Euler and Navier—Stokes Equations for Incompressible Fluids

  • Michael E. Taylor
Part of the Applied Mathematical Sciences book series (AMS, volume 117)


This chapter deals with equations describing motion of an incompressible fluid moving in a fixed compact space M, which it fills completely. We consider two types of fluid motion, with or without viscosity, and two types of compact space, a compact smooth Riemannian manifold with or without boundary. The two types of fluid motion are modeled by the Euler equation
$$\frac{\partial u} {\partial t} + {\nabla }_{u}u = -\text{ grad }p,\qquad \text{ div}u = 0,$$
for the velocity field u, in the absence of viscosity, and the Navier–Stokes equation
$$\frac{\partial u} {\partial t} + {\nabla }_{u}u = \nu \mathcal{L}u -\text{ grad }p,\qquad \text{ div }u = 0,$$
in the presence of viscosity. In (0.2), ν is a positive constant and \(\mathcal{L}\) is the second-order differential operator
$$\mathcal{L}u = \text{ div Def }u,$$
which on flat Euclidean space is equal to Δu, when div u = 0. If there is a boundary, the Euler equation has boundary condition nu = 0, that is, u is tangent to the boundary, while for the Navier–Stokes equation one poses the no-slip boundary condition u = 0 on ∂M.


Vector Field Stokes Equation Euler Equation Vortex Tube Compact Riemannian Manifold 
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  1. [BaC].
    H. Bahouri and J. Chemin, Equations de transport relatives a des champs de vecteurs non-lipschitziens et mecanique des fluides, Arch. Rat. Mech. Anal. 127(1994), 159–181.Google Scholar
  2. [BM].
    J. Ball and D. Marcus, Vorticity intensification and transition to turbulence in the three-dimensional Euler equations, Comm. Math. Phys. 147(1992), 371–394.Google Scholar
  3. [Bar].
    C. Bardos, Existence et unicité de la solution de l’équation d’Euler en dimension deux, J. Math. Anal. Appl. 40(1972), 769–790.Google Scholar
  4. [Bat].
    G. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967.MATHGoogle Scholar
  5. [BG].
    J. T. Beale and C. Greengard, Convergence of Euler–Stokes splitting of the Navier–Stokes equations, Preprint, 1992.Google Scholar
  6. [BKM].
    J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3-d Euler equations, Comm. Math. Phys. 94(1984), 61–66.Google Scholar
  7. [BeC].
    A. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys. 152(1993), 19–28.Google Scholar
  8. [BW].
    J. Bona and J. Wu, The zero-viscosity limit of the 2D Navier–Stokes equations, Stud. Appl. Math. 109 (2002), 265–278.MathSciNetMATHCrossRefGoogle Scholar
  9. [Bon].
    V. Bondarevsky, On the global regularity problem for 3-dimensional Navier–Stokes equations, Preprint, 1995.Google Scholar
  10. [BBr].
    J. Bourguignon and H. Brezis, Remarks on the Euler equations, J. Func. Anal. 15(1974), 341–363.Google Scholar
  11. [CKN].
    L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35(1982), 771–831.Google Scholar
  12. [Cha].
    D. Chae, Weak solutions of the 2-D Euler equations with initial vorticity in L(log L), J. Diff. Equ. 103(1993), 323–337.Google Scholar
  13. [Che1].
    J. Chemin, Remarques sur l’éxistence globale pour le système de Navier–Stokes incompressible, SIAM J. Math. Anal. 23 (1992), 20–28.MathSciNetMATHCrossRefGoogle Scholar
  14. [Che2].
    J. Chemin, Persistence des structures géometriques dans les fluides incompressibles bidimensionnels, Ann. Ecole Norm Sup. Paris 26(1993), 517–542.Google Scholar
  15. [Che3].
    J. Chemin, Fluides Parfaits Incompressibles, Asterisque #230, Société Math. de France, 1995.Google Scholar
  16. [ChL].
    J. Chemin and N. Lerner, Flot de champs de vecteurs non-lipschitziens et équations de Navier–Stokes, Publ. CNRS#1062, 1993.Google Scholar
  17. [Cho].
