Skip to main content
SpringerLink
Log in

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

  • Living reference work entry
  • First Online:
Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Abstract

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this chapter is to review recent progress on the mathematical analysis of this problem in each category.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  1. H. Abidi, R. Danchin, Optimal bounds for the inviscid limit of Navier-Stokes equations. Asymptot. Anal. 38, 35–46 (2004)

    MathSciNet  MATH  Google Scholar 

  2. S.N. Alekseenko, Solution of a degenerate linearized Navier-Stokes system with a homogeneous boundary condition, in Studies in Integro-Differential Equations, vol. 16 (Ilim, Frunze, 1983), pp. 243–257

    Google Scholar 

  3. S.N. Alekseenko, On vanishing viscosity in a linearized problem of the flow of an incompressible fluid. in Studies in Integro-Differential Equations, No. 19 (Russian) (Ilim, Frunze, 1986), pp. 288–304, 320

    Google Scholar 

  4. R. Alexandre, Y.-G. Wang, C.-J. Xu, T. Yang, Well-posedness of the prandtl equation in sobolev spaces. J. Am. Math. Soc. 28(3), 745–784 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. K. Asano, A note on the abstract Cauchy-Kowalewski theorem. Proc. Jpn. Acad. Ser. A Math. Sci. 64(4), 102–105 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. K. Asano, Zero-viscosity limit of the incompressible Navier-Stokes equations. II. Math. Anal. Fluid Plasma Dyn., Sūrikaisekikenkyūsho Kōkyūroku 656, 105–128 (1988)

    Google Scholar 

  7. C. Bardos, Existence et unicité de la solution de l’équation d’Euler en dimension deux. J. Math. Anal. Appl. 40, 769–790 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Bardos, M.C. Lopes Filho, D. Niu, H.J. Nussenzveig Lopes, E.S. Titi, Stability of two-dimensional viscous incompressible flows under three-dimensional perturbations and inviscid symmetry breaking. SIAM J. Math. Anal. 45(3), 1871–1885 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Beirão da Veiga, F. Crispo, Sharp inviscid limit results under Navier type boundary conditions. An L p theory. J. Math. Fluid Mech. 12(3), 397–411 2010

    Google Scholar 

  10. H. Beirão da Veiga, F. Crispo, Concerning the W k, p-inviscid limit for 3-D flows under a slip boundary condition. J. Math. Fluid Mech. 13(1), 117–135 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Beirão da Veiga, F. Crispo, The 3-D inviscid limit result under slip boundary conditions. A negative answer. J. Math. Fluid Mech. 14(1), 55–59 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Beirão da Veiga, F. Crispo, A missed persistence property for the Euler equations and its effect on inviscid limits. Nonlinearity 25(6), 1661–1669 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. H. Bellout, J. Neustupa, P. Penel, On the Navier-Stokes equation with boundary conditions based on vorticity. Math. Nachr. 269–270, 59–72 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. L.C. Berselli, S. Spirito, On the vanishing viscosity limit of 3D Navier-Stokes equations under slip boundary conditions in general domains. Commun. Math. Phys. 316(1), 171–198 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Bertozzi, P. Constantin, Global regularity for vortex patches. Commun. Math. Phys. 152(1), 19–28 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. J.L. Bona, J. Wu, The zero-viscosity limit of the 2D Navier-Stokes equations. Stud. Appl. Math. 109(4), 265–278 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Brunelli, C. Marchioro, Vanishing viscosity limit for a smoke ring with concentrated vorticity. J. Math. Fluid Mech. 13, 421–428 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. D. Bucur, E. Feireisl, Š. Nečasová, Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions. Arch. Ration. Mech. Anal. 197(1), 117–138 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. A.V. Busuioc, D. Iftimie, M.C. Lopes Filho, H.J. Nussenzveig Lopes. Incompressible Euler as a limit of complex fluid models with Navier boundary conditions. J. Differ. Equ. 252(1), 624–640 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. R.E. Caflisch, O.F. Orellana, Long time existence for a slightly perturbed vortex sheet. Commun. Pure Appl. Math. 39(6), 807–838 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. R.E. Caflisch, O.F. Orellana, Singular solutions and ill-posedness for the evolution of vortex sheets. SIAM J. Math. Anal. 20(2), 293–307 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  22. R.E. Caflisch, M. Sammartino, Vortex layers in the small viscosity limit, in WASCOM 2005-13th Conference on Waves and Stability in Continuous Media (World Scientific Publishing Company, Hackensack, 2006), pp. 59–70

