Hugo Beirão da Veiga

On the Global Integrability for Any Finite Power of the Full Gradient for a Class of Generalized Power Law Models p < 2

In the following we consider a class of non-linear systems that covers some well known generalized Navier-Stokes systems with shear dependent viscosity of power law type, p < 2 . We show that weak solutions to our class of systems have integrable gradient up to the boundary, with any finite exponent. This result extends, up the the boundary, some of the interior regularity results known in the literature for systems of the above power law type. Boundedness of the gradient, up to the boundary, remains an open problem.

Vorticity and Regularity for Solutions of Initial-boundary Value Problems for the Navier—Stokes Equations

In reference [7], among other side results, we prove that the solution of the evolution Navier—Stokes equations (1.1) under the Navier (or slip) boundary condition (1.2) is necessarily regular if the direction of the vorticity is 1/2-Holder continuous with respect to the space variables. In this notes we show the main steps in the proof and made some comments on the above problem under the non-slip boundary condition (3.2).