November  2019, 24(11): 5785-5802. doi: 10.3934/dcdsb.2019106

A note on the stochastic Ericksen-Leslie equations for nematic liquid crystals

1. 

Department of Mathematics, University of York, Heslington Road, York YO10 5DD, UK

2. 

Department of Mathematics and Information Technology, Montanuniversität Leoben, Franz Josef Straẞe 18, 8700 Leoben, Austria

3. 

Department of Mathematics and Applied Mathematics, University of Pretoria, Lynwood Road, Pretoria 0002, South Africa, Current Address: Department of Mathematics, University of York, Heslington Road, York YO10 5DD, UK

* Corresponding author: Paul André Razafimandimby

This article is part of a project that is currently funded by the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No. 791735 "SELEs". Part of this work was written while P. A. Razafimandimby was at the University of Pretoria; he is grateful to the funding he received from the National Research Foundation South Africa (Grant Numbers 109355 and 112084). He is also grateful to the European Mathematical Society for the EMS-Simons for Africa-Collaborative research grant which enables him to visit Montanuniversität Leoben, Austria

Received  July 2018 Published  November 2019 Early access  June 2019

Fund Project: E. Hausenblas is supported by the FWF-Austrian Science Fund through the Stand-Alone grant number P28010.

In this note we prove the existence and uniqueness of local maximal smooth solution of the stochastic simplified Ericksen-Leslie systems modelling the dynamics of nematic liquid crystals under stochastic perturbations.

Citation: Zdzisław Brzeźniak, Erika Hausenblas, Paul André Razafimandimby. A note on the stochastic Ericksen-Leslie equations for nematic liquid crystals. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5785-5802. doi: 10.3934/dcdsb.2019106
References:
[1]

K. Atkinson and W. Han, Theoretical Numerical Analysis. A Functional Analysis Framework, Third edition. Volume 39 of Texts in Applied Mathematics, Springer, Dordrecht, 2009. doi: 10.1007/978-1-4419-0458-4.

[2]

R. BeckerX. Feng and A. Prohl, Finite element approximations of the Ericksen-Leslie model for nematic liquid crystal flow,, SIAM J. Numer. Anal., 46 (2008), 1704-1731.  doi: 10.1137/07068254X.

[3]

H. BessaihZ. Brzeźniak and A. Millet, Splitting up method for the 2D stochastic Navier-Stokes equations,, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014), 433-470.  doi: 10.1007/s40072-014-0041-7.

[4]

Z. BrzeźniakS. Cerrai and M. Freidlin, Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise,, Probab. Theory Related Fields, 162 (2015), 739-793.  doi: 10.1007/s00440-014-0584-6.

[5]

Z. Brzeźniak and K. D. Elworthy, Stochastic differential equations on Banach manifolds, Methods Funct. Anal. Topology, 6 (2000), 43-84. 

[6]

Z. Brzeźniak and B. Ferrario, A note on stochastic Navier–Stokes equations with not regular multiplicative noise, Stoch. Partial Differ. Equ. Anal. Comput., 5 (2017), 53-80.  doi: 10.1007/s40072-016-0081-2.

[7]

Z. Brzeźniak, E. Hausenblas and P. Razafimandimby, Some results on the penalised nematic liquid crystals driven by multiplicative noise, arXiv preprint, arXiv: 1310.8641, (2016), 65 pages.

[8]

Z. Brzeźniak and A. Millet, On the stochastic Strichartz estimates and the stochastic nonlinear Schrödinger equation on a compact Riemannian manifold,, Potential Anal., 41 (2014), 269-315.  doi: 10.1007/s11118-013-9369-2.

[9]

C. CavaterraR. Rocca and H. Wu, Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows, J. Differential Equations, 255 (2013), 24-57.  doi: 10.1016/j.jde.2013.03.009.

[10] S. Chandrasekhar, Liquid Crystals, Cambridge University Press, 1992. 
[11]

B. Climent-EzquerraF. Guillén-González and M. A. Rojas-Medar, Reproductivity for a nematic liquid crystal model, Z. Angew. Math. Phys., 57 (2006), 984-998.  doi: 10.1007/s00033-005-0038-1.

[12]

B. Climent-Ezquerra and F. Guillén-González, A review of mathematical analysis of nematic and smectic-A liquid crystal models, European J. Appl. Math., 25 (2014), 133-153.  doi: 10.1017/S0956792513000338.

[13]

D. Coutand and S. Shkoller, Well-posdness of the full Ericksen-Leslie Model of nematic liquid crystals,, C.R. Acad. Sci. Paris. Série I, 333 (2001), 919-924.  doi: 10.1016/S0764-4442(01)02161-9.

