×

zbMATH — the first resource for mathematics

A refinement of the local Serrin-type regularity criterion for a suitable weak solution to the Navier-Stokes equations. (English) Zbl 1304.35502
This paper deals with the local regularity od suitable weak solutions to the Navier-Stokes equations. The author establish a new local regularity criterion which impose only a Serrin-type integrability condition on velocity in a backward neighbourhood of the point \((x_0, t_0)\), intersected with the exterior of a certain space-time paraboloid with vertex at point \((x_0, t_0)\).
Reviewer: Cheng He (Beijing)

MSC:
35Q30 Navier-Stokes equations
35D30 Weak solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Borchers, W.; Sohr, H., On the equations rot \(v\) = \(g\) and div \(u\) = \(f\) with zero boundary conditions, Hokkaido Math. J., 19, 67-87, (1990) · Zbl 0719.35014
[2] Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure Appl. Math., 35, 771-831, (1982) · Zbl 0509.35067
[3] Farwig, R., Kozono, H., Sohr, H.: Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin’s condition. Banach Center Publ. 81, Parabolic and Navier-Stokes Equations, Part 1. Warsaw, 175-184 (2008) · Zbl 1154.35416
[4] Galdi, G.P.: An Introduction to the Navier-Stokes initial-boundary value problem. In: Fundamental Directions in Mathematical Fluid Mechanics, (Eds. Galdi G.P., Heywood J. and Rannacher R.) series “Advances in Mathematical Fluid Mechanics”. Birkhauser, Basel, 1-98 (2000) · Zbl 0353.46018
[5] Kučera, P.; Skalák, Z., A note on the generalized energy inequality in the Navier-Stokes equations, Appl. Math., 48, 537-545, (2003) · Zbl 1099.35099
[6] Ladyzhenskaya, O.A.: Mathematical Problems in the Dynamics of Viscous Incompressible Fluid. Gordon and Breach, New York (1963) · Zbl 0954.35129
[7] Ladyzhenskaya, O.A.; Seregin, G.A., On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1, 356-387, (1999) · Zbl 0954.35129
[8] Lin, F., A new proof of the caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51, 241-257, (1998) · Zbl 0958.35102
[9] Nečas, J.; Neustupa, J., New conditions for local regularity of a suitable weak solution to the Navier-Stokes equations, J. Math. Fluid Mech., 4, 237-256, (2002) · Zbl 1010.35081
[10] Neustupa, J., A note on local interior regularity of a suitable weak solution to the Navier-Stokes problem, Discr. Cont. Dyn. Syst. Ser. S., 6, 1391-1400, (2013) · Zbl 1260.35125
[11] Neustupa, J., A removable singularity in a suitable weak solution to the Navier-Stokes equations, Nonlinearity, 25, 1695-1708, (2012) · Zbl 1245.35085
[12] Saks, R.S., Spectral problems for the curl and Stokes operators, Doklady Mathematics, 76, 724-728, (2007) · Zbl 1159.35391
[13] Scheffer, V., Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys., 55, 97-112, (1977) · Zbl 0357.35071
[14] Šebestová, I., Vejchodský, T.: Two-Sided Bounds for Eigenvalues of Differential Operators with Applications to Friedrichs, Poincaré, Trace and Similar Constants. Nečas Center for Mathematical Modelling, Preprint No. 2013-05, Prague (2013) · Zbl 1287.35050
[15] Seregin, G.; Šverák, V., On smoothness of suitable weak solutions to the Navier-Stokes equations, J. Math. Sci., 130, 4884-4892, (2005) · Zbl 1148.35344
[16] Seregin, G.A., Local regularity for suitable weak solutions of the Navier-Stokes equations, Russian Math. Surveys, 62, 595-614, (2007) · Zbl 1139.76018
[17] Sohr, H.: The Navier-Stokes Equations. An Elementary Functional Analytic Approach. Birkhäuser Advanced Texts, Basel, Boston, Berlin (2001) · Zbl 0983.35004
[18] Takahashi, S., On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscripta Math., 69, 237-254, (1990) · Zbl 0718.35022
[19] Talenti, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110, 353-372, (1976) · Zbl 0353.46018
[20] Temam, R.: Navier-Stokes Equations. North-Holland, Amsterdam, New York, Oxford (1977)
[21] Vasseur, A., A new proof of partial regularity of solutions to Navier-Stokes equations, Nonlin. Diff. Eq. Appl., 14, 753-785, (2007) · Zbl 1142.35066
[22] Wolf, J.: A new criterion for partial regularity of suitable weak solutions to the Navier-Stokes equations. Advances in Mathematical Fluid Mechanics (Eds. Rannacher R. and Sequeira A.) Springer, Berlin, 613-630 (2010) · Zbl 1374.35294
[23] Yurinsky, V.V., A lower bound for the principal eigenvalue of the Stokes operator in a random domain, Annales de l’Institut Henri Poincaré—Probabilités et Statistiques, 44, 1-18, (2008) · Zbl 1173.82333
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.