Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids Read chapter Read first chapter

2018 | OriginalPaper | Chapter

23. Modeling and Analysis of the Ericksen-Leslie Equations for Nematic Liquid Crystal Flows

Authors : Matthias Hieber, Jan W. Prüss

Published in: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This survey article discusses various aspects of modeling and analysis of the Ericksen-Leslie equations describing nematic liquid crystal flow both in the isothermal and non-isothermal situation. Of special interest is the development of thermodynamically consistent Ericksen-Leslie models in the general situation based on the entropy principle. The full analytical understanding of the dynamics of the underlying system is then based on this principle and gives rise to a rather complete understanding of the dynamics of this system. Furthermore, well-posedness and long-time behavior results in the weak and strong sense are described for the general Ericksen-Leslie system and various simplifications, ranging from the simplified and penalized model, with or without stretching terms, to the case of general Leslie stress both for incompressible and compressible fluids.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Equations for Viscoelastic Fluids
next chapter Classical Well-Posedness of Free Boundary Problems in Viscous Incompressible Fluid Mechanics
Literature
1.
R. Atkin, T. Sluchin, I.W. Stewart, Reflections on the life and work of Frank Matthews Leslie. J. Non-Newtonian. Fluid Mech. 119, 7–23 (2004)CrossRef
2.
J. Ball, A. Zarnescu, Orientability and energy minimization in liquid crystal models. Arch. Ration. Mech. Anal. 202, 493–535 (2011)MathSciNetCrossRef
3.
P. Biscari, A. DiCarlo, S. Turzi, Liquid relaxation: a new Parodi-like relation for nematic liquid crystals. arXiv:1509.09022v1
4.
5.
L. Cafarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)MathSciNetCrossRef
6.
C. Cavaterra, E. Rocca, H. Wu, Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows. J. Differ. Equ. 255, 24–57 (2013)MathSciNetCrossRef
7.
S. Chandrasekhar, Liquid Crystals (Cambridge University Press, Cambridge, 1992)CrossRef
8.
K. Chang, W, Ding, R. Ye, Finite time blow up of th heat flow of harmonic maps from surfaces. J. Differ. Geom. 36, 507–515 (1992)MathSciNetCrossRef
9.
D. Coutand, S. Shkoller, Well-posedness of the full Ericksen-Leslie model of nematic liquid crystals. C. R. Acad. Sci. Paris Sér. I Math. 333, 919–924 (2001)MathSciNetCrossRef
10.
M. Dai, M. Schonbek, Asymptotic behaviour of solutions to the liquid crystal system in \(H^{m}(\mathbb{R}^{3})\). SIAM J. Math. Anal. 46, 3131–3150 (2014)MathSciNetCrossRef
11.
F. De Anna, On the dynamics of some complex fluids, Ph.D. Thesis, Université de Bordeaux, 2016
12.
P.G. DeGennes, The Physics of Liquid Crystals (Oxford University Press, Oxford, 1974)
13.
P.G. DeGennes, J. Prost, The Physics of Liquid Crystals (Oxford University Press, Oxford, 1995)
14.
R. Denk, M. Hieber, J. Prüss, \(\mathbb{R}\)-boundedness, Fourier multipliersand problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166 (2003)
15.
, M. Doi, S. Edwards, The Theory of Polymer Dynamics (Oxford University Press, Oxford, 1986)
16.
J.L. Ericksen, Conservation laws fro liquid crystals. Trans. Soc. Rheol. 5, 22–34 (1961)
17.
J.L. Ericksen, Hydrostatic theory of liquid crystals. Arch. Ration. Mech. Anal. 9, 371–378 (1962)MathSciNetMATH
