Dissipative solutions to the model of a general compressible viscous fluid with the Coulomb friction law boundary condition

Abstract: We study a model of a general compressible viscous fluid subject to the Coulomb friction law boundary condition. For this model, we introduce a dissipative formulation and prove the existence of dissipative solutions. The proof of this result consists of a three-level approximation method: A Galerkin approximation, the classical parabolic regularization of the continuity equation as well as convex regularizations of the potential generating the viscous stress and the boundary terms incorporating the Coulomb friction law into the dissipative formulation. This approach combines the techniques already known from the proof of the existence of dissipative solutions to a model of general compressible viscous fluids under inflow-outflow boundary conditions as well as the proof of the existence of a weak solution to the incompressible Navier-Stokes equations under the Coulomb friction law boundary condition. It is the first time that this type of boundary condition is considered in the case of compressible flow.