Global existence for a 3D non-stationary Stokes flow with Coulomb's type friction boundary conditions

Abstract: In this paper we study non stationary viscous incompressible fluid flows with nonlinear boundary slip conditions given by a subdifferential property of friction type. More precisely we assume that the tangential velocity vanishes as long as the shear stress remains below a threshold, that may depend on the time and the position variables but also on the stress tensor, allowing to consider Coulomb's type friction laws. An existence and uniqueness theorem is obtained first when the shear stress threshold is a data and sharp estimates are derived for the velocity and pressure fields as well as for the stress tensor. Then an existence result is proved for the non-local Coulomb's friction case by using a successive approximation technique with respect to the shear stress threshold.