    A. Chorin, Vorticity and Turbulence, Springer, New York, 1994.MATHGoogle Scholar
  18. [ChM].
    A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer, New York, 1979.MATHCrossRefGoogle Scholar
  19. [CFe].
    P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier–Stokes equations, Indiana Math. J. 42(1993), 775–790.Google Scholar
  20. [CFo].
    P. Constantin and C. Foias, Navier–Stokes Equations, Chicago Lectures in Math., University of Chicago Press, 1988.MATHGoogle Scholar
  21. [CLM].
    P. Constantin, P. Lax, and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, CPAM 38(1985), 715–724.Google Scholar
  22. [Del].
    J. Delort, Existence de nappes de tourbillon en dimension deux, J. AMS 4(1991), 553–586.Google Scholar
  23. [DW].
    P. Deuring and W. von Wahl, Strong solutions of the Navier–Stokes system in Lipschitz bounded domains, Math. Nachr. 171(1995), 111–198.Google Scholar
  24. [DL].
    R. Diperna and P.-L. Lions, Ordinary differential equations, transport theory, and Sobolev spaces, Invent. Math. 98 (1989), 511–547.MathSciNetMATHCrossRefGoogle Scholar
  25. [DM].
    R. DiPerna and A. Majda, Concentration in regularizations for 2-D incompressible flow, CPAM 40(1987), 301–345.Google Scholar
  26. [Eb].
    D. Ebin, A concise presentation of the Euler equations of hydrodynamics, Comm. PDE 9(1984), 539–559.Google Scholar
  27. [EbM].
    D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math. 92(1970), 102–163.Google Scholar
  28. [EM].
    L. Evans and S. Müller, Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity, J. AMS 7(1994), 199–219.Google Scholar
  29. [FJR].
    E. Fabes, B. F. Jones, and N. Riviere, The initial boundary value problem for the Navier–Stokes equation with data in L p, Arch. Rat. Mech. Anal. 45(1972), 222–240.Google Scholar
  30. [FKV].
    E. Fabes, C. Kenig, and G. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 57(1988), 769–793.Google Scholar
  31. [Fed].
    P. Federbush, Navier and Stokes meet the wavelet, Comm. Math. Phys. 155(1993), 219–248.Google Scholar
  32. [Fer].
    A. Ferrari, On the blow-up of the 3-D Euler equation in a bounded domain, Comm. Math. Phys. 155(1993), 277–294.Google Scholar
  33. [FGT].
    C. Foias, C. Guillope, and R. Temam, Lagrangian representation of a flow, J. Diff. Equ. 57(1985), 440–449.Google Scholar
  34. [FT].
    C. Foias and R. Temam, Some analytical and geometric properties of the solutions of the evolution Navier–Stokes equations, J. Math. Pures et Appl. 58(1979), 339–368.Google Scholar
  35. [FK].
    H. Fujita and T. Kato, On the Navier–Stokes initial value problem, Arch. Rat. Mech. Anal. 16(1964), 269–315.Google Scholar
  36. [FM].
    H. Fujita and H. Morimoto, On fractional powers of the Stokes operator, Proc. Jpn. Acad. 16(1970), 1141–1143.Google Scholar
  37. [GM1].
    Y. Giga and T. Miyakawa, Solutions in L r of the Navier–Stokes initial value problem, Arch. Rat. Mech. Anal. 89(1985), 267–281.Google Scholar
  38. [GS].
    G. Grubb and V. Solonnikov, Boundary value problems for the nonstationary Navier–Stokes equations treated by pseudo-differential methods, Math. Scand. 69(1991), 217–290.Google Scholar
  39. [Hel].
    H. Helmholtz, On the integrals of the hydrodynamical equations that express vortex motion, Phil. Mag. 33(1887), 485–512.Google Scholar
  40. [Hop].
    E. Hopf, Uber die Anfangwertaufgabe fur die hydrodynamischen Grundgleichungen, Math. Nachr. 4(1951), 213–231.Google Scholar
  41. [HM].
