimensional spike-layers,, Discrete Contin. Dyn. Syst., 14, 187 (2006) · Zbl 1220.35050 · 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, 375 (1998) · Zbl 0915.76017 · 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., 516 (2009) · Zbl 1195.35151 |
| [102] | A. Mazzucato, Boundary layer associated with a class of 3D nonlinear plane parallel channel flows,, Indiana Univ. Math. J., 60, 1113 (2011) · Zbl 1426.76125 · doi:10.1512/iumj.2011.60.4479 |
| [103] | A. Mazzucato, Vanishing viscosity limits for a class of circular pipe flows,, Comm. Partial Differential Equations, 36, 328 (2011) · Zbl 1220.35121 · doi:10.1080/03605302.2010.505973 |
| [104] | A. L. Mazzucato, A nonconforming generalized finite element method for transmission problems,, SIAM J. Numer. Anal., 51, 555 (2013) · Zbl 1370.65069 · doi:10.1137/100816031 |
| [105] | A. L. Mazzucato, Quasi-optimal rates of convergence for the generalized finite element method in polygonal domains,, J. Comput. Appl. Math., 263, 466 (2014) · Zbl 1295.65115 · doi:10.1016/j.cam.2013.12.026 |
| [106] | N. Möes, A finite element method for crack growth without remeshing,, International Journal for Numerical Methods in Engineering, 46, 131 (1999) · Zbl 0955.74066 |
| [107] | O. A. Oleinik, <em>Mathematical Models in Boundary Layer Theory</em>,, Applied Mathematics and Mathematical Computation (1999) · Zbl 0928.76002 |
| [108] | R. E. O’Malley, On boundary value problems for a singularly perturbed differential equation with a turning point,, SIAM J. Math. Anal., 1, 479 (1970) · Zbl 0208.12301 · doi:10.1137/0501041 |
| [109] | R. E. O’Malley, <em>Introduction to Singular Perturbations</em>,, Applied Mathematics and Mechanics (1974) · Zbl 0287.34062 |
| [110] | R. E. O’Malley, <em>Singular Perturbation Analysis for Ordinary Differential Equations</em>,, Communications of the Mathematical Institute (1977) · Zbl 0426.34001 |
| [111] | R. E. O’Malley, <em>Singular Perturbation Methods for Ordinary Differential Equations</em>,, Applied Mathematical Sciences (1991) · Zbl 0743.34059 · doi:10.1007/978-1-4612-0977-5 |
| [112] | R. E. O’Malley, Singularly perturbed linear two-point boundary value problems,, SIAM Rev., 50, 459 (2008) · Zbl 1362.34092 · doi:10.1137/060662058 |
| [113] | C. H. Ou, Shooting method for nonlinear singularly perturbed boundary-value problems,, Stud. Appl. Math., 112, 161 (2004) · Zbl 1141.34328 · 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, 484 (1905) · JFM 36.0800.02 |
| [115] | L. Prandtl, <em>Gesammelte Abhandlungen Zur Angewandten Mechanik, Hydro- und Aerodynamik</em>,, Herausgegeben von Walter Tollmien (1961) · Zbl 0097.19701 |
| [116] | J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24, 921 (2007) · Zbl 1136.35070 · 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, 91 (1974) · Zbl 0339.76027 · 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, 217 (1974) · Zbl 0408.76020 · doi:10.1002/sapm1974533217 |
| [119] | H.-G. Roos, <em>Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems</em>,, Springer Series in Computational Mathematics (1996) · Zbl 0844.65075 · doi:10.1007/978-3-662-03206-0 |
| [120] | L. Ruan, Boundary layer for nonlinear evolution equations with damping and diffusion,, Discrete Contin. Dyn. Syst., 32, 331 (2012) · Zbl 1235.35151 · doi:10.3934/dcds.2012.32.331 |
| [121] | H. Schlichting, <em>Boundary Layer Theory</em>,, Translated by J. Kestin (1955) · Zbl 0065.18901 |
| [122] | S.-D. Shih, Asymptotic analysis of a singular perturbation problem,, SIAM J. Math. Anal., 18, 1467 (1987) · Zbl 0642.35006 · doi:10.1137/0518107 |
| [123] | E. Simonnet, Low-frequency variability in shallow-water models of the wind-driven ocean circulation, I. Steady-state solution,, J. Phys. Oceanogr., 33, 712 (2003) · 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, 1 (1980) · Zbl 0439.35017 · doi:10.1007/BF02566671 |
| [125] | D. R. Smith, <em>Singular-Perturbation Theory. An Introduction with Applications</em>,, Cambridge University Press (1985) · Zbl 0567.34055 |
