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 coefficients and with vacuum in one dimension. In other words, we give conditions on the weak solution constructed in [Q.S. Jiu and Z.P. Xin, Kinet. Relat. Models, 1(2): 313330, 2008] so that it is unique. The novelty consists of dealing with initial density rho(0) which contains vacuum. To do this we use the notion of relative entropy developed recently by Germain, Feireisl et al., and Mellet and Vasseur (see [P. Germain, J. Math. Fluid Mech., 13(1): 137-146, 2011], [E. Feireisl, A. Novotny, and S. Yongzhong, Indiana University Mathematical Journal, 60(2): 611-632, 2011], [A. Mellet and A. Vasseur, SIAM J. Math. Anal., 39(4): 1344-1365, 2007/08]) combined with a new formulation of the compressible system ([B. Haspot, Journal of Mathematical Fluid Mechanics, HAL Id: hal-00770248, arXiv:1304.4502, 1, 2013], [B. Haspot, Eprint Arxiv, hal-01081580, 2014]); more precisely we introduce a new effective velocity nu which makes the system parabolic on the density and hyperbolic on the velocity nu.