New conditions for local regularity of a suitable weak solution to the Navier-Stokes equation. (English) Zbl 1010.35081

Nečas, Jindřich; Neustupa, Jiří

J. Math. Fluid Mech. 4, No. 3, 237-256 (2002).

Summary: We formulate conditions which guarantee that a suitable weak solution \((v,p)\) to the Navier-Stokes equation (in the sense of L. Caffarelli, R. Kohn and L. Nirenberg [Commun. Pure Appl. Math. 35, 771-831 (1982; Zbl 0509.35067)]) cannot have a singularity at the point \((x_0,t_0)\in \mathbb{R}^3\times (0,T)\). The usual Prodi-Serrin condition on the velocity \(v\) is substantially replaced by an analogous condition imposed on the negative part \(p_-\) of the pressure \(p\).

MSC:

35Q30	Navier-Stokes equations
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

conditions for local regularity; suitable weak solution; Navier-Stokes equation; Prodi-Serrin condition