Science China Mathematics

, Volume 61, Issue 11, pp 2003–2016 | Cite as

Weak-strong uniqueness for the compressible Navier-Stokes equations with a hard-sphere pressure law

  • Eduard Feireisl
  • Yong LuEmail author
  • Antonín Novotný


We consider the Navier-Stokes equations with a pressure function satisfying a hard-sphere law. That means the pressure, as a function of the density, becomes infinite when the density approaches a finite critical value. Under some structural constraints imposed on the pressure law, we show a weak-strong uniqueness principle in periodic spatial domains. The method is based on a modified relative entropy inequality for the system. The main difficulty is that the pressure potential associated with the internal energy of the system is largely dominated by the pressure itself in the area close to the critical density. As a result, several terms appearing in the relative energy inequality cannot be controlled by the total energy.


Navier-Stokes equations hard-sphere pressure weak-strong uniqueness 


35Q35 76N10 


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The work of Eduard Feireisl and Yong Lu leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (Grant No. FP7/2007-2013) and European Research Council (ERC) Grant Agreement (Grant No. 320078). The Institute of Mathematics of the Academy of Sciences of the Czech Republic was supported by Rozvoj Výzkumné Organizace (RVO) (Grant No. 67985840).


  1. 1.
    Berthelin F, Degond P, Delitala M, et al. A model for the formation and evolution of traffic jams. Arch Ration Mech Anal, 2008, 187: 185–220MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berthelin F, Degond P, Le Blanc V, et al. A traffic-ow model with constraints for the modeling of traffic jams. Math Models Methods Appl Sci, 2008, 18: 1269–1298MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bogovski M E. Solution of some vector analysis problems connected with operators div and grad (in Russian). Trudy Sem S L Sobolev, 1980, 80: 5–40Google Scholar
  4. 4.
    Bresch D, Desjardins B, Zatorska E. Two-velocity hydrodynamics in fluid mechanics, part II: Existence of global k-entropy solutions to the compressible Navier-Stokes systems with degenerate viscosities. J Math Pures Appl (9), 2015, 104: 801–836MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bresch D, Perrin C, Zatorska E. Singular limit of a Navier-Stokes system leading to a free/congested zones two-phase model. C R Math Acad Sci Paris, 2014, 352: 685–690MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Carnahan N F, Starling K E. Equation of state for nonattracting rigid spheres. J Chem Phys, 1969, 51: 635–636CrossRefGoogle Scholar
  7. 7.
    Degond P, Hua J. Self-organized hydrodynamics with congestion and path formation in crowds. J Comput Phys, 2013, 237: 299–319MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Degond P, Hua J, Navoret L. Numerical simulations of the Euler system with congestion constraint. J Comput Phys, 2011, 230: 8057–8088MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Feireisl E, Jin B, Novotny A. Relative entropies, suitable weak solutions, and weak-strong uniqueness for the com-pressible Navier-Stokes system. J Math Fluid Mech, 2012, 14: 717–730MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Feireisl E, Lu Y, Malek J. On PDE analysis of flows of quasi-incompressible fluids. ZAMM Z Angew Math Mech, 2016, 96: 491–508MathSciNetCrossRefGoogle Scholar
  11. 11.
    Feireisl E, Novotny A, Sun Y. A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system. Arch Ration Mech Anal, 2014, 212: 219–239MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Feireisl E, Zhang P. Quasi-neutral limit for a model of viscous plasma. Arch Ration Mech Anal, 2010, 197: 271–295MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Galdi G P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, 2nd ed. Springer Monographs in Mathematics. New York: Springer, 2011CrossRefzbMATHGoogle Scholar
  14. 14.
    Germain P. Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. J Math Fluid Mech, 2010, 13: 137–146MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Maury B. Prise en compte de la congestion dans les modeles de mouvements de foules. Actes des Colloques Caen 2012-Rouen, 2011, Google Scholar
  16. 16.
    Perrin C, Zatorska E. Free/congested two-phase model from weak solutions to multi-dimensional compressible Navier-Stokes equations. Comm Partial Differential Equations, 2015, 40: 1558–1589MathSciNetCrossRefzbMATHGoogle Scholar

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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsAcademy of Sciences of the Czech RepublicPrahaCzech Republic
  2. 2.Department of MathematicsNanjing UniversityNanjingChina
  3. 3.IMATHUniversity of ToulonLa GardeFrance

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