June  2022, 15(3): 417-465. doi: 10.3934/krm.2021047

From kinetic to fluid models of liquid crystals by the moment method

1. 

Institut de Mathématiques de Toulouse; UMR5219, Université de Toulouse; CNRS UPS, F-31062 Toulouse Cedex 9, France

2. 

CEREMADE, CNRS, Université Paris-Dauphine, Université PSL, 75016 Paris, France

3. 

CNRS, Université de Poitiers, UMR 7348, Laboratoire de Mathématiques et Applications (LMA), 86000 Poitiers, France

4. 

Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA

* Corresponding author: Pierre Degond

In memory of Bob Glassey
PD is a visiting professor of the Department of Mathematics, Imperial College London, UK
AF acknowledges support from the Project EFI ANR-17-CE40-0030 of the French National Research Agency
JGL acknowledges support from the Department of Mathematics, Imperial College London, under Nelder Fellowship award and the National Science Foundation under Grants DMS-1812573 and DMS-2106988

Received  June 2021 Revised  October 2021 Published  June 2022 Early access  January 2022

This paper deals with the convergence of the Doi-Navier-Stokes model of liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number tending to zero. While the literature has investigated this problem by means of the Hilbert expansion method, we develop the moment method, i.e. a method that exploits conservation relations obeyed by the collision operator. These are non-classical conservation relations which are associated with a new concept, that of Generalized Collision Invariant (GCI). In this paper, we develop the GCI concept and relate it to geometrical and analytical structures of the collision operator. Then, the derivation of the limit model using the GCI is performed in an arbitrary number of spatial dimensions and with non-constant and non-uniform polymer density. This non-uniformity generates new terms in the Ericksen-Leslie model.

Citation: Pierre Degond, Amic Frouvelle, Jian-Guo Liu. From kinetic to fluid models of liquid crystals by the moment method. Kinetic and Related Models, 2022, 15 (3) : 417-465. doi: 10.3934/krm.2021047
References:
[1]

J. M. Ball, Axisymmetry of critical points for the Onsager functional, Phil. Trans. R. Soc. A., 379 (2021), Paper No. 20200110, 13 pp.

[2]

J. M. Ball, Mathematics and liquid crystals, Molecular Crystals and Liquid Crystals, 647 (2017), 1-27.  doi: 10.1080/15421406.2017.1289425.

[3]

J. M. Ball, E. Feireisl and F. Otto, Mathematical Thermodynamics of Complex Fluids, Lecture notes in Mathematics 2200, Springer, 2017.

[4]

C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations Ⅱ convergence proofs for the Boltzmann equation, Commun. Pure Appl. Math., 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.

[5]

C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. Ⅰ. Formal derivations, J. Stat. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.

[6]

L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gas-molekülen, Sitzungsberichte Akad. Wiss., Vienna, part Ⅱ, 66 (1872), 275-370.  doi: 10.1017/CBO9781139381420.023.

[7]

R. E. Caflisch, The fluid dynamical limit of the nonlinear Boltzmann equation, Commun. Pure Appl. Math., 33 (1980), 651-666.  doi: 10.1002/cpa.3160330506.

[8]

M. C. CaldererD. GolovatyF.-H. Lin and C. Liu, Time evolution of nematic liquid crystals with variable degree of orientation, SIAM J. Math. Anal., 33 (2002), 1033-1047.  doi: 10.1137/S0036141099362086.

[9]

M. C. Calderer and C. Liu, Liquid crystal flow: Dynamic and static configurations, SIAM J. Appl. Math., 60 (2000), 1925-1949.  doi: 10.1137/S0036139998336249.

[10]

C. Cercinani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer, 2013.

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S. Chapman, The kinetic theory of simple and composite gases: Viscosity, thermal conduction and diffusion, Proc. Roy. Soc., (London) A93 (1916/17), 1–20.

[12]

B. CharbonneauP. CharbonneauY. JinG. Parisi and F. Zamponi, Dimensional dependence of the Stokes–Einstein relation and its violation, The Journal of Chemical Physics, 139 (2013), 164502.  doi: 10.1063/1.4825177.

