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 sys-tem 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 ρ 0 which contains vacuum, indeed the most of the results exclude this situation in order to use the parabolic-ity 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.no