Very weak solutions for the Stokes equations

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 [9] for the Stokes equations has been intensively studied in the last years for the Navier-Stokes equations. However, a more rigorous study about the existence of traces for non regular vector fields is necessary in order to make a precise extension of the Stokes result to the Navier-Stokes case. Such study and a new and simpler proof of the existence of very weak solution for Stokes equations are made, based on density arguments and an adequate functional framework. We also obtain regularity results in fractional Sobolev spaces. All these results are obtained in the case of a bounded open set, connected of class $C^{1,1}$ of $R^3$ and can be extended to Laplace's equation and to other dimensions.

Domaines: Equations aux dérivées partielles [math.AP]

Chérif Amrouche, M.A. Rodriguez-Bellido. Very weak solutions for the Stokes equations. 2009. ⟨hal-00444237⟩