[13]

X. Chen and J.-G. Liu, Global weak entropy solution to Doi-Saintillan-Shelley model for active and passive rod-like and ellipsoidal particle suspensions, J. Diff. Eqs., 254 (2013), 2764-2802.  doi: 10.1016/j.jde.2013.01.005.

[14]

P. ConstantinI. G. Kevrekidis and E. S. Titi, Asymptotic states of a Smoluchowski equation, Arch. Ration. Mech. Anal., 174 (2004), 365-384.  doi: 10.1007/s00205-004-0331-8.

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P. Constantin and J. Vukadinovic, Note on the number of steady states for a two-dimensional Smoluchowski equation, Nonlinearity, 18 (2005), 441-443.  doi: 10.1088/0951-7715/18/1/022.

[16] P.-G. de Gennes and J. Prost, The Physics of Liquid Crystals, Oxford Univ. Press, 1993. 
[17]

P. DegondA. DiezA. Frouvelle and S. Merino-Aceituno, Phase transitions and macroscopic limits in a BGK model of body-attitude coordination, J. Nonlinear Sci., 30 (2020), 2671-2736.  doi: 10.1007/s00332-020-09632-x.

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P. DegondG. DimarcoT. B. N. Mac and N. Wang, Macroscopic models of collective motion with repulsion, Commun. Math. Sci., 13 (2015), 1615-1638.  doi: 10.4310/CMS.2015.v13.n6.a12.

[19]

P. DegondA. FrouvelleS. Merino-Aceituno and A. Trescases, Quaternions in collective dynamics, Multiscale Model. Simul., 16 (2018), 28-77.  doi: 10.1137/17M1135207.

[20]

P. DegondJ.-G. LiuS. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114.  doi: 10.4310/MAA.2013.v20.n2.a1.

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P. Degond and S. Merino-Aceituno, Nematic alignment of self-propelled particles: From particle to macroscopic dynamics, Math. Models Methods Appl. Sci., 30 (2020), 1935-1986.  doi: 10.1142/S021820252040014X.

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P. Degond, S. Merino-Aceituno, F. Vergnet and H. Yu, Coupled self-organized hydrodynamics and Stokes models for suspensions of active particles, J. Math. Fluid Mech., 21 (2019), Paper No. 6, 36 pp. doi: 10.1007/s00021-019-0406-9.

[23]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.

[24] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford Univ. Press, 1986. 
[25]

W. E and P. Zhang, A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit, Methods Appl. Anal., 13 (2006), 181-198.  doi: 10.4310/MAA.2006.v13.n2.a5.

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D. Enskog, Kinetische Theorie der Vorgänge in Mässig Verdünntent Gasen, 1, in Allgemeiner Teil, Almqvist & Wiksell, Uppsala, 1917.

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J. L. Ericksen, Liquid crystals with variable degree of orientation, Arch. Ration. Mech. Anal., 113 (1990), 97-120.  doi: 10.1007/BF00380413.

[28]

I. Fatkullin and V. Slastikov, A note on the Onsager model of nematic phase transitions, Commun. Math. Sci., 3 (2005), 21-26.  doi: 10.4310/CMS.2005.v3.n1.a2.

[29]

I. Fatkullin and V. Slastikov, Critical points of the Onsager functional on a sphere, Nonlinearity, 18 (2005) 2565–2580. doi: 10.1088/0951-7715/18/6/008.

[30]

A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Models Methods Appl. Sci., 22 (2012), 1250011, 40 pp. doi: 10.1142/S021820251250011X.

[31]

A. Frouvelle, Body-attitude alignment: First order phase transition, link with rodlike polymers through quaternions, and stability, arXiv: 2011.14891, 2021.

