Weak-Strong uniqueness for compressible Navier-Stokes system with degenerate viscosity coefficient and vacuum in one dimension

We prove weak-strong uniqueness results for the compressible Navier-Stokes system with degenerate viscosity coefficient and with vacuum in one dimension. In other words, we give conditions on the weak solution constructed in [20] so that it is unique. The main novelty consists in dealing with initial density ρ0 which contains vacuum, indeed the most of the results exclude this situation in order to use the parabolicity of the momentum equation (see [25]). To do this we use the notion of relative entropy developed recently by Germain and Feireisl et al (see [9, 8]) combined with a new formulation of the compressible system ([10, 12, 11, 13]), more precisely we introduce a new effective velocity which makes the system parabolic on the density and hyperbolic on this velocity.