Stationary Stokes, Oseen and Navier-Stokes equations with singular data

The concept of very weak solution introduced by Giga [20] for the Stokes equations has been hardly studied in the last years for either the Navier-Stokes equations or the Navier-Stokes type equations. We treat the stationary Stokes, Oseen and Navier-Stokes system in the case of a bounded open set, connected of class C 1,1 of R 3 . Taking the duality method introduced by Lions & Magenes in [28] and Giga in [20] up again for open sets of class C∞ (see also Necas [31] chapter 4 that consider the Hilbertian case p = 2 for general elliptic operators), we give a simpler proof of the existence of a very weak solution for stationary Oseen and Navier-Stokes equations when data are not regular enough, based on density arguments and a functional framework adequate for defining more rigourously the traces of non regular vector fields. In the stationary Navier-Stokes case, the results will be valid for external forces non necessarily small which let us extend the uniqueness class of solutions for these equations. Considering more regular data, regularity results in fractional Sobolev spaces will also be discussed for the three systems. All these results can be extended to other dimensions