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We prove weak-strong uniqueness results for the compressible Navier-Stokessystem with degenerate viscosity coefficient and with vacuum in one dimension.In other words, we give conditions on the weak solution constructed in\cite{Jiu} so that it is unique. The novelty consists in dealing with initialdensity $\rho_0$ which contains vacuum. To do this we use the notion ofrelative entropy developed recently by Germain, Feireisl et al and Mellet andVasseur (see \cite{PG,Fei,15}) combined with a new formulation of thecompressible system (\cite{cras,CPAM,CPAM1,para}) (more precisely we introducea new effective velocity which makes the system parabolic on the density andhyperbolic on this velocity).

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