| [126] | M. Stynes, Steady-state convection-diffusion problems,, Acta Numer., 14, 445 (2005) · Zbl 1115.65108 · doi:10.1017/S0962492904000261 |
| [127] | G. Fu Sun, Finite element methods on piecewise equidistant meshes for interior turning point problems,, Numer. Algorithms, 8, 111 (1994) · Zbl 0811.65059 · doi:10.1007/BF02145699 |
| [128] | R. Temam, Behaviour at time \(t=0\) of the solutions of semilinear evolution equations,, J. Differential Equations, 43, 73 (1982) · Zbl 0446.35057 · doi:10.1016/0022-0396(82)90075-4 |
| [129] | R. Temam, 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, 835 (2000) · Zbl 1050.76018 · doi:10.1002/1521-4001(200011)80:11/12<835::AID-ZAMM835>3.0.CO;2-9 |
| [130] | R. Temam, Boundary layers associated with incompressible Navier-Stokes equations: The noncharacteristic boundary case,, J. Differential Equations, 179, 647 (2002) · Zbl 0997.35042 · doi:10.1006/jdeq.2001.4038 |
| [131] | R. Temam, <em>Navier-Stokes Equations. Theory and Numerical Analysis</em>,, Reprint of the 1984 edition (1984) · Zbl 0568.35002 |
| [132] | R. Temam, Asymptotic analysis of the linearized Navier-Stokes equations in a channel,, Differential Integral Equations, 8, 1591 (1995) · Zbl 0832.35112 |
| [133] | R. Temam, Asymptotic analysis of Oseen type equations in a channel at small viscosity,, Indiana Univ. Math. J., 45, 863 (1996) · Zbl 0881.35097 · doi:10.1512/iumj.1996.45.1290 |
| [134] | R. Temam, Asymptotic analysis of the linearized Navier-Stokes equations in a general \(2\) D domain,, Asymptot. Anal., 14, 293 (1997) · Zbl 0889.35076 |
| [135] | R. Temam, Boundary layers for Oseen’s type equation in space dimension three,, Russian J. Math. Phys., 5, 227 (1997) · Zbl 0912.35125 |
| [136] | N. M. Temme, Analytical methods for an elliptic singular perturbation problem in a circle,, J. Comput. Appl. Math., 207, 301 (2007) · Zbl 1331.35117 · doi:10.1016/j.cam.2006.10.049 |
| [137] | M. Urano, Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity,, Discrete Contin. Dyn. Syst., 868 (2005) · Zbl 1195.35032 |
| [138] | F. Verhulst, <em>Methods and Applications of Singular Perturbations. Boundary Layers and Multiple Timescale Dynamics</em>,, Texts in Applied Mathematics (2005) · Zbl 1148.35006 · doi:10.1007/0-387-28313-7 |
| [139] | M. I. Višik, Regular degeneration and boundary layer for linear differential equations with small parameter,, Amer. Math. Soc. Transl. (2), 20, 239 (1962) · Zbl 0122.32402 |
| [140] | M. I. Višik, Regular degeneration and boundary layer for linear differential equations with small parameter,, Uspehi Mat. Nauk (N.S.), 12, 3 (1957) · Zbl 0087.29602 |
| [141] | T. von Kármán, Progress in the statistical theory of turbulence,, J. Marine Research, 7, 252 (1948) |
| [142] | L. Wang, Solutions with interior bubble and boundary layer for an elliptic problem,, Discrete Contin. Dyn. Syst., 21, 333 (2008) · Zbl 1160.35331 · doi:10.3934/dcds.2008.21.333 |
| [143] | L. Wang, Solutions with clustered bubbles and a boundary layer of an elliptic problem,, Discrete Contin. Dyn. Syst., 34, 2333 (2014) · Zbl 1279.35022 |
| [144] | W. Wasow, <em>Linear Turning Point Theory</em>,, Applied Mathematical Sciences (1985) · Zbl 0558.34049 · doi:10.1007/978-1-4612-1090-0 |
| [145] | R. Wong, On a boundary-layer problem,, Stud. Appl. Math., 108, 369 (2002) · Zbl 1152.34360 · doi:10.1111/1467-9590.01430 |
| [146] | R. Wong, On an internal boundary layer problem,, J. Comput. Appl. Math., 144, 301 (2002) · Zbl 1012.34049 · doi:10.1016/S0377-0427(01)00569-6 |
| [147] | R. Wong, On the Ackerberg-O’Malley resonance,, Stud. Appl. Math., 110, 157 (2003) · Zbl 1141.34331 · doi:10.1111/1467-9590.00235 |
| [148] | R. Wong, A singularly perturbed boundary-value problem arising in phase transitions,, European J. Appl. Math., 17, 705 (2006) · Zbl 1142.34013 · doi:10.1017/S095679250600670X |
| [149] | L. Zhang, Ph.D. Thesis, Indiana University,, in preparation (2015) |
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