[14]

M. Dai and M. Schonbek, Asymptotic behavior of solutions to the liquid crystal system in $H^m(R^3)$, SIAM J. Math. Anal., 46 (2014), 3131-3150.  doi: 10.1137/120895342.

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

K. D. Elworthy, Stochastic Differential Equations on Manifolds, London Math. Soc. LNS v 70, Cambridge University Press, 1982.

[17]

J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34.  doi: 10.1122/1.548883.

[18]

C. G. Gal and T. T. Medjo, On a regularized family of models for homogeneous incompressible two-phase flows, J. Nonlinear Sci., 24 (2014), 1033-1103.  doi: 10.1007/s00332-014-9211-z.

[19]

M. Grasselli and H. Wu, Long-time behavior for a hydrodynamic model on nematic liquid crystal flows with asymptotic stabilizing boundary condition and external force, SIAM J. Math. Anal., 45 (2013), 965-1002.  doi: 10.1137/120866476.

[20]

I. Gyöngy and N. V. Krylov, On stochastics equations with respect to semimartingales. Ⅱ. Itô formula in Banach spaces,, Stochastics, 6 (1981/82), 153-173.  doi: 10.1080/17442508208833202.

[21]

D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, EMS Textbooks in Mathematics. European Mathematical Society, Zürich, 2008.

[22]

M. HieberM. NesensohnJ. Prüss and K. Schade, Dynamics of nematic liquid crystal flows: The quasilinear approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 397-408.  doi: 10.1016/j.anihpc.2014.11.001.

[23]

M. Hieber and J. Prüss, Dynamics of the Ericksen-Leslie equations with general Leslie stress Ⅰ: the incompressible isotropic case, Math. Ann., 369 (2017), 977-996.  doi: 10.1007/s00208-016-1453-7.

[24]

M.-C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two,, Calculus of Variations, 40 (2011), 15-36.  doi: 10.1007/s00526-010-0331-5.

[25]

M.-C. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $ \mathbb{R}^2$, Adv. Math., 231 (2012), 1364-1400.  doi: 10.1016/j.aim.2012.06.009.

[26]

M.-C. HongJ. Li and Z. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in $ \mathbb{R}^3$, Comm. Partial Differential Equations, 39 (2014), 1284-1328.  doi: 10.1080/03605302.2013.871026.

[27]

W. Horsthemke and R. Lefever, Noise-induced Transitions. Theory and Applications in Physics, Chemistry, and Biology, Springer Series in Synergetics, 15. Springer-Verlag, Berlin, 1984.

[28]

J. HuangF. LinFa nghua 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.

[29]

T. HuangF. LinC. Liu and C. Wang, Finite time singularity of the nematic liquid crystal flow in dimension three, Arch. Ration. Mech. Anal., 221 (2016), 1223-1254.  doi: 10.1007/s00205-016-0983-1.

[30]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[31]

T. Kato and G. Ponce, Well posedness of the Euler and Navier–Stokes equations in the Lebesgues spaces $L^p_s(\mathbb{R}^2)$,, Rev. Mat. Iberoam., 2 (1986), 73-88.  doi: 10.4171/RMI/26.

[32] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990. 
[33]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.

[34]

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

[35]

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

[36]

F. Lin and C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361, 18 pp. doi: 10.1098/rsta.2013.0361.

[37]

F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals,, Chinese Annals of Mathematics, Series B., 31 (2010), 921-938.  doi: 10.1007/s11401-010-0612-5.

[38]

F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571.  doi: 10.1002/cpa.21583.

[39]

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

[40]

C. Liu and N. J. Walkington, Approximation of liquid crystal flows,, SIAM J. Numer. Anal., 37 (2000), 725-741.  doi: 10.1137/S0036142997327282.

[41]

R. Mikulevicius, On strong $ \mathrm{H}^{1}_2$-solutions of stochastic Navier-Stokes equation in a bounded domain, SIAM J. Math. Anal., 41 (2009), 1206-1230. doi: 10.1137/0807433747.

[42] E. M. Ouhabaz, Analysis of Heat Equations on Domains,, Volume 31 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, 2005. 
[43]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167.  doi: 10.1080/17442507908833142.

[44]

E. Pardoux, Equations aux Dérivées Partielles Stochastiques Monotones, Theèse de Doctorat, Université Paris-Sud, 1975.

[45]

F. Sagués and M. San Miguel, Dynamics of Fréedericksz transition in a fluctuating magnetic field,, Phys. Rev. A., 32 (1985), 1843-1851. 