    MATH  Google Scholar 

  23. M. Cannone, M.C. Lombardo, M. Sammartino, Well-posedness of Prandtl equations with non-compatible data. Nonlinearity 26(12), 3077–3100 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Cannone, M.C. Lombardo, M. Sammartino, On the Prandtl boundary layer equations in presence of corner singularities. Acta Appl. Math. 132, 139–149 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. D. Chae, P. Dubovskii, Functional and measure-valued solutions to the Euler equations for flows of incompressible fluids. Arch. Ration. Mech. Anal. 129, 385–396 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels. Ann. Sci. Ecole Norm. Sup. (4) 26(4), 517–542 (1993)

    Google Scholar 

  27. J.-Y. Chemin, A remark on the inviscid limit for two-dimensional incompressible fluids. Commun. Partial Differ. Equ. 21, 1771–1779 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. T. Clopeau, A. Mikelić, R. Robert, On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions. Nonlinearity 11(6), 1625–1636 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. P. Constantin, I. Kukavica, V. Vicol, On the inviscid limit of the Navier-Stokes equations. Proc. Am. Math. Soc. 143(7), 3075–3090 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. P. Constantin, J. Wu, Inviscid limit for vortex patches. Nonlinearity 8, 735–742 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. P. Constantin, J. Wu, The inviscid limit for non-smooth vorticity. Indiana Univ. Math. J. 45, 67–81 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. S.J. Cowley, L.M. Hocking, O.R. Tutty, The stability of solutions of the classical unsteady boundary-layer equation. Phys. Fluids 28, 441–443 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  33. R. Danchin, Poches de tourbillon visqueuses. J. Math. Pures Appl. 76, 609–647 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. R. Danchin, Persistance de structures géométriques et limite non visqueuse pour les fluides incompressibles en dimension quelconque. Bull. Soc. Math. Fr. 127, 179–227 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  35. N. Depauw, Poche de tourbillon pour Euler 2D dans un ouvert à bord. J. Math. Pures Appl. (9) 78(3), 313–351 (1999)

    Google Scholar 

  36. J. Duchon, R. Robert, Global vortex sheet solutions of Euler equations in the plane. J. Differ. Equ. 73(2), 215–224 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Dutrifoy, On 3-D vortex patches in bounded domains. Commun. Partial Differ. Equ. 28(7-8), 1237–1263 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. E. Weinan and B. Engquist, Blowup of solutions of the unsteady Prandtl’s equation. Commun. Pure Appl. Math. 50(12), 1287–1293 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Ebin, J. Marsden Groups of diffeomorphisms and the notion of an incompressible fluid. Ann. Math. 92, 102–163 (1970)

    Google Scholar 

  40. M. Fei, T. Tao, Z. Zhang, On the zero-viscosity limit of the Navier-Stokes equations in the half-space (Sep 2016). ArXiv e-prints

    Google Scholar 

  41. H. Feng, V. Šverák, On the Cauchy problem for axi-symmetric vortex rings. Arch. Ration. Mech. Anal. 215, 89–123 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. I. Gallagher, T. Gallay, Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity. Math. Ann. 332, 287–327 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. T. Gallay, Interaction of vortices in weakly viscous planar flows. Arch. Ration. Mech. Anal. 200, 445–490 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. T. Gallay, Y. Maekawa, Existence and stability of viscous vortices. in Handbook of Mathematical Analysis of Mechanics in Viscous Fluids. Springer.