[32]

Y. Giga and A. Novotnỳ (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, 2018. doi: 10.1007/978-3-319-13344-7.

[33]

H. Grad, On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.

[34]

D. Hilbert, Begründung der kinetischen Gastheorie, Mathematische Annalen, 72 (1912), 562-577.  doi: 10.1007/BF01456676.

[35]

J. HuangF. Lin and C. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in ${\mathbb R}^2$, Comm. Math. Phys., 331 (2014), 805-850.  doi: 10.1007/s00220-014-2079-9.

[36]

G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, Proceedings of the Royal Society A, 102 (1922), 161-179. 

[37]

N. Kuzuu and M. Doi, Constitutive equation for nematic liquid crystals under weak velocity gradient derived from a molecular kinetic equation, Journal of the Physical Society of Japan, 52 (1983), 3486-3494.  doi: 10.1143/JPSJ.52.3486.

[38]

F. M. Leslie, Some constitutive equations for anisotropic fluids, Quart. J. Mech. Appl. Math., 19 (1966), 357-370.  doi: 10.1093/qjmam/19.3.357.

[39]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.

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F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814.  doi: 10.1002/cpa.3160420605.

[41]

F.-H. Lin, On nematic liquid crystals with variable degree of orientation, Comm. Pure Appl. Math., 44 (1991), 453-468.  doi: 10.1002/cpa.3160440404.

[42]

F.-H. LinJ. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.  doi: 10.1007/s00205-009-0278-x.

[43]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503.

[44]

F.-H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 154 (2000), 135-156.  doi: 10.1007/s002050000102.

[45]

F.-H. LinC. Liu and P. Zhang, On a micro-macro model for polymeric fluids near equilibrium, Comm. Pure Appl. Math., 60 (2007), 838-866.  doi: 10.1002/cpa.20159.

[46]

F.-H. Lin and T. Zhang, Global small solutions to a complex fluid model in three dimensional, Arch. Ration. Mech. Anal., 216 (2015), 905-920.  doi: 10.1007/s00205-014-0822-1.

[47]

H. Liu, Global orientation dynamics for liquid crystalline polymers, Phys. D, 228 (2007), 122-129.  doi: 10.1016/j.physd.2007.02.008.

[48]

H. LiuH. Zhang and P. Zhang, Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential, Commun. Math. Sci., 3 (2005), 201-218.  doi: 10.4310/CMS.2005.v3.n2.a7.

[49]

C. LuoH. Zhang and P. Zhang, The structure of equilibrium solutions of the one-dimensional Doi equation, Nonlinearity, 18 (2005), 379-389.  doi: 10.1088/0951-7715/18/1/018.

[50]

W. Maier and A. Saupe, A simple molecular statistical theory of the nematic crystalline-liquid phase, I Z Naturf. A, 14 (1959), 882-889. 

[51]

J. C. Maxwell, On the dynamical theory of gases, Philos. Trans. Roy. Soc. London, 157 (1867), 49-88. 

[52]

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana Univ. Math. J., 59 (2010), 1333-1360.  doi: 10.1512/iumj.2010.59.4128.

[53]

L. Onsager, The effects of shape on the interaction of colloidal particles, Ann. N. Y. Acad. Sci., 51 (1949), 627-659. 

[54]

F. Otto and A. E. Tzavaras, Continuity of velocity gradients in suspensions of rod-like molecules, Comm. Math. Phys., 277 (2008), 729-758.  doi: 10.1007/s00220-007-0373-5.

[55]

T. VicsekA. CziròkE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

[56]

H. Wang and P. J. Hoffman, A unified view on the rotational symmetry of equilibiria of nematic polymers, dipolar nematic polymers, and polymers in higher dimensional space, Commun. Math. Sci., 6 (2008), 949-974.  doi: 10.4310/CMS.2008.v6.n4.a8.

[57]

Q. WangW. EC. Liu and P.-W. Zhang, Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential, Phys. Rev. E, 65 (2002), 051504.  doi: 10.1103/PhysRevE.65.051504.

[58]

W. WangL. Zhang and P. Zhang, Modeling and computation of liquid crystals, Acta Numer., 30 (2021), 765-851.  doi: 10.1017/S0962492921000088.