18.
J.L. Ericksen, D. Kinderlehrer (eds.), Theory and Applications of Liquid Crystals. The IMA Volumes in Mathematics and its Applications, vol. 5 (Springer, New York, 1987)MATH
19.
J. Fan, J. Li, Regularity criteria for the strong solution to the Ericksen-Leslie system in \(\mathbb{R}^{3}\). J. Math. Anal. Appl. 125, 695–703 (2015)MathSciNetCrossRef
20.
E. Feireisl, E. Rocca, G. Schimperna, On a non-isothermal model for nematic liquid crystals. Nonlinearity 24, 243–257 (2011)MathSciNetCrossRef
21.
E. Feireisl, M. Frémond, E. Rocca, G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals. Arch. Ration. Mech. Anal. 205, 651–672 (2012)MathSciNetCrossRef
22.
E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Evolution of non-isothermal Landau-de Gemmes nematic liquid crystal flows with singular potential. arXiv:1207.1643
23.
E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Nonisothermal nematic liquid crystal flows with Ball-Majumdar free energy. arXiv:1310.8474
24.
F.C. Frank, On the theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28 (1958)CrossRef
25.
J. Gao, Q. Tao, Z. Yao, Aysmptotic beaviour of solution to the incompressible nematic liquid crystal flow in \(\mathbb{R}^{3}\). arXiv:1412.0498v2
26.
G. Gray, Molecular Structure and Properties of Liquid Crystals (Academic Press, London/New York, 1962)
27.
J. Han, Y. Lou, W. Wang, P. Zhang, Z. Zhang, From microscopic theory to macroscopic theory: a systematic study on modeling of liquid crystals. Arch. Ration. Mech. Anal. 215, 741–809 (2015)MathSciNetCrossRef
28.
R. Hardt, D. Kinderlehrer, F. Lin, Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105, 547–570 (1986)MathSciNetCrossRef
29.
M. Hieber, M. Nesensohn, J. Prüss, K. Schade, Dynamics of nematic liquid crystals: the quasilinear approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 397–408 (2016)MathSciNetCrossRef
30.
M. Hieber, J. Prüss, Thermodynamic Consistent Modeling and Analysis of Nematic Liquid Crystal Flows. Springer Proceedings in Mathematics & Statistics (2016, to appear)CrossRef
31.
M. Hieber, J. Prüss, Dynamics of nematic liquid crystal flows I: general isotropic incompressible fluids. Math. Annalen, in press
32.
M. Hieber, J. Prüss, Dynamics of nematic liquid crystal flows II: general isotropic compressible fluids. Submitted
33.
M. Hieber, J. Saal, The Stokes equation in the L p -setting: well-posedness and regularity properties, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, ed. by Y. Giga, A. Novotny (Springer, Switzerland)
34.
Y. Hinemann, A survey of results and open problems for the hydrodynamic flow of nematic liquid crystals. Elect. J. Differ. Equ. Conf. 21, 149–172 (2014)MathSciNet
35.
Y. Hinemann, C.Y. Wang, Well-posedness of nematic liquid crystal flow in \(L_{uloc}^{3}(\mathbb{R}^{3})\). Arch. Ration. Mech. Anal. 210, 177–218 (2013)MathSciNetCrossRef
36.
M. Hong, J. Li, Z. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in \(\mathbb{R}^{3}\). Commun. Partial Differ. Equ. 39, 1284–1328 (2014)MathSciNetCrossRef
37.
X. Hu, D. Wang, Global solutions to the three-dimensional incompressible flow of liquid crystals. Commun. Math. Phys. 296, 861–880 (2010)MathSciNetCrossRef
38.
X. Hu, H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals. SIAM J. Math. Anal. 45, 2678–2699 (2013)MathSciNetCrossRef
39.
