On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem

Abstract

We establish regularity results up to the boundary for solutions to generalized Stokes and Navier-Stokes systems of equations in the stationary and evolutive cases. Generalized here means the presence of a shear dependent viscosity. We treat the case p ≥ 2. Actually, we are interested in proving regularity results in Lq.(ω) spaces for all the second order derivatives of the velocity and all the first order derivatives of the pressure. The main aim of the present paper is to extend our previous scheme, introduced in [3] and [4] for the flat-boundary case, to the case of curvilinear boundaries. © European Mathematical Society 2009.