[59]

W. WangP. Zhang and Z. Zhang, The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation, Comm. Pure Appl. Math., 68 (2015), 1326-1398.  doi: 10.1002/cpa.21549.

[60]

W. WangP. Zhang and Z. Zhang, Well-posedness of the Ericksen–Leslie system, Arch. Ration. Mech. Anal., 210 (2013), 837-855.  doi: 10.1007/s00205-013-0659-z.

[61]

H. Zhang and P. Zhang, Stable dynamic states at the nematic liquid crystals in weak shear flow, Phys. D, 232 (2007), 156-165.  doi: 10.1016/j.physd.2007.06.011.

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H. Zhang and P. Zhang, On the new multiscale rodlike model of polymeric fluids, SIAM J. Math. Anal., 40 (2008), 1246-1271.  doi: 10.1137/050640795.

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show all references

References:
[1]

J. M. Ball, Axisymmetry of critical points for the Onsager functional, Phil. Trans. R. Soc. A., 379 (2021), Paper No. 20200110, 13 pp.

[2]

J. M. Ball, Mathematics and liquid crystals, Molecular Crystals and Liquid Crystals, 647 (2017), 1-27.  doi: 10.1080/15421406.2017.1289425.

[3]

J. M. Ball, E. Feireisl and F. Otto, Mathematical Thermodynamics of Complex Fluids, Lecture notes in Mathematics 2200, Springer, 2017.

[4]

C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations Ⅱ convergence proofs for the Boltzmann equation, Commun. Pure Appl. Math., 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.

[5]

C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. Ⅰ. Formal derivations, J. Stat. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.

[6]

L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gas-molekülen, Sitzungsberichte Akad. Wiss., Vienna, part Ⅱ, 66 (1872), 275-370.  doi: 10.1017/CBO9781139381420.023.

[7]

R. E. Caflisch, The fluid dynamical limit of the nonlinear Boltzmann equation, Commun. Pure Appl. Math., 33 (1980), 651-666.  doi: 10.1002/cpa.3160330506.

[8]

M. C. CaldererD. GolovatyF.-H. Lin and C. Liu, Time evolution of nematic liquid crystals with variable degree of orientation, SIAM J. Math. Anal., 33 (2002), 1033-1047.  doi: 10.1137/S0036141099362086.

[9]

M. C. Calderer and C. Liu, Liquid crystal flow: Dynamic and static configurations, SIAM J. Appl. Math., 60 (2000), 1925-1949.  doi: 10.1137/S0036139998336249.

[10]

C. Cercinani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer, 2013.

[11]

S. Chapman, The kinetic theory of simple and composite gases: Viscosity, thermal conduction and diffusion, Proc. Roy. Soc., (London) A93 (1916/17), 1–20.

[12]

B. CharbonneauP. CharbonneauY. JinG. Parisi and F. Zamponi, Dimensional dependence of the Stokes–Einstein relation and its violation, The Journal of Chemical Physics, 139 (2013), 164502.  doi: 10.1063/1.4825177.

[13]

X. Chen and J.-G. Liu, Global weak entropy solution to Doi-Saintillan-Shelley model for active and passive rod-like and ellipsoidal particle suspensions, J. Diff. Eqs., 254 (2013), 2764-2802.  doi: 10.1016/j.jde.2013.01.005.

[14]

P. ConstantinI. G. Kevrekidis and E. S. Titi, Asymptotic states of a Smoluchowski equation, Arch. Ration. Mech. Anal., 174 (2004), 365-384.  doi: 10.1007/s00205-004-0331-8.

[15]

P. Constantin and J. Vukadinovic, Note on the number of steady states for a two-dimensional Smoluchowski equation, Nonlinearity, 18 (2005), 441-443.  doi: 10.1088/0951-7715/18/1/022.

