An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity.

*(English)* Zbl 1050.35073
Galdi, Giovanni P. (ed.) et al., Topics in mathematical fluid mechanics. Meeting on the occasion of Professor John G. Heywood sixtieth birthday, Capo Miseno, Italy, May 27–30, 2000. Rome: Aracne (ISBN 88-7999-410-7/hbk). Quad. Mat. 10, 163-183 (2002).

The Navier-Stokes initial-boundary value problem for a viscous incompressible fluid with homogeneous Dirichlet-type boundary conditions and nonhomogeneous initial velocity conditions is under consideration. The existence of a weak solution of this problem is known. The authors deal with the question whether the components of velocity are coupled in such a way that some information about a higher regularity of one of them alrealy implies the higher regularity of all of them. It was proved that the interior essential boundedness of one of the velocity components implies the interior regularity of all the components. The authors improve this result.

##### MSC:

35Q30 | Stokes and Navier-Stokes equations |

76D03 | Existence, uniqueness, and regularity theory |

76D05 | Navier-Stokes equations (fluid dynamics) |