American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Analytic integrability of a class of planar polynomial differential systems
  • DCDS-B Home
  • This Issue
  • Next Article
    Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one
October  2015, 20(8): 2611-2655. doi: 10.3934/dcdsb.2015.20.2611

Local well-posedness and small Deborah limit of a molecule-based $Q$-tensor system

Sirui Li 1, , Wei Wang 2, and  Pingwen Zhang 3,

1. 

School of Mathematical Sciences and LMAM, Peking University, Beijing, 100871, China
2. 

Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China
3. 

LMAM, CAPT and School of Mathematical Sciences, Peking University, Beijing, 100871

Received  October 2014 Revised  January 2015 Published  August 2015

In this paper, we consider a hydrodynamic $Q$-tensor system for nematic liquid crystal flow, which is derived from Doi-Onsager molecular theory by the Bingham closure. We first prove the existence and uniqueness of local strong solution. Furthermore, by taking Deborah number goes to zero and using the Hilbert expansion method, we present a rigorous derivation from the molecule-based $Q$-tensor theory to the Ericksen-Leslie theory.
Keywords: Ericksen-Leslie theory, local strong solution., Liquid crystals, Q-tensor theory.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 35A01, 76A15, 76D0.
Citation: Sirui Li, Wei Wang, Pingwen Zhang. Local well-posedness and small Deborah limit of a molecule-based $Q$-tensor system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2611-2655. doi: 10.3934/dcdsb.2015.20.2611
References:
[1]

H. Abels, G. Dolzmann and Y. Liu, Well-posedness of a fully-coupled Navier-Stokes/$Q$-tensor system with inhomogeneous boundary data, SIAM J. Math. Anal., 46 (2014), 3050-3077. doi: 10.1137/130945405.  Google Scholar

[2]

H. Abels, G. Dolzmann and Y. Liu, Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions,, preprint, ().   Google Scholar

[3]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Fundamental Principles of Mathematical Sciences, 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[4]

J. M. Ball and A. Majumdar, Nematic liquid crystals: From Maier-Saupe to a continuum theory, Mol. Cryst. Liq. Cryst., 525 (2010), 1-11. doi: 10.1080/15421401003795555.  Google Scholar

[5]

A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford Engrg. Sci. Ser., 36, Oxford University Press, Oxford, New York, 1994.  Google Scholar

[6]

P. G. De Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974. Google Scholar

[7]

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.  Google Scholar

[8]

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

[9]

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.  Google Scholar

[10]

J. Feng, C. V. Chaubal and L. G. Leal, Closure approximations for the Doi theory: Which to use in simulating complex flows of liquid-crystalline polymers?, J. Rheol., 42 (1998), 1095-1119. doi: 10.1122/1.550920.  Google Scholar

[11]

J. Feng, G. Sgalari and L. G. Leal, A theory for flowing nematic polymers with orientational distortion, J. Rheol., 44 (2000), 1085-1101. doi: 10.1122/1.1289278.  Google Scholar

[12]

J. Han, Y. Luo, W. Wang, P. Zhang and Z. Zhang, From microscopic theory to macroscopic theory: A systematic study on modeling for liquid crystals, Arch. Ration. Mech. Anal., 215 (2015), 741-809. doi: 10.1007/s00205-014-0792-3.  Google Scholar

[13]

J. Huang and S. Ding, Global well-posedness for a coupled incompressible Navier-Stokes and Q-tensor system,, preprint, ().   Google Scholar

[14]

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

[15]

N. Kuzuu and M. Doi, Constitutive equation for nematic liquid crystals under weak velocity gradient derived from a molecular kinetic equation, J. Phys. Soc. Japan, 52 (1983), 3486-3494. doi: 10.1143/JPSJ.52.3486.  Google Scholar

[16]

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

[17]

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.  Google Scholar

[18]

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

[19]

A. Majumdar and A. Zarnescu, Landau-De Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196 (2010), 227-280. doi: 10.1007/s00205-009-0249-2.  Google Scholar

[20]

N. J. Mottram and C. J. P. Newton, Introduction to $Q$-tensor theory,, preprint, ().   Google Scholar

[21]

