Advertisement

On the Existence of Weak Solutions to the Steady Compressible Flow of Nematic Liquid Crystals

  • Chunhua Jin
Article

Abstract

In this paper, we consider the existence of weak renormalized solutions for the steady compressible flow of nematic liquid crystals in a 3-D bounded domain with no-slip boundary condition. By using a three-level approximation scheme, we establish the existence of weak renormalized solutions under the hypothesis \(\gamma >1\) for the adiabatic constant.

Keywords

Steady flow nematic liquid crystals weak renormalized solution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bogovskii, M.E.: Solutions of some vector analysis problems connected with operators div and grad. Trudy Sem. S. L. Sobolev 80(1), 5–40 (1980)Google Scholar
  2. 2.
    Březina, J., Novotný, A.: On weak solutions of steady Navier–Stokes equations for monatomic gas. Comment. Math. Univ. Carolin. 49, 611–632 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Borchers, W., Sohr, H.: On the equation \(\text{ rot } v=g\) and \(\text{ div } u=f\) with zero boundary conditions. Hokkaido Math. J. 19, 67–87 (1990)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chandrasekhar, S.: Liquid Crystals, 2nd edn. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  5. 5.
    Ericksen, J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 23–34 (1961)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ericksen, J.L.: Hydrostatic theory of liquid crystal. Arch. Ration. Mech. Anal. 9, 371–378 (1962)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Eliezer, S., Ghatak, A., Hora, H.: An Introduction to Equations of States, Theory and Applications. Cambridge University Press, Cambridge (1986)zbMATHGoogle Scholar
  8. 8.
    Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics and Its Applications, vol. 26. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  9. 9.
    Feireisl, E.: Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel (2009)zbMATHCrossRefGoogle Scholar
  10. 10.
    de Gennes, P.G.: The Physics of Liquid Crystals. Oxford University Press, Oxford (1974)zbMATHGoogle Scholar
  11. 11.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)zbMATHCrossRefGoogle Scholar
  12. 12.
    Jiang, S., Zhou, C.: Existence of weak solutions to the three dimensional steady compressible Navier–Stokes equations. Ann. IHP Anal. Nonlineaire 28, 485–498 (2011)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Lions, P.L.: Mathematical Topics in Fluid Mechanics. Vol 2: Compressible Models, Oxford Science Publications, Calderon Press, New York (1998)Google Scholar
  14. 14.
    Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lin, F.H.: Nonlinear theory of defects in nematic liquid crystal: phase transition and flow phenomena. Commun. Pure Appl. Math. 42, 789C814 (1989)MathSciNetGoogle Scholar
  16. 16.
    Lin, F.H., Liu, C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48, 501–537 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Novo, S., Novotný, A.: On the existence of weak solutions to steady compressible Navier–Stokes equations when the density is not square integrable. J. Math. Kyoto Univ. 42, 531–550 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Novotný, A., Pokorny, M.: Weak and variational solutions to steady equations for compressible heat conducting fluids. SIAM J. Math. Anal. 43, 1158–1188 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Plotnikov, P., Sokolowski, J.: Concentrations of stationary solutions to compressible Navier–Stokes equations. Commun. Math. Phys. 258, 567–608 (2005)ADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Plotnikov, P.I., Weigant, W.: Steady 3D viscous compressible flows with adiabatic exponent \(\gamma \in (1, \infty )\). J. Math. Pures Appl. 104, 58–82 (2015)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Tartar, L.: Compensated compactness and applications to partila differential equations. In: Knopps, L.J. (ed.) Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, Research Notes in Math., vol. 39, pp. 136–211. Pitman (1975)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesSouth China Normal UniversityGuangzhouChina

Personalised recommendations