Skip to main content
SpringerLink
Log in

Non-isothermal General Ericksen–Leslie System: Derivation, Analysis and Thermodynamic Consistency

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We derive a model describing the evolution of a nematic liquid–crystal material under the action of thermal effects. The first and second laws of thermodynamics lead to an extension of the general Ericksen–Leslie system where the Leslie stress tensor and the Oseen–Frank energy density are considered in their general forms. The work postulate proposed by Ericksen–Leslie is traduced in terms of entropy production.We finally analyze the global-in-time well-posedness of the system for small initial data in the framework of Besov spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bahouri H., Chemin J.-Y., Danchin R.: Fourier Analysis and Nonlinear Partial Differential Equations,Grundlehren derMathematis chen Wissenschaften (Fundamental Principles of Mathematical Sciences), Vol. 343. Springer, Heidelberg (2011)

    Google Scholar 

  2. Cavaterra C., Rocca E., Wu H.: Global weak solution and blow–up criterion of the general Ericksen–Leslie system for nematic liquid crystal flows. J. Differ. Equ. 255, 1432–1807 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Danchin R.: Global existence in critical spaces for flows of compressible viscous and heat-conductive gases. Arch. Ration. Mech. Anal. 160, 1 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Danchin R.: Local theory in critical spaces for compressible viscous and heat conductive gases. Commun. Partial Differ. Equ. 26, 7–8 (2011) 2183–1233

    MathSciNet  Google Scholar 

  5. Danchin R., Mucha P. B.: A Lagrangian approach for the incompressible Navier–Stokes equations with variable density. Commun. Pure Appl. Math. 65, 1458–1480 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. De Anna F.: Global solvability of the inhomogeneous Ericksen–Leslie system with only bounded density. Anal. Appl. 0, 1–51 (2016)

    MathSciNet  Google Scholar 

  7. Ericksen J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 23–34 (1961)

    Article  MathSciNet  Google Scholar 

  8. Ericksen J.L.: Hydrostatic theory of liquid crystals. Arch. Rational Mech. Anal. 9, 371–378 (1962)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Feireisl E., Frémond M., Rocca E., Schimperna G.: A new approach to non isothermal models for nematic liquid crystals. Arch. Ration. Mech. Anal. 205, 651–672 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Feireisl E., Rocca E., Schimperna G.: On a non-isothermal model for the nematic liquid crystals. Nonlinearity, 24, 243–257 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Feireisl E., Rocca E., Schimperna G., Zarnescu A.: Nonisothermal nematic liquid crystal flows with the Ball–Majumdar free energy. Annali di Matematica Pura ed Applicata 194, 1269–1299 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Feireisl E., Rocca E., Schimperna G., Zarnescu A.: Evolution of non–isothermal Landau–de Gennes nematic liquid crystals flows with singular potential. Commun.Math. Sci. 12, 317–343 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Feireisl E.: On a non-isothermal model for the nematic liquid crystals. Nonlinearity, 24, 243–257 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Frank F.C.: On the theory of liquid crystals. Discuss. Faraday Soc. 25, 19 (1958)

    Article  Google Scholar 

  15. Hineman, J.L., Wang, C.: Well-posedness of nematic liquid crystal flow in \({{L}^{3}_{{\rm uloc}}(\mathbb{R}^{3}})\). Arch. Ration. Mech. Anal. 210, 210–177 (2013)

  16. Jeffery G.: The motion of ellipsolidal particles immersed in a viscous fluid. R. Soc. Proc. 102, 102–161 (1922)

    Article  Google Scholar 

  17. Liu C., Wu H., Xu X.: On the general Ericksen–Leslie system: Parodi’s relation, well–posedness and stability. Arch. Ration. Mech. Anal. 208, 59–107 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Málek, J., Pruša, V.: Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham, 1–70, 2016

  19. Hieber, M., Nesensohn, M., Prüss, J., Schade,K.: Dynamics of nematic liquid crystal flows: The quasilinear approach. Annales de l’institut Henri Poincare (C) Non Linear Analysis, 0 2014

  20. Hieber, M., Prüss, J.: Dynamics of the Ericksen–Leslie equations with general Leslie stress I: the incompressible isotropic case. Math. Ann., 1432–1807 2016

  21. Hieber, M., Prüss, J.: Modeling and Analysis of the Ericksen–Leslie Equations for Nematic Liquid Crystal Flows, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (Eds. Giga Y., Novotny A.) Springer (in press)

  22. Hieber, M., Prüss, J.: Thermodynamic consistent modeling and analysis of nematic liquid crystal flows. Proceedings in Mathematics and Statistics, Springer, 2016 (to appear)

  23. Huang J., Lin F.-H., Wang C.: Regularity and existence of global solutions to the Ericksen–Leslie system in \({\mathbb{R}^{2}}\) . Commun. Math. Phys. 331, 805–850 (2014)

    Article  ADS  MATH  Google Scholar 

  24. Leslie F.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal., 28(4), 265–283 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lin F.-H., Liu C.: Existence of solutions for the Ericksen–Leslie system. Arch. Ration. Mech. Anal. 154, 135–156 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lin F.-H., Liu C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48, 501–537 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lin F.-H, Wang C.: Global existence of weak solutions of the nematic liquid crystal flow in dimension three. Commun. Pure Appl. Math. 69, 1532–1571 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Onsager L.: Reciprocal relation in irreversible process I. Phys. Rev. 37, 405–426 (1931)

    Article  ADS  MATH  Google Scholar 

  29. Parodi O.: Stress tensor for a nematic liquid crystal. J. Phys. 31, 581–584 (1970)

    Article  Google Scholar 

  30. Sonnet A.M., Virga E.G.: Theory of Flow Phenomena in Liquid Crystals Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 343. Springer, New York (2012)

    Google Scholar 

  31. Stewart I.W.: The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Liquid Crystals Book Series. CRC Press, Boca Raton (2004)

    Google Scholar 

Download references

Acknowledgements

The authors express their sincere appreciation to Professor Marius Paicu and Professor Arghir Zarnescu for constructive suggestions and discussions. The work Progressed substantially at theDepartment ofMathematics of the Penn StateUniversity. We thank deeply the Department of Mathematics there for their generous support and for providing a stimulating environment in which to work. The work of the second author has been partially supported by the NSF (Grants DMS-1714401 and DMS-1412005).

Author information

Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, US

    Francesco De Anna

  2. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, US

    Chun Liu

Authors
  1. Francesco De Anna
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco De Anna.

Additional information

Communicated by F. Lin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

De Anna, F., Liu, C. Non-isothermal General Ericksen–Leslie System: Derivation, Analysis and Thermodynamic Consistency. Arch Rational Mech Anal 231, 637–717 (2019). https://doi.org/10.1007/s00205-018-1287-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-018-1287-4

Search

Navigation