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):313–330, 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. Novotný, 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 $v$ which makes the system parabolic on the density and hyperbolic on the velocity $v$.

Keywords

fluid mechanics, weak-strong uniqueness, relative entropy

2010 Mathematics Subject Classification

35A02, 35Axx, 35D30, 35D35, 35Q30

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Published 24 February 2017