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.
Similar content being viewed by others
References
Bergh, J. and Löfström, J., “Interpolation Spaces,” Springer-Verlag, Berlin-Heidelberg-New York, 1976
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)
Fabes, E., Lewis, J., and Riviere, N.,Singular integrals and hydrodynamic potentials, Amer. J. Math.99, 601–625 (1977)
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)
Giga, Y. and Kohn, R.,Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math.42, 845–884 (1989)
Hopf, E.Über die anfangswertaufgabe für die hydrodynamischen grundgleichungen, Math. Nachr.4, 213–231 (1951)
Hunt, R.,On L(p,q) spaces, Enseignement Math.12, 249–276 (1966)
Ladyzenskaya, O., Ural'ceva, N., and Solonnikov, V., “Linear and Quasi-Linear Equations of Parabolic Type,” Amer. Math. Soc., Providence RI, 1968
Leray, J.,Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math.63, 193–248 (1934)
Lions, J. and Magenes, E., “Non-Homogeneous Boundary Value Problems and Applications I,” Springer Grundlehren, Berlin-Heidelberg-New York, 1972
Nirenberg, L.,On elliptic partial differential equations, Ann. Scuola Normale Pisa Ser III13, 115–162 (1959)
Ohyama, T.,Interior regularity of weak solutions of the time-dependent Navier-Stokes equation, Proc. Japan Acad.36, 273–277 (1960)
Reed, M. and Simon, B., “Method of Modern Mathematical Physics II,” Academic Press, New York-San Francisco-London, 1975
Serrin, J.,On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal.9, 187–195 (1962)
Sohr, H.,Zur regularitätstheorie der instantionären Gleichungen von Navier-Stokes, Math. Z.184, 359–375 (1983)
Struwe, M.,On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math.41, 437–458 (1988)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Takahashi, S. On interior regularity criteria for weak solutions of the navier-stokes equations. Manuscripta Math 69, 237–254 (1990). https://doi.org/10.1007/BF02567922
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02567922