    Google Scholar 

  45. T. Gallay, C.E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation. Commun. Math. Phys. 255, 97–129 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  46. P. Gamblin, X. Saint Raymond, On three-dimensional vortex patches. Bull. Soc. Math. Fr. 123(3), 375–424 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  47. F. Gargano, M. Sammartino, V. Sciacca, Singularity formation for Prandtl’s equations. Phys. D 238(19), 1975–1991 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. D. Gérard-Varet, E. Dormy. On the ill-posedness of the Prandtl equation. J. Am. Math. Soc. 23(2), 591–609 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. D. Gérard-Varet, Y. Maekawa, N. Masmoudi. Gevrey stability of Prandtl expansions for 2D Navier-Stokes flows (July 2016). ArXiv e-prints

    Google Scholar 

  50. D.Gérard-Varet, N. Masmoudi, Well-posedness for the Prandtl system without analyticity or monotonicity. Ann. Scient. Éc. Norm. Sup. 48(4), 1273–1325 (2015)

    Google Scholar 

  51. D.Gérard-Varet, T. Nguyen, Remarks on the ill-posedness of the Prandtl equation. Asymptot. Anal. 77(1–2), 71–88 (2012)

    Google Scholar 

  52. G.-M. Gie, Asymptotic expansion of the Stokes solutions at small viscosity: the case of non-compatible initial data. Commun. Math. Sci. 12(2), 383–400 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. G.-M. Gie, C.-Y. Jung, Vorticity layers of the 2D Navier-Stokes equations with a slip type boundary condition. Asymptot. Anal. 84(1–2), 17–33 (2013)

    MathSciNet  MATH  Google Scholar 

  54. G.-M. Gie, J.P. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions. J. Differ. Equ. 253(6), 1862–1892 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  55. G.-M. Gie, J.P. Kelliher, M.C. Lopes, A.L. Mazzucato, H.J. Nussenzveig Lopes, The vanishing viscosity limit for some symmetric flows 2016, preprint

    Google Scholar 

  56. Y. Giga, T. Miyakawa, Navier-Stokes flow in \(\mathbb{R}^{3}\) with measures as initial vorticity and Morrey spaces. Commun. Partial Differ. Equ. 14(5), 577–618 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  57. Y. Giga, T. Miyakawa, H. Osada. Two-dimensional Navier-Stokes flow with measures as initial vorticity. Arch. Ration. Mech. Anal. 104, 223–250 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  58. K.K. Golovkin, Vanishing viscosity in the cauchy problem for equations of hydrodynamics. Trudy Mat. Inst. Steklov. 92, 31–49 (1966)

    MathSciNet  Google Scholar 

  59. E. Grenier, On the nonlinear instability of Euler and Prandtl equations. Commun. Pure Appl. Math. 53(9), 1067–1091 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  60. E. Grenier, Y. Guo, T. Nguyen, Spectral instability of characteristic boundary layer flows (June 2014). ArXiv e-prints

    Google Scholar 

  61. Y. Guo, T. Nguyen, A note on Prandtl boundary layers. Commun. Pure Appl. Math. 64(10), 1416–1438 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  62. D. Han, A.L. Mazzucato, D. Niu, X. Wang, Boundary layer for a class of nonlinear pipe flow. J. Differ. Equ. 252(12), 6387–6413 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  63. D. Han, X. Wang, Initial-boundary layer associated with the nonlinear Darcy-Brinkman system. J. Differ. Equ. 256(2), 609–639 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  64. T. Hmidi, Régularité h’́oldérienne des poches de tourbillon visqueuses. J. Math. Pures Appl. 84, 1455–1495 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  65. T. Hmidi, Poches de tourbillon singulières dans un fluide faiblement visqueux. Rev. Mat. Iberoam. 22, 489–543 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  66. L. Hong, J. Hunter, Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations. Commun. Math. Sci. 1, 293–316 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  67. C. Huang, Remarks on regularity of non-constant vortex patches. Commun. Appl. Anal. 3(4), 449–459 (1999)

