Lubrication Theory and Viscous Shallow-Water Equations

  • Didier BreschEmail author
  • Mathieu Colin
  • Xi Lin
  • Pascal Noble
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 17)


In this proceeding dedicated to Enrique Fernández-Cara, we indicate how to derive rigorously lubrication equations studied in [A.L. Bertozzi, M.C. Pugh, CPAM (1998)] from the viscous shallow-water equations with drag terms and surface tension effect in the one-dimensional in space case. We consider strong and weak solutions and propose simple adaptation of recent relative entropy tools already developed by the first and third author. Note that the viscous shallow-water equation involves height dependent viscosity which vanishes if height vanishes. We choose such presentation because Enrique Fernández-Cara. worked with Francisco Guillén on non-homogeneous incompressible Navier-Stokes equations with density dependent viscosities and in some sense we want to show that in the compressible setting more general test functions are available so it helps to cover degenerate viscosity when height vanishes contrarily to the incompressible setting where such case is an open question.


Shallow-water equations Navier-Stokes equations Global weak solutions Lubrication models 

2000 MR Subject Classification




Research of P. Noble was partially supported by French ANR project no. ANR-09-JCJC-0103-01. Research of M. Colin, X. Lin, D. Bresch has been partially supported by the ANR Project Viscap.


  1. 1.
    Bertozzi, A.L., Grün, G., Witelski, T.P.: Dewetting films: bifurcations and concentrations. Nonlinearity 14, 1569–1592 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bertozzi, A.L., Pugh, M.C.: Long-wave instabilities and saturation in thin film equations. Commun. Pure Appl. Math. 51, 625–661 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brenier, Y., Duan, X.: From conservative to dissipative systems through quadratic change of time, with application to the curve-shortening flow. ArXiv (2017)Google Scholar
  4. 4.
    Bresch, D., Colin, M., Msheik, K., Xi, L.: On a lubrication equation in one-dimension in space (2018, in preparation)Google Scholar
  5. 5.
    Bresch, D., Desjardins, B.: Weak solutions via the total energy formulation and their qualitative properties - density dependent viscosities. In: Y. Giga, Novotný, A. (eds.) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Berlin (2017)Google Scholar
  6. 6.
    Bresch, D., Desjardins, B.: Existence of global weak solutions for a 2D viscous shallow water model and convergence to the quasigeostrophic model. Commun. Math. Phys. 238(1–2), 211–223 (2003)CrossRefGoogle Scholar
  7. 7.
    Bresch, D., Desjardins, B.: Quelques modèles diffusifs capillaires de type Korteweg. C. R. Acad. Sci. Paris Section Mécanique 332(11), 881–886 (2004)zbMATHGoogle Scholar
  8. 8.
    Bresch, D., Desjardins, B., Lin, C.K.: On some compressible fluid models: Korteweg, lubrication and shallow water systems. Commun. Part. Differ. Equ. 28(3–4), 1009–1037 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bresch, D., Gisclon, M., Lacroix-Violet, I.: On Navier-Stokes-Korteweg and Euler-Korteweg systems: application to quantum fluids models (2017, submitted)Google Scholar
  10. 10.
    Bresch, D., Jabin, P.-E.: Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamical unstable pressure and anisotropy viscous stress tensor. Ann. Math. 188, 577–684 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bresch, D., Jabin, P.-E.: Global weak solutions of PDEs for compressible media: a compactness criterion to cover new physical situations. In: F. Colombini, D. Del Santo, D. Lannes (eds.) Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics. Springer INdAM-series, Spécial Issue Dedicated to G. Métivier, pp. 33–54. Springer, Cham (2017)Google Scholar
  12. 12.
    Bresch, D., Noble, P.: Mathematical derivation of viscous shallow-water equations with zero surface tension. Indiana Univ. J. 60(4), 1137–1269 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bresch, D., Noble, P., Vila, J.-P.: Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and applications. C.R. Acad. Sci. Paris 354(1), 45–49 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bresch, D., Noble, P., Vila, J.-P.: Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and various applications. In: ESAIM Proceedings (2017). MathSciNetCrossRefGoogle Scholar
  15. 15.
    Danchin, R., Mucha, P.: The incompressible Navier-Stokes equations in vacuum. ArXiv:1705.06061 (2017)Google Scholar
  16. 16.
    Feireisl, E., Novotny, A., Petzeltova, H.: On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Fernández-Cara, E.: Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete Contin. Dyn. Syst. Ser. S 5, 1021–1090 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fernández-Cara, E., Guillén-Gonzalez, F.: Some new existence results for the variable density Navier-Stokes Ann. Fac. Sci. Toulouse Math. Ser. 6 2(2), 185–204 (1993)Google Scholar
  19. 19.
    Fernández-Cara, E., Guillén-Gonzalez, F.: The existence of nonhomogeneous, viscous and incompressible flow in unbounded domains. Commun. Part. Differ. Equ. 17(7 & 8), 1253–1265 (1992)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Gerbeau, J.F., Perthame, B.: Derivation of viscous Saint-Venant system for laminar shallow water: Numerical validation. Discrete Contin. Dyn. Syst. 1, 89–102 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Haspot, B.: Weak-Strong uniqueness for compressible Navier-Stokes system with degenerate viscosity coefficient and vacuum in one dimension. Commun. Math. Sci. (2017, to appear)Google Scholar
  22. 22.
    Kazhikhov, A.: Resolution of boundary value problems for non homogeneous viscous fluids. Dokl. Akad. Nauk 216, 1008–1010 (1974)Google Scholar
  23. 23.
    Kitavtsev, G., Laurencot, P., Niethammer, B.: Weak solutions to lubrication equations in the presence of strong slippage. Methods Appl. Anal. 18, 183–202 (2011)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Li, J., Xin, Z.P.: Global existence of weak solutions to the barotropic compressible Navier-Stokes flows with degenerate viscosities. (2015, submitted) (see arXiv:1504.06826)Google Scholar
  25. 25.
    Lions, P.-L.: Mathematical Topics in Fluid Mechanics, vol. 1. Oxford Lecture Series in Mathematics and Its Applications. The Clarendon Press, Oxford University Press, New York (1996). Incompressible Models, Oxford Science PublicationsGoogle Scholar
  26. 26.
    Oron, A., Davis, S.H., Bankoff, S.G.: Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69(3), 931–980 (1997)CrossRefGoogle Scholar
  27. 27.
    Rousset, F.: Solutions faibles de léquation de Navier-Stokes des fluides compressibles [d’après A. Vasseur et C. Yu]. Séminaire Bourbaki, no 135 (2016–2017)Google Scholar
  28. 28.
    Simon, J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density and pressure. SIAM J. Math. Anal. 21(5), 1093–1117 (1990)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Vasseur, A., Yu, C.: Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations. Invent. Math. 206, 935–974 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Didier Bresch
    • 1
    Email author
  • Mathieu Colin
    • 2
  • Xi Lin
    • 2
  • Pascal Noble
    • 3
  1. 1.Université Grenoble AlpesUniversité Savoie Mont Blanc, CNRS, LAMAChambéryFrance
  2. 2.Equipe INRIA, CARDAMOM, Institut Mathématiques de BordeauxÉquipe EDP 351TalenceFrance
  3. 3.Université de LyonUniversité Lyon 1, Institut Camille Jordan, UMR, CNRSVilleurbanne CedexFrance

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