T. Huang, F. Lin, C. Liu, C. Wang, Finite time singularities of the nematic liquid crystal flow in dimension three. arXiv:1504.01080v1
40.
J. Huang, F. Lin, C. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in \(\mathbb{R}^{2}\). Commun. Math. Phys. 331, 805–850 (2014)MathSciNetCrossRef
41.
T. Huang, C. Wang, Blow up criterion for nematic liquid crystal flow. Commun. Partial Differ. Equ. 37, 875–884 (2012)MathSciNetCrossRef
42.
T. Huang, C. Wang, H. Wen, Strong solutions for compressible nematic liquid crystal flow. J. Differ. Equ. 252, 2222–2265 (2012)MathSciNetCrossRef
43.
T. Huang, C. Wang, H. Wen, Blow up criterion for compressible nematic liquid crystal flows in dimension three. Arch. Ration. Mech. Anal. 204, 285–311 (2012)MathSciNetCrossRef
44.
F. Jiang, J. Song, D. Wang, On multidimensional compressible flows for nematic liquid crystals with large initial energy in a bounded domain. J. Funct. Anal. 265, 1711–1756 (2013)CrossRef
45.
M. Köhne, J. Prüss, M. Wilke, On quasilinear parabolic evolution equations in L p -spaces. J. Evol. Equ. 10, 443–463 (2010)MathSciNetCrossRef
46.
J. LeCrone, J. Prüss, M. Wilke, On quasilinear parabolic evolution equations in weighted L p -spaces II. J. Evol. Equ. 14, 509–533 (2014)MathSciNetCrossRef
47.
O. Lehmann, Über fliessende Krystalle. Z. Physik. Chemie 4, 462–472 (1889)
48.
49.
50.
J. Li, Global strong solutions to inhomogeneous incompressible nematic liquid crystal flow. Methods Appl. Anal. 22, 201–220 (2015)MathSciNetMATH
51.
J. Li, Z. Xu, J. Zhang, Global well-posedness with large oscillations and vacuum to the three dimensional equations of compressible liquid crystal flows. arXiv:1204:4966v1
52.
X. Li, D. Wang, Global solution to the incompressible flow of liquid crystals. J. Differ. Equ. 252, 745–767 (2012)MathSciNetCrossRef
53.
X. Li, D. Wang, Global strong solution to the density dependent incompressible flow of liquid crystals. Trans. Am. Math. Soc. 367, 2301–2338 (2015)MathSciNetCrossRef
54.
F. Lin, Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena. Commun. Pure Appl. Math. 42, 789–814 (1989)MathSciNetCrossRef
55.
F. Lin, On nematic liquid crystals with variable degree of freedom. Commun. Pure Appl. Math. 44, 453–468 (1991)CrossRef
56.
57.
F. Lin, Ch. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48, 501–537 (1995)MathSciNetCrossRef
58.
F. Lin, Ch. Liu, Partial regularities of the nonlinear dissipative system modeling the flow of liquid crystals. Discret. Contin. Dyn. Syst. 2, 1–22 (1996)
59.
F. Lin, Ch. Liu, Existence of solutions for the Ericksen-Leslie system. Arch. Ration. Mech. Anal. 154, 135–156 (2000)MathSciNetCrossRef
60.
61.
62.
F. Lin, C. Wang, The Analysis of Harmonic Maps and Their Heat Flows (World Scientific, Singapore, 2008)CrossRef
63.
F. Lin, C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three. arXiv:1408.4146v1
64.
F. Lin, C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals. Philos. Trans. R. Soc. Lon. Ser. A, Math. Phys. Eng. Sci 372, 20130361 (2014)MathSciNetCrossRef
65.
H. Liu, H. Zhang, P. Zhang, Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential. Commun. Math. Sci. 3, 201–218 (2005)MathSciNetCrossRef
66.
W. Ma, H. Gong, J. Li, Global strong solutions to incompressible Ericksen-Leslie system in \(\mathbb{R}^{3}\). Nonlinear Anal. 109, 230–235 (2014)MathSciNetCrossRef
67.