    MathSciNet  MATH  Google Scholar 

  68. C. Huang, Singular integral system approach to regularity of 3d vortex patches. Indiana Univ. Math. J. 50(1), 509–552 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  69. D. Iftimie, M.C. Lopes Filho, H.J. Nussenzveig Lopes, Incompressible flow around a small obstacle and the vanishing viscosity limit. Commun. Math. Phys. 287(1), 99–115 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  70. D. Iftimie, G. Planas, Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity 19(4), 899–918 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  71. D. Iftimie, F. Sueur, Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions. Arch. Ration. Mech. Anal. 199(1), 145–175 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  72. M. Ignatova, V. Vicol, Almost global existence for the Prandtl boundary layer equations. Arch. Ration. Mech. Anal. 220, 809–848 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  73. W. Jäger, A. Mikelić, On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equ. 170(1), 96–122 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  74. T. Kato, Nonstationary flows of viscous and ideal fluids in R 3. J. Funct. Anal. 9, 296–305 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  75. T. Kato, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary, in Seminar on Nonlinear Partial Differential Equations, Berkeley, 1983. Volume 2 of Mathematical Sciences Research Institute Publications (Springer, New York, 1984), pp. 85–98

    Google Scholar 

  76. T. Kato, The Navier-Stokes equation for an incompressible fluid in \(\mathbb{R}^{2}\) with a measure as the initial vorticity. Differ. Integral Equ. 7, 949–966 (1994)

    MathSciNet  MATH  Google Scholar 

  77. J.P. Kelliher, Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM J. Math. Anal. 38(1), 210–232 (2006, electronic)

    Google Scholar 

  78. J.P. Kelliher, On Kato’s conditions for vanishing viscosity. Indiana Univ. Math. J. 56(4), 1711–1721 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  79. J.P. Kelliher, Vanishing viscosity and the accumulation of vorticity on the boundary. Commun. Math. Sci. 6(4), 869–880 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  80. J. P. Kelliher. Observations on the vanishing viscosity limit (Sept 2014). ArXiv e-prints

    Google Scholar 

  81. J.P. Kelliher, M.C.L. Filho, H.J.N. Lopes, Vanishing viscosity limit for an expanding domain in space. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(6), 2521–2537 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  82. J.P. Kelliher, R. Temam, X. Wang, Boundary layer associated with the Darcy-Brinkman-Boussinesq model for convection in porous media. Physica D 240(7), 619–628 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  83. I. Kukavica, N. Masmoudi, V. Vicol, T.K. Wong, On the local well-posedness of the Prandtl and hydrostatic Euler equations with multiple monotonicity regions. SIAM J. Math. Anal. 46(6), 3865–3890 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  84. I. Kukavica, V. Vicol, On the local existence of analytic solutions to the Prandtl boundary layer equations. Commun. Math. Sci. 11(1), 269–292 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  85. I. Kukavica, V. Vicol, F. Wang. The van Dommelen and Shen singularity in the Prandtl equations (Dec 2015). ArXiv e-prints

    Google Scholar 

  86. C. Lacave, A.L. Mazzucato, The vanishing viscosity limit in the presence of a porous medium. Math. Ann. 365 (2016) n. 3, 1527–1557

    Article  MathSciNet  MATH  Google Scholar 

  87. E. Lauga, M. Brenner, H. Stone, Microfluidics: the no-slip boundary condition, in Springer Handbook of Experimental Fluid Mechanics, ed. by C. Tropea, A. Yarin, J. F. Foss (Springer, 2007), pp. 1219–1240

    Google Scholar 

  88. G. Lebeau, Régularité du problème de Kelvin-Helmholtz pour l’équation d’euler 2D, in Séminaire: Equations aux Dérivées Partielles, 2000-2001, Exp. No. II, Séminaire: Équations aux Dérivées Partielles (École Polytechnique, Palaiseau, 2001), p. 12

    Google Scholar 

  89. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires (Dunod; Gauthier-Villars, Paris, 1969)

    MATH  Google Scholar 

  90. J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Lecture Notes in Mathematics, vol. 323 (Springer, Berlin/New York, 1973)

    Google Scholar 

  91. P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. 1. Volume 3 of Oxford Lecture Series in Mathematics and Its Applications (The Clarendon Press/Oxford University Press, New York, 1996)

