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

We establish regularity results up to the boundary for solutions to generalized Stokes and Navier-Stokes systems of equations in the stationary and in the evolutive cases. Generalized here means the presence of a shear dependent viscosity. We treat here the case $p\geq 2$. Actually, we are interested in proving regularity results in $L^q(\Omega)$ 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 references \cite{bvlali} and \cite{bvcubo} for the flat-boundary case, to the case of curvilinear boundaries.