[46]

M. San Miguel, Nematic liquid crystals in a stochastic magnetic field: Spatial correlations, Phys. Rev. A, 32 (1985), 3811-3813. 

[47]

S. Shkoller, Well-posedness and global attractors for liquid crystal on Riemannian manifolds,, Communication in Partial Differential Equations, 27 (2002), 1103-1137.  doi: 10.1081/PDE-120004895.

[48]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 41, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983.

[49]

M. Wang and W. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system,, Calc. Var. Partial Differential Equations, 51 (2014), 915-962.  doi: 10.1007/s00526-013-0700-y.

[50]

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.

[51]

M. WangW. Wang and Z. Zhang, On the uniqueness of weak solution for the 2-D Ericksen-Leslie system, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 919-941.  doi: 10.3934/dcdsb.2016.21.919.

[52]

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.

show all references

References:
[1]

K. Atkinson and W. Han, Theoretical Numerical Analysis. A Functional Analysis Framework, Third edition. Volume 39 of Texts in Applied Mathematics, Springer, Dordrecht, 2009. doi: 10.1007/978-1-4419-0458-4.

[2]

R. BeckerX. Feng and A. Prohl, Finite element approximations of the Ericksen-Leslie model for nematic liquid crystal flow,, SIAM J. Numer. Anal., 46 (2008), 1704-1731.  doi: 10.1137/07068254X.

[3]

H. BessaihZ. Brzeźniak and A. Millet, Splitting up method for the 2D stochastic Navier-Stokes equations,, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014), 433-470.  doi: 10.1007/s40072-014-0041-7.

[4]

Z. BrzeźniakS. Cerrai and M. Freidlin, Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise,, Probab. Theory Related Fields, 162 (2015), 739-793.  doi: 10.1007/s00440-014-0584-6.

[5]

Z. Brzeźniak and K. D. Elworthy, Stochastic differential equations on Banach manifolds, Methods Funct. Anal. Topology, 6 (2000), 43-84. 

[6]

Z. Brzeźniak and B. Ferrario, A note on stochastic Navier–Stokes equations with not regular multiplicative noise, Stoch. Partial Differ. Equ. Anal. Comput., 5 (2017), 53-80.  doi: 10.1007/s40072-016-0081-2.

[7]

Z. Brzeźniak, E. Hausenblas and P. Razafimandimby, Some results on the penalised nematic liquid crystals driven by multiplicative noise, arXiv preprint, arXiv: 1310.8641, (2016), 65 pages.

[8]

Z. Brzeźniak and A. Millet, On the stochastic Strichartz estimates and the stochastic nonlinear Schrödinger equation on a compact Riemannian manifold,, Potential Anal., 41 (2014), 269-315.  doi: 10.1007/s11118-013-9369-2.

[9]

C. CavaterraR. Rocca and H. Wu, Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows, J. Differential Equations, 255 (2013), 24-57.  doi: 10.1016/j.jde.2013.03.009.

[10] S. Chandrasekhar, Liquid Crystals, Cambridge University Press, 1992. 
[11]

B. Climent-EzquerraF. Guillén-González and M. A. Rojas-Medar, Reproductivity for a nematic liquid crystal model, Z. Angew. Math. Phys., 57 (2006), 984-998.  doi: 10.1007/s00033-005-0038-1.

[12]

B. Climent-Ezquerra and F. Guillén-González, A review of mathematical analysis of nematic and smectic-A liquid crystal models, European J. Appl. Math., 25 (2014), 133-153.  doi: 10.1017/S0956792513000338.

[13]

D. Coutand and S. Shkoller, Well-posdness of the full Ericksen-Leslie Model of nematic liquid crystals,, C.R. Acad. Sci. Paris. Série I, 333 (2001), 919-924.  doi: 10.1016/S0764-4442(01)02161-9.

[14]

M. Dai and M. Schonbek, Asymptotic behavior of solutions to the liquid crystal system in $H^m(R^3)$, SIAM J. Math. Anal., 46 (2014), 3131-3150.  doi: 10.1137/120895342.

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

K. D. Elworthy, Stochastic Differential Equations on Manifolds, London Math. Soc. LNS v 70, Cambridge University Press, 1982.

[17]

J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34.  doi: 10.1122/1.548883.

[18]

C. G. Gal and T. T. Medjo, On a regularized family of models for homogeneous incompressible two-phase flows, J. Nonlinear Sci., 24 (2014), 1033-1103.  doi: 10.1007/s00332-014-9211-z.