M. Paicu and A. Zarnescu, Energy dissipation and regularity for a coupled Navier-Stokes and $Q$-tensor system, Arch. Ration. Mech. Anal., 203 (2012), 45-67. doi: 10.1007/s00205-011-0443-x.  Google Scholar

[22]

M. Paicu and A. Zarnescu, Global existence and regularity for the full coupled Navier-Stokes and $Q$-tensor system, SIAM J. Math. Anal., 43 (2011), 2009-2049. doi: 10.1137/10079224X.  Google Scholar

[23]

O. Parodi, Stress tensor for a nematic liquid crystal, J. Phys. France, 31 (1970), 581-584. doi: 10.1051/jphys:01970003107058100.  Google Scholar

[24]

T. Qian and P. Sheng, Generalized hydrodynamic equations for nematic liquid crystals, Phys. Rev. E, 58 (1998), 7475-7485. doi: 10.1103/PhysRevE.58.7475.  Google Scholar

[25]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, Birkhäuser Verlag, Basel, Boston, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[26]

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

[27]

W. Wang, P. 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.  Google Scholar

[28]

W. Wang, P. 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.  Google Scholar

[29]

W. Wang, P. Zhang and Z. Zhang, Rigorous derivation from Landau-de Gennes theory to Ericksen-Leslie theory, SIAM J. Math. Anal., 47 (2015), 127-158. doi: 10.1137/13093529X.  Google Scholar

show all references

References:
[1]

H. Abels, G. Dolzmann and Y. Liu, Well-posedness of a fully-coupled Navier-Stokes/$Q$-tensor system with inhomogeneous boundary data, SIAM J. Math. Anal., 46 (2014), 3050-3077. doi: 10.1137/130945405.  Google Scholar

[2]

H. Abels, G. Dolzmann and Y. Liu, Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions,, preprint, ().   Google Scholar

[3]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Fundamental Principles of Mathematical Sciences, 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[4]

J. M. Ball and A. Majumdar, Nematic liquid crystals: From Maier-Saupe to a continuum theory, Mol. Cryst. Liq. Cryst., 525 (2010), 1-11. doi: 10.1080/15421401003795555.  Google Scholar

[5]

A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford Engrg. Sci. Ser., 36, Oxford University Press, Oxford, New York, 1994.  Google Scholar

[6]

P. G. De Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974. Google Scholar

[7]

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.  Google Scholar

[8]

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

[9]

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.  Google Scholar

[10]

J. Feng, C. V. Chaubal and L. G. Leal, Closure approximations for the Doi theory: Which to use in simulating complex flows of liquid-crystalline polymers?, J. Rheol., 42 (1998), 1095-1119. doi: 10.1122/1.550920.  Google Scholar

[11]

J. Feng, G. Sgalari and L. G. Leal, A theory for flowing nematic polymers with orientational distortion, J. Rheol., 44 (2000), 1085-1101. doi: 10.1122/1.1289278.  Google Scholar

[12]

J. Han, Y. Luo, W. Wang, P. Zhang and Z. Zhang, From microscopic theory to macroscopic theory: A systematic study on modeling for liquid crystals, Arch. Ration. Mech. Anal., 215 (2015), 741-809. doi: 10.1007/s00205-014-0792-3.  Google Scholar

[13]

J. Huang and S. Ding, Global well-posedness for a coupled incompressible Navier-Stokes and Q-tensor system,, preprint, ().   Google Scholar

[14]

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

[15]

N. Kuzuu and M. Doi, Constitutive equation for nematic liquid crystals under weak velocity gradient derived from a molecular kinetic equation, J. Phys. Soc. Japan, 52 (1983), 3486-3494. doi: 10.1143/JPSJ.52.3486.  Google Scholar

[16]

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

[17]

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.  Google Scholar

[18]

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

[19]

A. Majumdar and A. Zarnescu, Landau-De Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196 (2010), 227-280. doi: 10.1007/s00205-009-0249-2.  Google Scholar

[20]

N. J. Mottram and C. J. P. Newton, Introduction to $Q$-tensor theory,, preprint, ().   Google Scholar

[21]