    Google Scholar 

  92. C.-J. Liu, Y.-G. Wang, Derivation of Prandtl boundary layer equations for the incompressible Navier-Stokes equations in a curved domain. Appl. Math. Lett. 34, 81–85 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  93. C.J. Liu, Y.-G. Wang, T. Yang, A well-posedness theory for the Prandtl equations in three space variables (May 2014). ArXiv e-prints

    Google Scholar 

  94. C.-J. Liu, Y.-G. Wang, T. Yang, On the ill-posedness of the Prandtl equations in three-dimensional space. Arch. Ration. Mech. Anal. 220, 83–108 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  95. M.C. Lombardo, M. Cannone, M. Sammartino, Well-posedness of the boundary layer equations. SIAM J. Math. Anal. 35(4), 987–1004 (2003, electronic)

    Google Scholar 

  96. M.C. Lombardo, M. Sammartino, Zero viscosity limit of the Oseen equations in a channel. SIAM J. Math. Anal. 33(2), 390–410 (2001, electronic)

    Google Scholar 

  97. M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussenzveig Lopes, Vanishing viscosity limit for incompressible flow inside a rotating circle. Physica D 237(10–12), 1324–1333 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  98. M.C. Lopes Filho, A.L. Mazzucato, H.J. Nussenzveig Lopes, M. Taylor, Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows. Bull. Braz. Math. Soc. (N.S.) 39(4), 471–513 (2008)

    Google Scholar 

  99. M.C. Lopes Filho, H.J. Nussenzveig Lopes, G. Planas, On the inviscid limit for two-dimensional incompressible flow with Navier friction condition. SIAM J. Math. Anal. 36(4), 1130–1141 (2005, electronic)

    Google Scholar 

  100. M.C. Lopes Filho, H.J. Nussenzveig Lopes, S. Schochet, A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution. Trans. Am. Math. Soc. 359(9), 4125–4142 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  101. M.C. Lopes Filho, H.J. Nussenzveig Lopes, E.S. Titi, A. Zang, Approximation of 2D Euler equations by the second-grade fluid equations with Dirichlet boundary conditions. J. Math. Fluid Mech. 17(2), 327–340 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  102. Y. Maekawa, On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane. Commun. Pure Appl. Math. 67(7), 1045–1128 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  103. A. Majda, Vorticity and the mathematical theory of incompressible fluid flow. Commun. Pure Appl. Math. 39, S187–S220 (1986)

    Article  MathSciNet  Google Scholar 

  104. A.J. Majda, A.L. Bertozzi, Vorticity and Incompressible Flow. Volume 27 of Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2002)

    Google Scholar 

  105. C. Marchioro, On the vanishing viscosity limit for two-dimensional Navier-Stokes equations with singular initial data. Math. Methods Appl. Sci. 12, 463–470 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  106. C. Marchioro, On the inviscid limit for a fluid with a concentrated vorticity. Commun. Math. Phys. 196, 53–65 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  107. C. Marchioro, Vanishing viscosity limit for an incompressible fluid with concentrated vorticity. J. Math. Phys. 48, 065302 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  108. C. Marchioro, M. Pulvirenti, Vortices and localization in Euler flows. Commun. Math. Phys. 154, 49–61 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  109. N. Masmoudi, Examples of singular limits in hydrodynamics, in Handbook of Differential Equations: Evolutionary Equations, Vol. III (Elsevier/North-Holland, Amsterdam, 2007), pp. 195–275

