zbMATH — the first resource for mathematics

A note on the generalized energy inequality in the Navier-Stokes equations. (English) Zbl 1099.35099
The authors prove the existence of a suitable weak solution such that the generalized energy inequality is satisfied for all non-negative smooth test functions. The usual definition requires that the test functions must have compact support.
The construction is based on a special approximation which provides smooth solutions and thus the solution itself is an acceptable test function for the approximate problem.

35Q35 PDEs in connection with fluid mechanics
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI EuDML
[1] L. Caffarelli, R. Kohn and L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982), 771-831. · Zbl 0509.35067 · doi:10.1002/cpa.3160350604
[2] Y. Giga, H. Sohr: Abstract \(L^p\) estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102 (1991), 72-94. · Zbl 0739.35067 · doi:10.1016/0022-1236(91)90136-S
[3] P. Kučera, Z. Skalák: Generalized energy inequality for suitable weak solutions of the Navier-Stokes equations. Proceedings of seminar Topical Problem of Fluid Mechanics 2003, Institute of Thermomechanics AS CR, J. Příhoda, K. Kozel (eds.), Prague, 2003, pp. 61-66.
[4] A. Kufner, O. John, S. Fučík: Function Spaces. Academia, Prague, 1979.
[5] J. Neustupa, A. Novotný, P. Penel: A remark to interior regularity of a suitable weak solution to the Navier-Stokes equations. Preprint, University of Toulon-Var, 1999.
[6] G. A. Seregin: Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary. J. Math. Fluid Mech. 4 (2002), 1-29. · Zbl 0997.35044 · doi:10.1007/s00021-002-8533-z
[7] Z. Skalák, P. Kučera: Remark on regularity of weak solutions to the Navier-Stokes equations. Comment. Math. Univ. Carolin. 42 (2001), 111-117. · Zbl 1115.35104
[8] R. Temam: Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland Publishing Company, Amsterdam-New York-Oxford. Revised edition, 1979. · Zbl 0426.35003
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.