In this chapter, we consider the Stokes and Navier-Stokes systems of the previous two chapters, but we allow non-smooth coefficients. We first define the volume (or Newtonian) and layer potentials for the Stokes system with non-smooth coefficients in a Lipschitz subdomain of a compact Riemannian manifold and prove some of their main properties by using a variational approach based on the Nec̆as-Babus̆ka-Brezzi method. We then combine the well-posedness of the Dirichlet problem for the Stokes system with a fixed point theorem and show the existence of a weak solution for the Dirichlet problem for our non-smooth coefficient Navier-Stokes system in \(L^2\) -based Sobolev spaces in a Lipschitz domain on a closed Riemannian manifold of dimension 2 or 3. This is the last chapter of the third part.