[16] P.-G. de Gennes and J. Prost, The Physics of Liquid Crystals, Oxford Univ. Press, 1993. 
[17]

P. DegondA. DiezA. Frouvelle and S. Merino-Aceituno, Phase transitions and macroscopic limits in a BGK model of body-attitude coordination, J. Nonlinear Sci., 30 (2020), 2671-2736.  doi: 10.1007/s00332-020-09632-x.

[18]

P. DegondG. DimarcoT. B. N. Mac and N. Wang, Macroscopic models of collective motion with repulsion, Commun. Math. Sci., 13 (2015), 1615-1638.  doi: 10.4310/CMS.2015.v13.n6.a12.

[19]

P. DegondA. FrouvelleS. Merino-Aceituno and A. Trescases, Quaternions in collective dynamics, Multiscale Model. Simul., 16 (2018), 28-77.  doi: 10.1137/17M1135207.

[20]

P. DegondJ.-G. LiuS. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114.  doi: 10.4310/MAA.2013.v20.n2.a1.

[21]

P. Degond and S. Merino-Aceituno, Nematic alignment of self-propelled particles: From particle to macroscopic dynamics, Math. Models Methods Appl. Sci., 30 (2020), 1935-1986.  doi: 10.1142/S021820252040014X.

[22]

P. Degond, S. Merino-Aceituno, F. Vergnet and H. Yu, Coupled self-organized hydrodynamics and Stokes models for suspensions of active particles, J. Math. Fluid Mech., 21 (2019), Paper No. 6, 36 pp. doi: 10.1007/s00021-019-0406-9.

[23]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.

[24] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford Univ. Press, 1986. 
[25]

W. E and P. Zhang, A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit, Methods Appl. Anal., 13 (2006), 181-198.  doi: 10.4310/MAA.2006.v13.n2.a5.

[26]

D. Enskog, Kinetische Theorie der Vorgänge in Mässig Verdünntent Gasen, 1, in Allgemeiner Teil, Almqvist & Wiksell, Uppsala, 1917.

[27]

J. L. Ericksen, Liquid crystals with variable degree of orientation, Arch. Ration. Mech. Anal., 113 (1990), 97-120.  doi: 10.1007/BF00380413.

[28]

I. Fatkullin and V. Slastikov, A note on the Onsager model of nematic phase transitions, Commun. Math. Sci., 3 (2005), 21-26.  doi: 10.4310/CMS.2005.v3.n1.a2.

[29]

I. Fatkullin and V. Slastikov, Critical points of the Onsager functional on a sphere, Nonlinearity, 18 (2005) 2565–2580. doi: 10.1088/0951-7715/18/6/008.

[30]

A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Models Methods Appl. Sci., 22 (2012), 1250011, 40 pp. doi: 10.1142/S021820251250011X.

[31]

A. Frouvelle, Body-attitude alignment: First order phase transition, link with rodlike polymers through quaternions, and stability, arXiv: 2011.14891, 2021.

[32]

Y. Giga and A. Novotnỳ (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, 2018. doi: 10.1007/978-3-319-13344-7.

[33]

H. Grad, On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.

[34]

D. Hilbert, Begründung der kinetischen Gastheorie, Mathematische Annalen, 72 (1912), 562-577.  doi: 10.1007/BF01456676.

[35]

J. HuangF. Lin and C. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in ${\mathbb R}^2$, Comm. Math. Phys., 331 (2014), 805-850.  doi: 10.1007/s00220-014-2079-9.

[36]

G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, Proceedings of the Royal Society A, 102 (1922), 161-179. 

[37]

N. Kuzuu and M. Doi, Constitutive equation for nematic liquid crystals under weak velocity gradient derived from a molecular kinetic equation, Journal of the Physical Society of Japan, 52 (1983), 3486-3494.  doi: 10.1143/JPSJ.52.3486.

[38]

F. M. Leslie, Some constitutive equations for anisotropic fluids, Quart. J. Mech. Appl. Math., 19 (1966), 357-370.  doi: 10.1093/qjmam/19.3.357.

[39]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.

[40]

F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814.  doi: 10.1002/cpa.3160420605.

