A regularization of incompressible Stokes problem with Tresca friction condition

In the present article, we introduce and study a model addressing the Stokes problem with non-linear boundary conditions of the Tresca type. We suggest a new procedure for regularizing incompressible fluid, i.e. we assume that the divergence $\mathrm{\nabla }\cdot \mathbf{u}\in [-\epsilon ,\epsilon ]$ which leads to class of constrained elliptic variational inequalities. We use a fixed point strategy to show the existence and uniqueness of a solution and we reformulate the problem as an equivalent constrained minimization problem. An ADMM is applied to the minimization problem and some algorithm are provided.