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

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Fort-Faible.1D(1).pdf (235.6Kb)
Date
2017
Dewey
Analyse
Sujet
fluid mechanics; weak-strong uniqueness; relative entropy
Journal issue
Communications in Mathematical Sciences
Volume
15
Number
3
Publication date
02-2017
Article pages
587-591
Publisher
International Press
DOI
http://dx.doi.org/10.4310/CMS.2017.v15.n3.a1
URI
https://basepub.dauphine.fr/handle/123456789/20062
Collections
  • CEREMADE : Publications
Metadata
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Author
Haspot, Boris
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Article accepté pour publication ou publié
Abstract (EN)
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 ρ0 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.

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