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

Chérif Amrouche (1) , M.A. Rodriguez-Bellido (2)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 EDAN - Departamento de Ecuaciones Differenciales y Analysis numérico [Sevilla]

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 C1;1 of R3. Taking the duality method introduced by Lions & Magenes in [28] and Giga in [20] up again for open sets of class C1 (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 de¯ning more rigourously the traces of non regular vector ¯elds. 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, regu- larity results in fractional Sobolev spaces will also be discussed for the three systems. All these results can be extended to other dimensions.

Mots clés: Stokes equations; Oseen equations; Navier-Stokes equations; Very weak solutions; Stationary Solutions

Domaines: Equations aux dérivées partielles [math.AP]

Chérif Amrouche, M.A. Rodriguez-Bellido. Stationary Stokes, Oseen and Navier-Stokes equations with singular data. 2010. ⟨hal-00549166⟩