On interior regularity criteria for weak solutions of the Navier-Stokes equations. (English) Zbl 0718.35022

In this paper some results on interior regularity of weak solution for certain parabolic system in spaces \(L^{p,q}(Q)\) are obtained under minimal regularity assumptions on the coefficients.
Applying the above results for the vorticity equations, there are obtained new local interior regularity criteria for weak solutions of the time-dependent Navier-Stokes equations for an incompressible medium with the adherence property to the smooth boundary of the considered domain in \(R^ n\) (n\(\geq 3).\)
A priori estimates for weak solutions of a Cauchy problem of some linear parabolic nonhomogeneous system with nonregular coefficients in \(R^ n\times (0,T)\) are also obtained using cut-off functions instead of traces in Sobolev spaces of negative order.


35D10 Regularity of generalized solutions of PDE (MSC2000)
35Q30 Navier-Stokes equations
Full Text: DOI EuDML


[1] Bergh, J. and Löfström, J., ”Interpolation Spaces,” Springer-Verlag, Berlin-Heidelberg-New York, 1976 · Zbl 0344.46071
[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) · Zbl 0509.35067
[3] Fabes, E., Lewis, J., and Riviere, N.,Singular integrals and hydrodynamic potentials, Amer. J. Math.99, 601–625 (1977) · Zbl 0374.44006
[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) · Zbl 0577.35058
[5] Giga, Y. and Kohn, R.,Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math.42, 845–884 (1989) · Zbl 0703.35020
[6] Hopf, E.Über die anfangswertaufgabe für die hydrodynamischen grundgleichungen, Math. Nachr.4, 213–231 (1951) · Zbl 0042.10604
[7] Hunt, R.,On L(p,q) spaces, Enseignement Math.12, 249–276 (1966) · Zbl 0181.40301
[8] Ladyzenskaya, O., Ural’ceva, N., and Solonnikov, V., ”Linear and Quasi-Linear Equations of Parabolic Type,” Amer. Math. Soc., Providence RI, 1968
[9] Leray, J.,Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math.63, 193–248 (1934) · JFM 60.0726.05
[10] Lions, J. and Magenes, E., ”Non-Homogeneous Boundary Value Problems and Applications I,” Springer Grundlehren, Berlin-Heidelberg-New York, 1972 · Zbl 0223.35039
[11] Nirenberg, L.,On elliptic partial differential equations, Ann. Scuola Normale Pisa Ser III13, 115–162 (1959) · Zbl 0088.07601
[12] Ohyama, T.,Interior regularity of weak solutions of the time-dependent Navier-Stokes equation, Proc. Japan Acad.36, 273–277 (1960) · Zbl 0100.22404
[13] Reed, M. and Simon, B., ”Method of Modern Mathematical Physics II,” Academic Press, New York-San Francisco-London, 1975 · Zbl 0308.47002
[14] Serrin, J.,On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal.9, 187–195 (1962) · Zbl 0106.18302
[15] Sohr, H.,Zur regularitätstheorie der instantionären Gleichungen von Navier-Stokes, Math. Z.184, 359–375 (1983) · Zbl 0506.35084
[16] Struwe, M.,On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math.41, 437–458 (1988) · Zbl 0632.76034
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