    T. Hughes and J. Marsden, A Short Course in Fluid Mechanics, Publish or Perish Press, Boston, 1976.MATHGoogle Scholar
  42. [Kt1].
    T. Kato, On classical solutions of two dimensional nonstationary Euler equations, Arch. Rat. Mech. Anal. 25(1967), 188–200.Google Scholar
  43. [Kt2].
    T. Kato, Nonstationary flows of viscous and ideal fluids in \({\mathbb{R}}^{3}\), J. Funct. Anal. 9(1972), 296–305.Google Scholar
  44. [Kt3].
    T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Springer LNM 448(1974), 25–70.Google Scholar
  45. [Kt4].
    T. Kato, Strong L p-solutions to the Navier–Stokes equations in \({\mathbb{R}}^{m}\), with applications to weak solutions, Math. Zeit. 187(1984), 471–480.Google Scholar
  46. [Kt5].
    T. Kato, Strong solutions of the Navier–Stokes equation in Morrey spaces, Bol. Soc. Bras. Mat. 22(1992), 127–155.Google Scholar
  47. [Kt6].
    T. Kato, The Navier–Stokes equation for an incompressible fluid in \({\mathbb{R}}^{2}\) with a measure as the initial vorticity, Diff. Integr. Equ. 7 (1994), 949–966.MATHGoogle Scholar
  48. [Kt7].
    T. Kato, Remarks on the zero viscosity limit for nonstationary Navier–Stokes flows with boundary, Seminar on PDE (S. S. Chern, ed.), Springer, New York, 1984.Google Scholar
  49. [KL].
    T. Kato and C. Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal. 56(1984), 15–28.Google Scholar
  50. [KP].
    T. Kato and G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, CPAM 41(1988), 891–907.Google Scholar
  51. [Kel].
    J. Kelliher, Vanishing viscosity and the accumulation of vorticity on the boundary, Commun. Math. Sci. 6 (2008), 869–880.MathSciNetMATHGoogle Scholar
  52. [Lad].
    O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.MATHGoogle Scholar
  53. [Lam].
    H. Lamb, Hydrodynamics, Dover, New York, 1932.MATHGoogle Scholar
  54. [Ler].
    J. Leray, Etude de diverses équations integrales non linéaires et de quelques problèmes que pose d’hydrodynamique, J. Math. Pures et Appl. 12(1933), 1–82.Google Scholar
  55. [Li].
    J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Lineaires, Dunod, Paris, 1969.MATHGoogle Scholar
  56. [LMN].
    M. Lopes Filho, A. Mazzucato, and H. Nessenzveig Lopes, Vanishing viscosity limits for incompressible flow inside rotating circles, Phys. D. Nonlin. Phenom. 237 (2008), 1324–1333.MATHCrossRefGoogle Scholar
  57. [LMNT].
    M. Lopes Filho, A. Mazzucato, H. Nussenzveig Lopes, and M. Taylor, Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows, Bull. Braz. Math. Soc. 39 (2008), 471–513.MathSciNetMATHCrossRefGoogle Scholar
  58. [Mj].
    A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sci. #53, Springer, New York, 1984.Google Scholar
  59. [Mj2].
    A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, CPAM 38(1986), 187–220.Google Scholar
  60. [Mj3].
    A. Majda, Mathematical fluid dynamics: The interaction of nonlinear analysis and modern applied mathematics, Proc. AMS Centennial Symp. (1988), 351–394.Google Scholar
  61. [Mj4].
    A. Majda, Vorticity, turbulence, and acoustics in fluid flow, SIAM Rev. 33(1991), 349–388.Google Scholar
  62. [Mj5].
    A. Majda, Remarks on weak solutions for vortex sheets with a distinguished sign, Indiana Math. J. 42(1993), 921–939.Google Scholar
  63. [MP].
    C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Springer, New York, 1994.MATHCrossRefGoogle Scholar
  64. [Mat].
    S. Matsui, Example of zero viscosity limit for two-dimensional nonstationary Navier–Stokes flow with boundary, Jpn. J. Indust. Appl. Math. 11 (1994), 155–170.MathSciNetMATHCrossRefGoogle Scholar
  65. [MT1].
