Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berthelin F, Degond P, Delitala M, et al. A model for the formation and evolution of traffic jams. Arch Ration Mech Anal, 2008, 187: 185–220
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: 1269–1298
Bogovski M E. Solution of some vector analysis problems connected with operators div and grad (in Russian). Trudy Sem S L Sobolev, 1980, 80: 5–40
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: 801–836
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: 685–690
Carnahan N F, Starling K E. Equation of state for nonattracting rigid spheres. J Chem Phys, 1969, 51: 635–636
Degond P, Hua J. Self-organized hydrodynamics with congestion and path formation in crowds. J Comput Phys, 2013, 237: 299–319
Degond P, Hua J, Navoret L. Numerical simulations of the Euler system with congestion constraint. J Comput Phys, 2011, 230: 8057–8088
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: 717–730
Feireisl E, Lu Y, Malek J. On PDE analysis of flows of quasi-incompressible fluids. ZAMM Z Angew Math Mech, 2016, 96: 491–508
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: 219–239
Feireisl E, Zhang P. Quasi-neutral limit for a model of viscous plasma. Arch Ration Mech Anal, 2010, 197: 271–295
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, 2011
Germain P. Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. J Math Fluid Mech, 2010, 13: 137–146
Maury B. Prise en compte de la congestion dans les modeles de mouvements de foules. Actes des Colloques Caen 2012-Rouen, 2011, https://doi.org/docplayer.fr/32954222-Prise-en-compte-de-la-congestion-dans-les-modeles-de-mouvements-de-foules.html
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: 1558–1589
Acknowledgements
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Feireisl, E., Lu, Y. & Novotný, A. Weak-strong uniqueness for the compressible Navier-Stokes equations with a hard-sphere pressure law. Sci. China Math. 61, 2003–2016 (2018). https://doi.org/10.1007/s11425-017-9272-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-017-9272-7