Stationary Stokes, Oseen and Navier–Stokes Equations with Singular Data

Abstract

The concept of very weak solution introduced by Giga (Math Z 178:287–329, 1981) for the Stokes equations has hardly been studied in recent years for either the Navier–Stokes equations or the Navier–Stokes type equations. We treat the stationary Stokes, Oseen and Navier–Stokes systems in the case of a bounded open set, connected of class $${\mathcal{C}^{1,1}}$$ of $${\mathbb{R}^3}$$ . Taking up once again the duality method introduced by Lions and Magenes (Problemes aus limites non-homogenes et applications, vols. 1 & 2, Dunod, Paris, 1968) and Giga (Math Z 178:287–329, 1981) for open sets of class $${\mathcal{C}^{\infty}}$$ [see also chapter 4 of Necas (Les methodes directes en theorie des equations elliptiques. (French) Masson et Cie, Ed., Paris; Academia, Editeurs, Prague, 1967), which considers 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 not necessarily small, which lets 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.