    Book  MATH  Google Scholar 

  110. N. Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system. Commun. Math. Phys. 270(3), 777–788 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  111. N. Masmoudi, F. Rousset, Uniform regularity for the Navier-Stokes equation with Navier boundary condition. Arch. Ration. Mech. Anal. 203(2), 529–575 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  112. N. Masmoudi, T. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Commun. Pure Appl. Math. 68, 1683–1741 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  113. S. Matsui, Example of zero viscosity limit for two-dimensional nonstationary Navier-Stokes flows with boundary. Jpn J. Indust. Appl. Math. 11(1), 155–170 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  114. S. Matsui, T. Shirota, On separation points of solutions to Prandtl boundary layer problem. Hokkaido Math. J. 13, 92–108 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  115. A. Mazzucato, D. Niu, X. Wang, Boundary layer associated with a class of 3D nonlinear plane parallel channel flows. Indiana Univ. Math. J. 60(4), 1113–1136 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  116. A. Mazzucato, M. Taylor, Vanishing viscosity limits for a class of circular pipe flows. Commun. Partial Differ. Equ. 36(2), 328–361 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  117. A.L. Mazzucato, M.E. Taylor, Vanishing viscosity plane parallel channel flow and related singular perturbation problems. Anal. PDE 1(1), 35–93 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  118. F.J. McGrath, Nonstationary plane flow of viscous and ideal fluids. Arch. Ration. Mech. Anal. 27, 329–348 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  119. A. Mikelić, Š. Nečasová, M. Neuss-Radu, Effective slip law for general viscous flows over an oscillating surface. Math. Methods Appl. Sci. 36(15), 2086–2100 (2013)

    MathSciNet  MATH  Google Scholar 

  120. A. Mikelić, L. Paoli, Homogenization of the inviscid incompressible fluid flow through a 2D porous medium. Proc. Am. Math. Soc. 127(7), 2019–2028 (1999)

    Article  MATH  Google Scholar 

  121. O.A. Oleĭnik, On the system of equations of the boundary layer theory. Zh. bychisl. matem. i matem. fiz. 3(3), 489–507 (1963)

    MathSciNet  Google Scholar 

  122. O.A. Oleĭnik, On the mathematical theory of boundary layer for an unsteady flow of incompressible fluid. J. Appl. Math. Mech. 30, 951–974 (1967/1966)

    Article  MathSciNet  MATH  Google Scholar 

  123. O.A. Oleĭnik, On the stability of solutions of the system of boundary layer equations for a nonstationary flow of an incompressible fluid. PMM 30(3), 417–423 (1966)

    MATH  Google Scholar 

  124. O.A. Oleĭnik, V.N. Samokhin, Mathematical Models in Boundary Layer Theory. Volume 15 of Applied Mathematics and Mathematical Computation (Chapman & Hall/CRC, Boca Raton, 1999)

    Google Scholar 

  125. M. Paddick, Stability and instability of Navier boundary layers. Differ. Integral Equ. 27, 893–930 (2014)

    MathSciNet  MATH  Google Scholar 

  126. L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verh. III Intern. Math. Kongr. Heidelberg, 1904, pp. 485–491

    Google Scholar 

  127. W.M. Rusin, On the inviscid limit for the solutions of two-dimensional incompressible Navier-Stokes equations with slip-type boundary conditions. Nonlinearity 19(6), 1349–1363 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  128. M. Sammartino, R.E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. I, II. Commun. Math. Phys. 192(2), 433–491 (1998)

    Google Scholar 

  129. H. Schlichting, K. Gersten, Boundary-Layer Theory, Enlarged edn. (Springer, Berlin, 2000). With contributions by Egon Krause and Herbert Oertel, Jr., Translated from the ninth German edition by Katherine Mayes

    Google Scholar 

  130. F. Sueur, A Kato type theorem for the inviscid limit of the Navier-Stokes equations with a moving rigid body. Commun. Math. Phys. 316(3), 783–808 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  131. F. Sueur, On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain. J. Math. Fluid Mech. 16(1), 163–178 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  132. F. Sueur, Viscous profiles of vortex patches. J. Inst. Math. Jussieu 14(1), 1–68 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  133. C. Sulem, P.-L. Sulem, C. Bardos, U. Frisch, Finite time analyticity for the two- and three-dimensional Kelvin-Helmholtz instability. Commun. Math. Phys. 80(4), 485–516 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  134. H.S.G. Swann, The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R 3. Trans. Am. Math. Soc. 157, 373–397 (1971)

