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The Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations

  • Helmut Abels
  • Johannes Daube
  • Christiane Kraus
  • Dietmar Kröner
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 236)

Abstract

We investigate the sharp-interface limit for the Navier–Stokes–Korteweg model, which is an extension of the compressible Navier–Stokes equations. By means of compactness arguments, we show that solutions of the Navier–Stokes–Korteweg equations converge to solutions of a physically meaningful free-boundary problem. Assuming that an associated energy functional converges in a suitable sense, we obtain the sharp-interface limit at the level of weak solutions.

Keywords

Sharp-interface limit Diffuse-interface model Liquid–vapour flow Navier–Stokes–Korteweg system Free-boundary problem 

MSC (2010)

35B40 76T10 35Q35 35R35 

References

  1. 1.
    H. Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. Commun. Math. Phys. 289(1), 45–73 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems (Clarendon Press, Oxford, 2000)zbMATHGoogle Scholar
  3. 3.
    S. Benzoni-Gavage, R. Danchin, S. Descombes, Well-posedness of one-dimensional Korteweg models. Electron. J. Differ. Equ. 59, 1–35 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    S. Benzoni-Gavage, R. Danchin, S. Descombes, On the well-posedness for the Euler–Korteweg model in several space dimensions. Indiana Univ. Math. J. 56(4), 1499–1579 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    D. Bresch, B. Desjardins, C.K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Commun. Partial Differ. Equ. 28(3–4), 843–868 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    X. Chen, Global asymptotic limit of solutions of the Cahn–Hilliard equation. J. Differ. Geom. 44(2), 262–311 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    R. Danchin, B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 18(1), 97–133 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Daube, Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations. Ph.D. thesis, University of Freiburg (2016). https://freidok.uni-freiburg.de/data/11679
  9. 9.
    R.J. DiPerna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)MathSciNetCrossRefGoogle Scholar
  10. 10.
    E. Feireisl, A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids (Springer Science & Business Media, Berlin, 2009)CrossRefGoogle Scholar
  11. 11.
    B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type. J. Math. Fluid Mech. 13(2), 223–249 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    H. Hattori, D. Li, The existence of global solutions to a fluid dynamic model for materials for Korteweg type. J. Partial Differ. Equ. 9(4), 323–342 (1996)MathSciNetzbMATHGoogle Scholar
  13. 13.
    K. Hermsdörfer, C. Kraus, D. Kröner, Interface conditions for limits of the Navier–Stokes–Korteweg model. Interfaces Free Bound. 13(2), 239–254 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Köhne, J. Prüss, M. Wilke, Qualitative behaviour of solutions for the two-phase Navier–Stokes equations with surface tension. Math. Ann. 356(2), 737–792 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    D.J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l’on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité. Arch. Néerl. 2(6), 1–24 (1901)zbMATHGoogle Scholar
  16. 16.
    M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25(4), 679–696 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Kotschote, Strong well-posedness for a Korteweg-type model for the dynamics of a compressible non-isothermal fluid. J. Math. Fluid Mech. 12(4), 473–484 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    S. Luckhaus, L. Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions. Arch. Ration. Mech. Anal. 107(1), 71–83 (1989)MathSciNetCrossRefGoogle Scholar
  19. 19.
    J. Simon, Compact sets in the space \(L^p(0, T;B)\). Ann. Math. Pura Appl. 146(4), 65–96 (1987)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Abteilung für Angewandte MathematikUniversität FreiburgFreiburgGermany
  3. 3.Weierstraß-InstitutBerlinGermany

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