Non-isothermal Navier-Stokes system with mixed boundary conditions and friction law: Uniqueness and regularity properties

We consider an unsteady non-isothermal fluid flow subjected to non-homogeneous Dirichlet conditions on a part of the boundary and Tresca's friction law on the other part. For this problem an existence result has been proved recently in Boukrouche et al. (2014) but uniqueness has been left as an open question. Starting from the approximation of the problem based on a regularization of the free boundary condition due to friction combined with a special penalty method, we establish some sharp a priori estimates leading to better regularity properties for the velocity field and to the uniqueness of the solutions. Finally we study the regularity of the pressure field and of the stress tensor.