Global existence of weak solution for the 2-D Ericksen–Leslie system

  • Meng WangEmail author
  • Wendong Wang


We prove the global existence of weak solution for two dimensional Ericksen–Leslie system with the Leslie stress and general Ericksen stress under the physical constrains on the Leslie coefficients. We also prove the local well-posedness of the Ericksen–Leslie system in two and three spatial dimensions.

Mathematics Subject Classification

35Q35 35A01 76A15 76D03 



The authors would like to thank Professor Zhifei ZHANG for his suggestion to consider the problem and his some valuable discussions with them. Part of this work is carried out when the first author is visiting Math. Department of Princeton University. Meng is partially supported by NSFC 10931001, 11371316, and Chen-su star project by Zhejiang University. Wendong is supported by “the Fundamental Research Funds for the Central Universities”, NSFC 11301048 and the Institute of Mathematical Sciences of Chinese University of Hong Kong.


  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Pure and Applied Mathematics (Amsterdam), vol. 140. Elsevier/Academic Press, Amsterdam, xiv+305 pp (2003)Google Scholar
  2. 2.
    Arnold, V.I.: Ordinary Differential Equations. The MIT Press, Massachusetts (1978). ISBN 0-262-51018-9Google Scholar
  3. 3.
    Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343. Springer, Heidelberg, xvi+523 pp (2011)Google Scholar
  4. 4.
    Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler Equations. Commun. Math. Phys. 94, 61–66 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ericksen, J.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 22–34 (1961)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ericksen, J.: Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113, 97–120 (1990)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Giaquinta, M., Modica, G., Soucek, J.: Cartesian currents in the calculus of variations, part II. Variational Integrals, A Series of Modern Surveys in Mathematics, vol. 38. Springer, Berlin (1998)Google Scholar
  8. 8.
    Hardt, R., Kinderlehrer, D., Lin, F.-H.: Existence and partial regularity of static liquid crystal configuration. Commun. Math. Phys. 105, 547–570 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hong, M.-C.: Global existence of solutions of the simplified Ericksen–Leslie system in dimension two. Calc. Var. Partial Differ. Equ. 40, 15–36 (2011)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hong, M.-C., Xin, Z.-P.: Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in \(\mathbb{R}^2\). Adv. Math. 231, 1364–1400 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hong, M.-C., Li, J.-K., Xin, Z.-P.: Blow-up criteria of strong solutions to the Ericksen–Leslie system in \(\mathbb{R}^3\) (2013) [arxiv: 1303.4488v2[math.AP]]Google Scholar
  12. 12.
    Huang, T., Wang, C.: Blow up criterion for nematic liquid crystal flows. Commun. Partial Differ. Equ. 37, 875–884 (2012)CrossRefzbMATHGoogle Scholar
  13. 13.
    Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lin, F.-H., Liu, C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48, 501–537 (1995)CrossRefzbMATHGoogle Scholar
  15. 15.
    Lin, F.-H., Liu, C.: Partial regularity of the nonlinear dissipative systems modeling the fow of liquid crystals. Discrete Contin. Dyn. Syst. 2, 1–22 (1996)zbMATHGoogle Scholar
  16. 16.
    Lin, F.-H., Liu, C.: Existence of solutions for the Ericksen-Leslie system. Arch. Ration. Mech. Anal. 154, 135–156 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lin, F.-H., Lin, J., Wang, C.: Liquid crystal flows in two dimensions. Arch. Ration. Mech. Anal. 197, 297–336 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lin, F.-H., Wang, C.: On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. Chin. Ann. Math. Ser. B 31, 921–938 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Struwe, M.: On the evolution of harmonic maps of Riemannian surfaces. Commun. Math. Helv. 60, 558–581 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Struwe, M.: The existence of surfaces of constant mean curvature with free boundaries. Acta Math. 160(1–2), 19–64 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schoen, R., Uhlenbeck, K.: Regularity of minimizing harmonic maps into the sphere. Invent. Math. 78, 89–100 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, C.: Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data. Arch. Ration. Mech. Anal. 200, 1–19 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wu, H., Xu, X., Liu, C.: On the general Ericksen Leslie system: Parodi’s relation, well-posedness and stability. Arch. Ration. Mech. Anal. 208, 59–107 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, W., Zhang, P., Zhang, Z.: Well-posedness of the Ericksen–Leslie system (2012) [arxiv: 1208.6107v1[math.AP]]Google Scholar
  25. 25.
    Xu, X., Zhang, Z.: Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows. J. Differ. Equ. 252, 1169–1181 (2012)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang UniversityHangzhouPeople’s Republic of China
  2. 2.School of Mathematical SciencesDalian University of TechnologyDalianPeople’s Republic of China
  3. 3.The Institute of Mathematical SciencesThe Chinese University of Hong KongShatin, N.T.Hong Kong

Personalised recommendations