[41]

F.-H. Lin, On nematic liquid crystals with variable degree of orientation, Comm. Pure Appl. Math., 44 (1991), 453-468.  doi: 10.1002/cpa.3160440404.

[42]

F.-H. LinJ. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.  doi: 10.1007/s00205-009-0278-x.

[43]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503.

[44]

F.-H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 154 (2000), 135-156.  doi: 10.1007/s002050000102.

[45]

F.-H. LinC. Liu and P. Zhang, On a micro-macro model for polymeric fluids near equilibrium, Comm. Pure Appl. Math., 60 (2007), 838-866.  doi: 10.1002/cpa.20159.

[46]

F.-H. Lin and T. Zhang, Global small solutions to a complex fluid model in three dimensional, Arch. Ration. Mech. Anal., 216 (2015), 905-920.  doi: 10.1007/s00205-014-0822-1.

[47]

H. Liu, Global orientation dynamics for liquid crystalline polymers, Phys. D, 228 (2007), 122-129.  doi: 10.1016/j.physd.2007.02.008.

[48]

H. LiuH. Zhang and P. Zhang, Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential, Commun. Math. Sci., 3 (2005), 201-218.  doi: 10.4310/CMS.2005.v3.n2.a7.

[49]

C. LuoH. Zhang and P. Zhang, The structure of equilibrium solutions of the one-dimensional Doi equation, Nonlinearity, 18 (2005), 379-389.  doi: 10.1088/0951-7715/18/1/018.

[50]

W. Maier and A. Saupe, A simple molecular statistical theory of the nematic crystalline-liquid phase, I Z Naturf. A, 14 (1959), 882-889. 

[51]

J. C. Maxwell, On the dynamical theory of gases, Philos. Trans. Roy. Soc. London, 157 (1867), 49-88. 

[52]

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana Univ. Math. J., 59 (2010), 1333-1360.  doi: 10.1512/iumj.2010.59.4128.

[53]

L. Onsager, The effects of shape on the interaction of colloidal particles, Ann. N. Y. Acad. Sci., 51 (1949), 627-659. 

[54]

F. Otto and A. E. Tzavaras, Continuity of velocity gradients in suspensions of rod-like molecules, Comm. Math. Phys., 277 (2008), 729-758.  doi: 10.1007/s00220-007-0373-5.

[55]

T. VicsekA. CziròkE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

[56]

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Figure 1.  Graphical representation of the function $ \lambda \mapsto \rho^n(\lambda) $ (after [56]). (a) case $ n = 2 $. (b) case $ n \geq 3 $. The portions of the curves that correspond to stable equilibria are in blue, the unstable ones, in green.
Figure 2.  Graphical representation of Condition (101). The ambient three-dimensional space in the figure represents the flat space $ {\mathcal S}_0^n $ in which $ {\mathcal U}_0^n $ is an imbedded manifold represented by a surface. $ {\mathcal N} $ is a submanifold of $ {\mathcal U}_0^n $ depicted as the curvy blue line. It endows $ {\mathcal U}_0^n $ of a fiber bundle structure of base $ {\mathcal N} $. Let $ \Sigma \in {\mathcal U}_0^n $. It projects (in the bundle sense) onto $ A_\Omega \in {\mathcal N} $ and so, belongs to the fiber $ {\mathcal F}_\Omega $ represented by the curvy red line. The tangent space to $ {\mathcal N} $ at $ A_\Omega $, $ T_{A_\Omega} {\mathcal N} $ is represented by the magenta straight line. Its orthogonal $ (T_{A_\Omega} {\mathcal N})^\bot $ is the gray-shaded plane on the figure. It contains $ {\mathcal F}_\Omega $ by virtue of Lemma 5.6 (ii). Then, condition (101) means that the GCI associated with $ (\eta,\Sigma) $ are the functions $ \psi $ that cancel $ L_{\eta \Sigma} f $ for all $ f $ whose Q-tensor $ Q_f $ (represented by the point Q on the figure) belongs to $ (T_{A_\Omega} {\mathcal N})^\bot $
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