[19]

M. Grasselli and H. Wu, Long-time behavior for a hydrodynamic model on nematic liquid crystal flows with asymptotic stabilizing boundary condition and external force, SIAM J. Math. Anal., 45 (2013), 965-1002.  doi: 10.1137/120866476.

[20]

I. Gyöngy and N. V. Krylov, On stochastics equations with respect to semimartingales. Ⅱ. Itô formula in Banach spaces,, Stochastics, 6 (1981/82), 153-173.  doi: 10.1080/17442508208833202.

[21]

D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, EMS Textbooks in Mathematics. European Mathematical Society, Zürich, 2008.

[22]

M. HieberM. NesensohnJ. Prüss and K. Schade, Dynamics of nematic liquid crystal flows: The quasilinear approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 397-408.  doi: 10.1016/j.anihpc.2014.11.001.

[23]

M. Hieber and J. Prüss, Dynamics of the Ericksen-Leslie equations with general Leslie stress Ⅰ: the incompressible isotropic case, Math. Ann., 369 (2017), 977-996.  doi: 10.1007/s00208-016-1453-7.

[24]

M.-C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two,, Calculus of Variations, 40 (2011), 15-36.  doi: 10.1007/s00526-010-0331-5.

[25]

M.-C. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $ \mathbb{R}^2$, Adv. Math., 231 (2012), 1364-1400.  doi: 10.1016/j.aim.2012.06.009.

[26]

M.-C. HongJ. Li and Z. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in $ \mathbb{R}^3$, Comm. Partial Differential Equations, 39 (2014), 1284-1328.  doi: 10.1080/03605302.2013.871026.

[27]

W. Horsthemke and R. Lefever, Noise-induced Transitions. Theory and Applications in Physics, Chemistry, and Biology, Springer Series in Synergetics, 15. Springer-Verlag, Berlin, 1984.

[28]

J. HuangF. LinFa nghua 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.

[29]

T. HuangF. LinC. Liu and C. Wang, Finite time singularity of the nematic liquid crystal flow in dimension three, Arch. Ration. Mech. Anal., 221 (2016), 1223-1254.  doi: 10.1007/s00205-016-0983-1.

[30]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[31]

T. Kato and G. Ponce, Well posedness of the Euler and Navier–Stokes equations in the Lebesgues spaces $L^p_s(\mathbb{R}^2)$,, Rev. Mat. Iberoam., 2 (1986), 73-88.  doi: 10.4171/RMI/26.

[32] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990. 
[33]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.

[34]

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

[35]

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

[36]

F. Lin and C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361, 18 pp. doi: 10.1098/rsta.2013.0361.

[37]

F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals,, Chinese Annals of Mathematics, Series B., 31 (2010), 921-938.  doi: 10.1007/s11401-010-0612-5.

[38]

F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571.  doi: 10.1002/cpa.21583.

[39]

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

[40]

C. Liu and N. J. Walkington, Approximation of liquid crystal flows,, SIAM J. Numer. Anal., 37 (2000), 725-741.  doi: 10.1137/S0036142997327282.

[41]

R. Mikulevicius, On strong $ \mathrm{H}^{1}_2$-solutions of stochastic Navier-Stokes equation in a bounded domain, SIAM J. Math. Anal., 41 (2009), 1206-1230. doi: 10.1137/0807433747.

[42] E. M. Ouhabaz, Analysis of Heat Equations on Domains,, Volume 31 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, 2005. 
[43]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167.  doi: 10.1080/17442507908833142.

[44]

E. Pardoux, Equations aux Dérivées Partielles Stochastiques Monotones, Theèse de Doctorat, Université Paris-Sud, 1975.

[45]

F. Sagués and M. San Miguel, Dynamics of Fréedericksz transition in a fluctuating magnetic field,, Phys. Rev. A., 32 (1985), 1843-1851. 

[46]

M. San Miguel, Nematic liquid crystals in a stochastic magnetic field: Spatial correlations, Phys. Rev. A, 32 (1985), 3811-3813. 

[47]

S. Shkoller, Well-posedness and global attractors for liquid crystal on Riemannian manifolds,, Communication in Partial Differential Equations, 27 (2002), 1103-1137.  doi: 10.1081/PDE-120004895.

[48]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 41, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983.

[49]

M. Wang and W. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system,, Calc. Var. Partial Differential Equations, 51 (2014), 915-962.  doi: 10.1007/s00526-013-0700-y.

[50]

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.

[51]

M. WangW. Wang and Z. Zhang, On the uniqueness of weak solution for the 2-D Ericksen-Leslie system, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 919-941.  doi: 10.3934/dcdsb.2016.21.919.

[52]

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.

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