On Stokes' Problem

We consider the Stokes problem of viscous hydrodynamics in bounded and exterior Lipschitz domains Omega of R-m (>= 2) with boundary datum in L-2(partial derivative Omega). We show that this problem has a unique very weak solution in bounded domains. As far as exterior domains are concerned, we prove that a very weak solution exists such that u = p(k) + O(r(-1-m-k)) at infinity, with p(k) a Stokes's polynomial of degree k, if and only if the data satisfy a suitable compatibility condition. In particular, we derive the well-known Stokes' paradox of hydrodynamics for very weak solutions. We use this results to prove the existence of a very weak solution to the Navier-Stokes problem in bounded and exterior Lipschitz domains of R-3 by requiring that the boundary datum belongs to L-8/3(partial derivative Omega).