Weak entropy boundary conditions for isentropic gas dynamics via kinetic relaxation

We consider a kinetic BGK model relaxing to isentropic gas dynamics previously introduced by the authors, but with Dirichlet boundary condition on the incoming velocities. We pass to the limit as the relaxation parameter tends to zero by compensated compactness inside the domain, and obtain that the limit satisses entropy inequalities on the boundary involving weak traces of entropy uxes. Our method is very general and could be applied to any entropy satisfying BGK model as soon as we have strong compactness of the macroscopic variables inside the domain.