Advertisement

Initial value problems for the Navier-Stokes equations with neumann conditions

  • Gerd Grubb
General Qualitative Theory
Part of the Lecture Notes in Mathematics book series (LNM, volume 1530)

Keywords

Dirichlet Problem Besov Space Neumann Problem Trace Operator Stokes Operator 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B-L].
    J. Bergh and J. Löfström, “Interpolation spaces,” Springer Verlag, Berlin, New York, 1976.CrossRefzbMATHGoogle Scholar
  2. [F-L-R].
    E. B. Fabes, J. E. Lewis and N. M. Riviere, Boundary value problems for the Navier-Stokes equations, Amer. J. Math. 99 (1977), 626–668.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [Gi1].
    Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L r-spaces, Math. Zeitschr. 178 (1981), 297–329.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [Gi2].
    —, Domains of fractional powers of the Stokes operator in L r spaces, Arch. Rat. Mech. Anal. 89 (1985), 251–265.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Gi3].
    —, The nonstationary Navier-Stokes system with some first order boundary conditions, Proc. Jap. Acad. 58 (1982), 101–104.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Gi4].
    —, The Navier-Stokes initial value problem in L p and related problems, “Nonlinear partial differential equations in applied sciences (Tokyo 1982),” North Holland Math. Studies 81, Amsterdam-New York, 1983, pp. 37–54.Google Scholar
  7. [Gi5].
    —, Weak and strong solutions of the Navier-Stokes initial value problem, Publ. RIMS Kyoto Univ. 19 (1983), 887–910.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [G-M].
    Y. Giga and T. Miyakawa, Solutions in L r of the Navier-Stokes initial value problem, Arch. Rat. Mech. Anal. 89 (1985), 267–281.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Gri].
    P. Grisvard, Equations différentielles abstraites, Ann. Ecole Norm. Sup. 2 (Série 4) (1969), 311–395.MathSciNetzbMATHGoogle Scholar
  10. [G1].
    G. Grubb, “Functional Calculus of Pseudo-Differential Boundary Problems,” Progress in Math. Vol. 65, Birkhäuser, Boston, 1986.CrossRefzbMATHGoogle Scholar
  11. [G2].
    —, Pseudo-differential boundary problems in L p spaces, Comm. P. D. E. 15 (1990), 289–340.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [G3].
    —, Parabolic pseudodifferential boundary problems in anisotropic L p spaces, with applications to Navier-Stokes problems, preprint, August 1991.Google Scholar
  13. [G4].
    —, Solution dans les espaces de Sobolev L p anisotropes des problèmes aux limites pseudo-différentiels paraboliques et des problèmes de Stokes, C. R. Acad. Sci. Paris 312, Série I (1991), 89–92.MathSciNetzbMATHGoogle Scholar
  14. [G-K].
    G. Grubb and N. J. Kokholm, Parameter-dependent pseudodifferential boundary problems in global L p Sobolev spaces, preprint, August 1991.Google Scholar
  15. [G-S1].
    G. Grubb and V. A. Solonnikov, Reduction of basic initial-boundary value problems for the Stokes equation to initial-boundary value problems for systems of pseudodifferential equations, Zap. Nauchn. Sem. L.O.M.I. 163 (1987), 37–48;=J. Soviet Math. 49 (1990), 1140–1147.MathSciNetzbMATHGoogle Scholar
  16. [G-S2].
    —, Solution of parabolic pseudo-differential initial-boundary value problems, J. Diff. Equ. 87 (1990), 256–304.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [G-S3].
    —, Reduction of basic initial-boundary value problems for the Navier-Stokes equations to nonlinear parabolic systems of pseudodifferential equations, Zap. Nauchn. Sem. L.O.M.I. 171 (1989), 36–52; English transl. available as Copenh. Univ. Math. Dept. Report Ser. 1989 no. 5.zbMATHGoogle Scholar
  18. [G-S4].
    —, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods, Math. Scand. 69 (1991).Google Scholar
  19. [H-W].
    J. Heywood and O. Walsh, A counter example concerning the pressure in the Navier-Stokes equations, as t → 0+, preprint, August 1991 (presented at the Oberwolfach meeting on Navier-Stokes equations).Google Scholar
  20. [Se].
    R. T. Seeley, Interpolation in L p with boundary conditions, Studia Math. 44 (1972), 47–60.MathSciNetzbMATHGoogle Scholar
  21. [S-W].
    H. Sohr and W. von Wahl, On the singular set and the uniqueness of weak solutions of the Navier-Stokes equations, Manuscripta Math. 49 (1984), 27–59.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [S].
    V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes systems, Zap. Nauchn. Sem. LOMI 38 (1973), 153–231;=J. Soviet Math. 8 (1977), 467–529.MathSciNetzbMATHGoogle Scholar
  23. [T].
    H. Triebel, “Interpolation theory, function spaces, differential operators,” North-Holland Publ. Co., Amsterdam, New York, 1978.zbMATHGoogle Scholar
  24. [W].
    W. von Wahl, Regularity questions for the Navier-Stokes equations, “Approximation methods for Navier-Stokes problems,” Lecture Note 771, Springer Verlag, Berlin, Heidelberg, 1980.Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Gerd Grubb
    • 1
  1. 1.Mathematics DepartmentUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations