On the foundations of the Navier-Stokes equations

The foundations of the Navier-Stokes equations, describing the dynamics of a viscous and incompressible fluid, are reconsidered. By reviewing the original ideas of Navier, of a 'microscopic' nature, and those used by Stokes, of phenomenological character, the equations are rederived following similar arguments and a modern language. The underlying assumptions in each case are discussed. Macroscopically, we merely rephrase the well known arguments in a somewhat newer version. The microscopical counterpart is studied under the optics of Linear Response Theory. Using the Mori-Zwanzig projection operator formalism, the appropriate Generalized Langevin equation is analyzed for the mass particle and linear momentum densities. Sufficient hypotheses are introduced into the, otherwise exact, microscopical description in order to derive the well established macroscopic theory. A brief discussion dwells mainly with the two sets of assumptions (Author)