A. Majumdar, A. Zarnescu, Landau de Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond. Arch. Ration. Mech. Anal. 196, 227–280 (2010)MathSciNetCrossRef
68.
W. Maier, A. Saupe, Eine einfache molekulare Theorie des nematischen kristallflüsigen Zustandes. Z. Naturf. A 13, 564–566 (1958)
69.
I. Müller, Thermodynamics. Interaction of Mechanics and Mathematics (Pitman, Boston, 1985)MATH
70.
71.
L. Onsager, The effects of shape on the interaction of colloidal particles. Ann. N.Y. Acad. Sci. 51, 627–659 (1949)CrossRef
72.
C.W. Oseen, The theory of liquid crystals. Trans. Faraday Soc. 29, 883–899 (1933)CrossRef
73.
O. Parodi, Stress tensor for a nematic liquid crystal. J. Phys. 31, 581–584 (1970)CrossRef
74.
J. Prüss, Maximal regularity for evolution equations in L p -spaces. Conf. Semin. Mat. Univ. Bari 285, 1–39 (2002, 2003)
75.
J. Prüss, G. Simonett, Maximal regularity for evolution equations in weighted L p -spaces. Arch. Math. (Basel) 82, 415–431 (2004)MathSciNetCrossRef
76.
J. Prüss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics Birkhäuser, 2016CrossRef
77.
J. Prüss, G. Simonett, R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems. J. Differ. Equ. 246, 3902–3931 (2009)MathSciNetCrossRef
78.
K. Schade, Y. Shibata, On strong dynamics of compressible nematic liquid crystals. SIAM J. Math. Anal. 47, 3963–3992 (2015)MathSciNetCrossRef
79.
B. Seguin, E. Fried, Statistical foundations of liquid crystal theory. I. Discrete systems of rod-like molecules. Arch. Ration. Mech. Anal. 206, 1039–1072 (2012)CrossRef
80.
B. Seguin, E. Fried, Statistical foundations of liquid crystal theory. II. Macroscopic balance laws. Arch. Ration. Mech. Anal. 207, 1–37 (2013)MATH
81.
I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals. The Liquid Crystal Book Series (Taylor and Francis, London/New York, 2004)
82.
H. Sun, Ch. Liu, On energetic variational approaches in modeling the nematic liquid crystal flows. Discret Cont. Dyn. Syst. 23, 455–475 (2009)MathSciNetCrossRef
83.
E.G. Virga, Variational Theories for Liquid Crystals (Chapman-Hall, London, 1994)CrossRef
84.
H. Wu, X. Xu, Ch. Liu, On the general Ericksen-Leslie system: Parodi’s relation, well-posedness and stability. Arch. Ration. Mech. Anal. 208, 59–107 (2013)MathSciNetCrossRef
85.
C. Wang, 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)MathSciNetCrossRef
86.
W. Wang, P. Zhang, Z. Zhang, Well-posedness of the Ericksen-Leslie system. Arch. Ration. Mech. Anal. 210, 837–855 (2013)MathSciNetCrossRef
87.
W. Wang, P. Zhang, Z. Zhang, The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation. Commun. Pure Appl. Math. 68, 1326–1398 (2015)MathSciNetCrossRef
88.
W. Wang, P. Zhang, Z. Zhang, Rigorous derivation from Landau-De Gennes theory to Ericksen-Leslie theory. SIAM J. Math. Anal. 47, 127–158 (2015)MathSciNetCrossRef
89.
H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discret. Contin. Dyn. Syst. 26, 379–396 (2010)MathSciNetCrossRef
90.
H. Zocher, The effect of a magnetic field on the nematic state. Trans. Faraday Soc. 29, 945–957 (1933)CrossRef
Metadata
Title
Modeling and Analysis of the Ericksen-Leslie Equations for Nematic Liquid Crystal Flows
Authors
Matthias Hieber
Jan W. Prüss
Copyright Year
2018
Book
Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Print ISBN: 978-3-319-13343-0

Electronic ISBN: 978-3-319-13344-7

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-13344-7_26

Premium Partner

    Image Credits