    A. Mazzucato and M. Taylor, Vanishing viscosity plane parallel channel flow and related singular perturbation problems, Anal. PDE 1 (2008), 35–93.MathSciNetMATHCrossRefGoogle Scholar
  66. [MT2].
    A. Mazzucato and M. Taylor, Vanishing viscosity limits for a class of circular pipe flows, Preprint, 2010.Google Scholar
  67. [MF].
    R. von Mises and K. O. Friedrichs, Fluid Dynamics, Appl. Math. Sci. 5, Springer, New York, 1971.Google Scholar
  68. [MiT].
    M. Mitrea and M. Taylor, Navier–Stokes equations on Lipschitz domains in Riemannian manifolds, Math. Ann. 321 (2001), 955–987.MathSciNetMATHCrossRefGoogle Scholar
  69. [Mon].
    S. Monniaux, Navier–Stokes equations in arbitrary domains: the Fujita–Kato scheme, Math. Res. Lett. 13 (2006), 455–461.MathSciNetMATHGoogle Scholar
  70. [OO].
    H. Ockendon and J. Ockendon, Viscous Flow, Cambridge University Press, Cambridge, 1995.MATHCrossRefGoogle Scholar
  71. [OT].
    H. Ockendon and A. Tayler, Inviscid Fluid Flow, Appl. Math. Sci. #43, Springer, New York, 1983.Google Scholar
  72. [PT].
    L. Prandtl and O. Tietjens, Applied Hydro- and Aerodynamics, Dover, New York, 1934.Google Scholar
  73. [Saf].
    P. Saffman, Vortex Dynamics, Cambridge University Press, Cambridge, 1992.MATHGoogle Scholar
  74. [Sch].
    H. Schlichting, Boundary Layer Theory, 8th ed., Springer, New York, 2000.MATHGoogle Scholar
  75. [Se1].
    J. Serrin, Mathematical principles of classical fluid dynamics, Encycl. of Physics, Vol. 8, pt. 1, pp. 125–263, Springer, New York, 1959.Google Scholar
  76. [Se2].
    J. Serrin, The initial value problem for the Navier–Stokes equations, in Non-linear Problems, R.E.Langer, ed., University of Wisc. Press, Madison, Wisc., 1963, pp. 69–98.Google Scholar
  77. [Sol1].
    V. Solonnikov, On estimates of the tensor Green’s function for some boundary-value problems, Dokl. Akad. Nauk SSSR 130(1960), 988–991.Google Scholar
  78. [Sol2].
    V. Solonnikov, Estimates for solutions of nonstationary Navier–Stokes equations, J. Sov. Math. 8(1977), 467–529.Google Scholar
  79. [T1].
    M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.MATHCrossRefGoogle Scholar
  80. [T2].
    M. Taylor, Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations, Comm. PDE 17(1992), 1407–1456.Google Scholar
  81. [Tem].
    R. Temam, Navier–Stokes Equations, North-Holland, New York, 1977.MATHGoogle Scholar
  82. [Tem2].
    R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal. 20(1975), 32–43.Google Scholar
  83. [Tem3].
    R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, Penn., 1983.Google Scholar
  84. [VD].
    M. Van Dyke, An Album of Fluid Motion, Parabolic, Stanford, Calif., 1982.Google Scholar
  85. [vWa].
    W. von Wahl, The Equations of Navier–Stokes and Abstract Parabolic Equations, Vieweg & Sohn, Braunschweig, 1985.Google Scholar
  86. [W].
    X. Wang, A Kato type theorem on zero viscosity limit of Navier–Stokes flows, Indiana Univ. Math. J. 50 (2001), 223–241.MathSciNetMATHCrossRefGoogle Scholar
  87. [Wol].
    W. Wolibner, Un théorème d’existence du mouvement plan d’un fluide parfait, homogene, incompressible, pendant un temps infiniment long, Math. Zeit. 37(1933), 698–726.Google Scholar
  88. [Yud].
    V. Yudovich, Non-stationary flow of an ideal incompressible fluid, J. Math. Math. Phys. 3(1963), 1032–1066.Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

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