    MathSciNet  MATH  Google Scholar 

  135. R. Temam, X. Wang, Asymptotic analysis of Oseen type equations in a channel at small viscosity. Indiana Univ. Math. J. 45(3), 863–916 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  136. R. Temam, X. Wang, Boundary layers for Oseen’s type equation in space dimension three. Russ. J. Math. Phys. 5(2), 227–246 (1997/1998)

    MathSciNet  MATH  Google Scholar 

  137. R. Temam, X. Wang, On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(3-4), 807–828 (1997/1998). Dedicated to Ennio De Giorgi

    Google Scholar 

  138. R. Temam, X. Wang, Boundary layers in channel flow with injection and suction. Appl. Math. Lett. 14(1), 87–91 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  139. R. Temam, X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case. J. Differ. Equ. 179(2), 647–686 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  140. L.L. Van Dommelen, S.F. Shen, The spontaneous generation of the singularity in a separating laminar boundary layer. J. Comput. Phys. 38, 125–140 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  142. M. Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Sci. Ecole Norm. Sup. 32, 769–812 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  143. M.I. Vishik, L.A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter. Uspekki Mat. Nauk 12, 3–122 (1957)

    MathSciNet  MATH  Google Scholar 

  144. X. Wang, A Kato type theorem on zero viscosity limit of Navier-Stokes flows. Indiana Univ. Math. J. 50(Special Issue), 223–241 (2000/2001). Dedicated to Professors Ciprian Foias and Roger Temam (IN, Bloomington)

    Google Scholar 

  145. Y.-G. Wang, Z. Xin, Zero-viscosity limit of the linearized compressible Navier-Stokes equations with highly oscillatory forces in the half-plane. SIAM J. Math. Anal. 37(4), 1256–1298 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  146. S. Wu, Mathematical analysis of vortex sheets. Commun. Pure Appl. Math. 59(8), 1065–1206 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  147. Y. Xiao, Z. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Commun. Pure Appl. Math. 60(7), 1027–1055 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  148. Y. Xiao, Z. Xin, J. Wu, Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition. J. Funct. Anal. 257(11), 3375–3394 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  149. X. Xie, C. Li, Vanishing viscosity limit for viscous magnetohydrodynamic equations with a slip boundary condition. Appl. Math. Sci. (Ruse) 5(41–44), 1999–2011 (2011)

    Google Scholar 

  150. Z. Xin, T. Yanagisawa, Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane. Commun. Pure Appl. Math. 52(4), 479–541 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  151. Z. Xin, L. Zhang, On the global existence of solutions to the Prandtl’s system. Adv. Math. 181, 88–133 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  152. V.I. Yudovich, Non-stationary flows of an ideal incompressible fluid. Z̆. Vyčisl. Mat. i Mat. Fiz. 3, 1032–1066 (1963)

    Google Scholar 

  153. V.I. Yudovich Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett. 2, 27–38 (1995)

    Google Scholar 

  154. P. Zhang, Z. Zhang, Long time well-posedness of Prandtl system with small and analytic initial data. J. Funct. Anal. 270, 2591–2615 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Y. Maekawa was partially supported by the Grant-in-Aid for Young Scientists (B) 25800079. A. Mazzucato was partially supported by the US National Science Foundation Grant DMS-1312727.

Author information

Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science Kyoto University, Kitashirakawa Oiwake-cho, 606-8502, Kyoto, Japan

    Yasunori Maekawa

  2. Mathematics Department, Eberly College of Science, The Pennsylvania State University, Office: 324 McAllister Building, 16802, University Park, State College, PA, USA

    Anna Mazzucato

Authors
  1. Yasunori Maekawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Mazzucato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yasunori Maekawa .

Editor information

Editors and Affiliations

  1. Tokyo, Tokyo, Japan

    Yoshikazu Giga

  2. Toulon, Var, France

    Antonin Novotny

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this entry

Cite this entry

Maekawa, Y., Mazzucato, A. (2016). The Inviscid Limit and Boundary Layers for Navier-Stokes Flows. In: Giga, Y., Novotny, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-10151-4_15-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-10151-4_15-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Cham

  • Online ISBN: 978-3-319-10151-4

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation