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.


Vector Field Weak Solution Euler Equation Incompressible Fluid Vortex Tube 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BaC]
    H. Bahouri and J. Chemin, Equations de transport relatives a des champs de vecteurs non-lipsehitziens et mécanique des fluides, Arch. Rat. Mech. Anal. 127(1994), 159–181.MathSciNetMATHCrossRefGoogle 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.MathSciNetCrossRefGoogle 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.MathSciNetMATHCrossRefGoogle Scholar
  4. [Bat]
    G. Batchelor, An Introduction to Fluid Dynamics, Cambridge Univ. 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.MathSciNetMATHCrossRefGoogle Scholar
  7. [BeC]
    A. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math.Phys. 152(1993), 19–28.MathSciNetMATHCrossRefGoogle Scholar
  8. [Bon]
    V. Bondarevsky, On the global regularity problem for 3-dimensional Navier-Stokes equations, Preprint, 1995.Google Scholar
  9. [BBr]
    J. Bourguignon and H. Brezis, Remarks on the Euler equations, J. Func. Anal. 15(1974), 341–363.MathSciNetMATHCrossRefGoogle Scholar
  10. [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.MathSciNetMATHCrossRefGoogle Scholar
  11. [Cha]
    D. Chae, Weak solutions of the 2-D Euler equations with initial vorticity in L(log L), J. Diff. Eqs. 103(1993), 323–337.MathSciNetMATHCrossRefGoogle Scholar
  12. [Chel]
    J. Chemin, Remarques sur l’existence globale pour le système de Navier-Stokes incompressible, Preprint, 1990.Google Scholar
  13. [Che2]
    J. Chemin, Persistence des structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Ecole Norm Sup. Paris 26(1993), 517–542.MathSciNetMATHGoogle Scholar
  14. [Che3]
    J. Chemin, Une facette mathématique de la mécanique des fluides, I, Publ. CNRS #1055, Paris, 1993.Google Scholar
  15. [ChL]
    J. Chemin and N. Lerner, Flot de champs de vecteurs non-lipschitziens et équations de Navier-Stokes, Publ. CNRS #1062, 1993.Google Scholar
  16. [Cho]
    A. Chorin, Voracity and Turbulence, Springer-Verlag, New York, 1994.Google Scholar
  17. [ChM]
    A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, New York, 1979.MATHCrossRefGoogle Scholar
  18. [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.MathSciNetMATHCrossRefGoogle Scholar
  19. [CFo]
    P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Math., Univ. of Chicago Press, 1988.MATHGoogle Scholar
  20. [CLM]
    P. Constantin, P. Lax, and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, CPAM 38(1985), 715–724.MathSciNetMATHGoogle Scholar
  21. [Del]
    J. Delort, Existence de nappes de tourbillon en dimension deux, J. AMS 4(1991), 553–586.MathSciNetMATHGoogle Scholar
  22. [DW]
    P. Deuring and W. von Wahl, Strong solutions of the Navier-Stokes system in Lipschitz bounded domains, Math. Nachr. 171(1995), 111–198.MathSciNetMATHCrossRefGoogle Scholar
  23. [DM]
    R. DiPerna and A. Majda, Concentration in regularisations for 2-D incompressible flow, CPAM 40(1987), 301–345.MathSciNetMATHGoogle Scholar
  24. [Eb]
    D. Ebin, A concise presentation of the Euler equations of hydrodynamics, Comm. PDE 9(1984), 539–559.MathSciNetMATHCrossRefGoogle Scholar
  25. [EbM]
    D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Math. 92(1970), 102–163.MathSciNetMATHCrossRefGoogle Scholar
  26. [EM]
    L. Evans and S. Müller, Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity, J. AMS 7(1994), 199–219.MATHGoogle Scholar
  27. [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.MathSciNetMATHCrossRefGoogle Scholar
  28. [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.MathSciNetMATHCrossRefGoogle Scholar
  29. [Fed]
    P. Federbush, Navier and Stokes meet the wavelet, Comm. Math. Phys. 155(1993), 219–248.MathSciNetMATHCrossRefGoogle Scholar
  30. [Fer]
    A. Ferrari, On the blow-up of the 3-D Euler equation in a bounded domain, Comm. Math. Phys. 155(1993), 277–294.MathSciNetMATHCrossRefGoogle Scholar
  31. [FGT]
    C. Foias, C. Guillope, and R. Temam, Lagrangian representation of a flow, J. Diff. Eqs. 57(1985), 440–449.MathSciNetMATHCrossRefGoogle Scholar
  32. [FT]
    C. Foias and R. Temam, Some analytical and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures etAppl. 58(1979), 339–368.MathSciNetMATHGoogle Scholar
  33. [FK]
    H. Fujita and T. Kato, On the Navier-Stokes initial value problem, Arch. Rat. Mech.Anal. 16(1964), 269–315.MathSciNetMATHGoogle Scholar
  34. [FM]
    H. Fujita and H. Morimoto, On fractional powers of the Stokes operator, Proc. Japan Acad. 16(1970), 1141–1143.MathSciNetCrossRefGoogle Scholar
  35. [GM1]
    Y. Giga and T. Miyakawa, Solutions in Lr of the Navier-Stokes initial value problem, Arch. Rat. Mech. Anal. 89(1985), 267–281.MathSciNetMATHCrossRefGoogle Scholar
  36. [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.MathSciNetMATHGoogle Scholar
  37. [Hei]
    H. Heimholte, On the integrals of the hydrodynamical equations that express vortex motion, Phil. Mag. 33(1887), 485–512.Google Scholar
  38. [Hop]
    E. Hopf, Über die Anfangwertaufgabe fur die hydrodynamischen Grundgleichungen, Math. Nachr. 4(1951), 213–231.MathSciNetMATHCrossRefGoogle Scholar
  39. [HM]
    T. Hughes and J. Marsden, A Short Course in Fluid Mechanics, Publish or Perish Press, Boston, 1976.MATHGoogle Scholar
  40. [Ktl]
    T. Kato, On classical solutions of two dimensional nonstationary Euler equations, Arch. Rat. Mech. Anal. 25(1967), 188–200.MATHCrossRefGoogle Scholar
  41. [Kt2]
    T. Kato, Nonstationary flows of viscous and ideal fluids in ℝ3, J. Func. Anal. 9(1972), 296–305.MATHCrossRefGoogle Scholar
  42. [Kt3]
    T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Springer LNM 448(1974), 25–70.Google Scholar
  43. [Kt4]
    T. Kato, Strong L p-solutions to the Navier-Stokes equations in R m, with applications to weak solutions, Math. Zeit. 187(1984), 471–480.MATHCrossRefGoogle Scholar
  44. [Kt5]
    T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Bras. Mat. 22(1992), 127–155.MATHCrossRefGoogle Scholar
  45. [Kt6]
    T. Kato, The Navier-Stokes equation for an incompressible fluid in R 2 with a measure as the initial vorticity, Preprint, 1993.Google Scholar
  46. [KL]
    T. Kato and C. Lai, Nonlinear evolution equations and the Euler flow, J. Func.Anal. 56(1984), 15–28.MathSciNetMATHCrossRefGoogle Scholar
  47. [KP]
    T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, CPAM 41(1988), 891–907.MathSciNetMATHGoogle Scholar
  48. [Lad]
    O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.MATHGoogle Scholar
  49. [Lam]
    H. Lamb, Hydrodynamics, Dover, New York, 1932.MATHGoogle Scholar
  50. [Ler]
    J. Leray, Etude de diverses équations integrales non linéaires et de quelques problèmes que pose d’hydrodynamique, J. Math. Pures etAppl. 12(1933), 1–82.MATHGoogle Scholar
  51. [Li]
    J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites NonLinéaires, Dunod, Paris, 1969.MATHGoogle Scholar
  52. [Mj]
    A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sci. #53, Springer-Verlag, 1984.MATHCrossRefGoogle Scholar
  53. [Mj2]
    A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, CPAM 38(1986), 187–220.Google Scholar
  54. [Mj3]
    A. Majda, Mathematical fluid dynamics: The interaction of nonlinear analysis and modern applied mathematics, Proc. AMS Centennial Symp. (1988), 351–394.Google Scholar
  55. [Mj4]
    A. Majda, Vorticity, turbulence, and acoustics in fluid flow, SIAMReview 33(1991), 349–388.MathSciNetMATHGoogle Scholar
  56. [Mj5]
    A. Majda, Remarks on weak solutions for vortex sheets with a distinguished sign, Indiana Math. J. 42(1993), 921–939.MathSciNetMATHCrossRefGoogle Scholar
  57. [MP]
    C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonvis-cous Fluids, Springer-Verlag, New York, 1994.CrossRefGoogle Scholar
  58. [MF]
    R. von Mises and K. O. Friedrichs, Fluid Dynamics, Appl. Math. Sci. 5, Springer-Verlag, New York, 1971.MATHCrossRefGoogle Scholar
  59. [OO]
    H. Ockendon and J. Ockendon, Viscous Flow, Cambridge Univ. Press, Cambridge, 1995.MATHCrossRefGoogle Scholar
  60. [OT]
    H. Ockendon and A. Tayler, Inviscid Fluid Flow, Appl. Math. Sci. #43, Springer-Verlag, New York, 1983.Google Scholar
  61. [Safl]
    P. Saffman, Vortex Dynamics, Cambridge Univ. Press, Cambridge, 1992.MATHGoogle Scholar
  62. [Se1]
    J. Serrin, Mathematical principles of classical fluid dynamics, Encycl. of Physics, Vol. 8, pt. 1, pp. 125–263, Springer-Verlag, New York, 1959.Google Scholar
  63. [Se2]
    J. Serrin, The initial value problem for the Navier-Stokes equations, in Non-linear Problems, R.E. Langer, ed., Univ. of Wise. Press, Madison, Wise, 1963, pp. 69–98.Google Scholar
  64. [So11]
    V. Solonnikov, On estimates of the tensor Green’s function for some boundary-value problems, Dokl. Akad. Nauk SSSR 130(1960), 988–991.MathSciNetGoogle Scholar
  65. [Sol2]
    V. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, 7. Soviet Math. 8(1977), 467–529.MATHCrossRefGoogle Scholar
  66. [T1]
    M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.MATHCrossRefGoogle Scholar
  67. [T2]
    M. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. PDE 17(1992), 1407–1456.MATHCrossRefGoogle Scholar
  68. [Tern]
    R. Temam, Navier-Stokes Equations, North-Holland, New York, 1977.MATHGoogle Scholar
  69. [Tem2]
    R. Temam, On the Euler equations of incompressible perfect fluids, J. Func. Anal. 20(1975), 32–43.MathSciNetMATHCrossRefGoogle Scholar
  70. [VD]
    M. Van Dyke, An Album of Fluid Motion, Parabolic Press, Stanford, Calif., 1982.Google Scholar
  71. [vWa]
    W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations, Vieweg & Sohn, Braunschweig, 1985.Google Scholar
  72. [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.MathSciNetCrossRefGoogle Scholar
  73. [Yud]
    V. Yudovich, Non-stationary flow of an ideal incompressible fluid, J. Math, and Math. Phys. 3(1963), 1032–1066.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Michael E. Taylor
    • 1
  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

Personalised recommendations