M. Paicu and A. Zarnescu, Energy dissipation and regularity for a coupled Navier-Stokes and $Q$-tensor system, Arch. Ration. Mech. Anal., 203 (2012), 45-67. doi: 10.1007/s00205-011-0443-x.  Google Scholar

[22]

M. Paicu and A. Zarnescu, Global existence and regularity for the full coupled Navier-Stokes and $Q$-tensor system, SIAM J. Math. Anal., 43 (2011), 2009-2049. doi: 10.1137/10079224X.  Google Scholar

[23]

O. Parodi, Stress tensor for a nematic liquid crystal, J. Phys. France, 31 (1970), 581-584. doi: 10.1051/jphys:01970003107058100.  Google Scholar

[24]

T. Qian and P. Sheng, Generalized hydrodynamic equations for nematic liquid crystals, Phys. Rev. E, 58 (1998), 7475-7485. doi: 10.1103/PhysRevE.58.7475.  Google Scholar

[25]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, Birkhäuser Verlag, Basel, Boston, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[26]

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

[27]

W. Wang, P. 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.  Google Scholar

[28]

W. Wang, P. 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.  Google Scholar

[29]

W. Wang, P. Zhang and Z. Zhang, Rigorous derivation from Landau-de Gennes theory to Ericksen-Leslie theory, SIAM J. Math. Anal., 47 (2015), 127-158. doi: 10.1137/13093529X.  Google Scholar

[1]

Zdzisław Brzeźniak, Erika Hausenblas, Paul André Razafimandimby. A note on the stochastic Ericksen-Leslie equations for nematic liquid crystals. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5785-5802. doi: 10.3934/dcdsb.2019106
[2]

Jishan Fan, Tohru Ozawa. Regularity criteria for a simplified Ericksen-Leslie system modeling the flow of liquid crystals. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 859-867. doi: 10.3934/dcds.2009.25.859
[3]

Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475
[4]

Stefano Bosia. Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2012, 11 (2) : 407-441. doi: 10.3934/cpaa.2012.11.407
[5]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[6]

Yuning Liu, Wei Wang. On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3879-3899. doi: 10.3934/dcdsb.2018115
[7]

Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591
[8]

Jinrui Huang, Wenjun Wang, Huanyao Wen. On $ L^p $ estimates for a simplified Ericksen-Leslie system. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1485-1507. doi: 10.3934/cpaa.2020075
[9]

Jihoon Lee. Scaling invariant blow-up criteria for simplified versions of Ericksen-Leslie system. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 381-388. doi: 10.3934/dcdss.2015.8.381
[10]

Etienne Emmrich, Robert Lasarzik. Weak-strong uniqueness for the general Ericksen—Leslie system in three dimensions. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4617-4635. doi: 10.3934/dcds.2018202
[11]

Apala Majumdar. The Landau-de Gennes theory of nematic liquid crystals: Uniaxiality versus Biaxiality. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1303-1337. doi: 10.3934/cpaa.2012.11.1303
[12]

Meng Wang, Wendong Wang, Zhifei Zhang. On the uniqueness of weak solution for the 2-D Ericksen--Leslie system. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 919-941. doi: 10.3934/dcdsb.2016.21.919
[13]

Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357
[14]

Paolo Barbante, Aldo Frezzotti, Livio Gibelli. A kinetic theory description of liquid menisci at the microscale. Kinetic & Related Models, 2015, 8 (2) : 235-254. doi: 10.3934/krm.2015.8.235
[15]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349
[16]

Neil S. Trudinger. On the local theory of prescribed Jacobian equations. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1663-1681. doi: 10.3934/dcds.2014.34.1663
[17]

M. Carme Calderer, Carlos A. Garavito Garzón, Baisheng Yan. A Landau--de Gennes theory of liquid crystal elastomers. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 283-302. doi: 10.3934/dcdss.2015.8.283
[18]

Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623
[19]

G. Leugering, Marina Prechtel, Paul Steinmann, Michael Stingl. A cohesive crack propagation model: Mathematical theory and numerical solution. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1705-1729. doi: 10.3934/cpaa.2013.12.1705
[20]

Thi-Bich-Ngoc Mac. Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3013-3027. doi: 10.3934/dcdsb.2015.20.3013

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences