manuscripta mathematica

, Volume 69, Issue 1, pp 237–254

On interior regularity criteria for weak solutions of the navier-stokes equations

  • Shuji Takahashi
Article

Abstract

We are concerned with the behavior of weak solutions of the Navier-Stokes equations near possible singularities. We shall show that if a weak solution is in some Lebesgue space or small in some Lorentz space locally, it does not blowup there. Our basic idea is to estimate integral formulas for vorticity which satisfies parabolic equations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bergh, J. and Löfström, J., “Interpolation Spaces,” Springer-Verlag, Berlin-Heidelberg-New York, 1976MATHGoogle Scholar
  2. 2.
    Caffarelli, L., Kohn, R., and Nirenberg, L.,Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math.35, 771–831 (1982)MATHMathSciNetGoogle Scholar
  3. 3.
    Fabes, E., Lewis, J., and Riviere, N.,Singular integrals and hydrodynamic potentials, Amer. J. Math.99, 601–625 (1977)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Giga, Y.,Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations62, 186–212 (1986)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Giga, Y. and Kohn, R.,Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math.42, 845–884 (1989)MATHMathSciNetGoogle Scholar
  6. 6.
    Hopf, E.Über die anfangswertaufgabe für die hydrodynamischen grundgleichungen, Math. Nachr.4, 213–231 (1951)MATHMathSciNetGoogle Scholar
  7. 7.
    Hunt, R.,On L(p,q) spaces, Enseignement Math.12, 249–276 (1966)MATHMathSciNetGoogle Scholar
  8. 8.
    Ladyzenskaya, O., Ural'ceva, N., and Solonnikov, V., “Linear and Quasi-Linear Equations of Parabolic Type,” Amer. Math. Soc., Providence RI, 1968Google Scholar
  9. 9.
    Leray, J.,Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math.63, 193–248 (1934)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Lions, J. and Magenes, E., “Non-Homogeneous Boundary Value Problems and Applications I,” Springer Grundlehren, Berlin-Heidelberg-New York, 1972MATHGoogle Scholar
  11. 11.
    Nirenberg, L.,On elliptic partial differential equations, Ann. Scuola Normale Pisa Ser III13, 115–162 (1959)MathSciNetGoogle Scholar
  12. 12.
    Ohyama, T.,Interior regularity of weak solutions of the time-dependent Navier-Stokes equation, Proc. Japan Acad.36, 273–277 (1960)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Reed, M. and Simon, B., “Method of Modern Mathematical Physics II,” Academic Press, New York-San Francisco-London, 1975Google Scholar
  14. 14.
    Serrin, J.,On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal.9, 187–195 (1962)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Sohr, H.,Zur regularitätstheorie der instantionären Gleichungen von Navier-Stokes, Math. Z.184, 359–375 (1983)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Struwe, M.,On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math.41, 437–458 (1988)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Shuji Takahashi
    • 1
  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan

Personalised recommendations