On the existence of Leray-Hopf Weak Solutions to the Navier-Stokes equations

We give a rather short and self contained presentation of the global existence results for Leray-Hopf weak solutions to the three dimensional incompressible Navier-Stokes equations. We give a unified treatment in terms of the domains and the relative boundary conditions and in terms of the approximation methods. Precisely, we consider the case of the whole space, the flat torus, and the case of a general bounded domain with smooth boundary and we consider as approximation schemes the Leray approximation method, the Faedo-Galerkin method, the semi-discretization in time and the approximation by adding a Smagorinsky term. We focus mainly on developing a unified treatment especially in the compactness argument needed to show that approximations converge to the weak solutions.