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

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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

Abstract
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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 and 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.
[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, ().
[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.
[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.
[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.
[6]	P. G. De Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974.
[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.
[8]	J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 23-34. doi: 10.1122/1.548883.
[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.
[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.
[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.
[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.
[13]	J. Huang and S. Ding, Global well-posedness for a coupled incompressible Navier-Stokes and Q-tensor system,, preprint, ().
[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.
[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.
[16]	F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.
[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.
[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.
[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.
[20]	N. J. Mottram and C. J. P. Newton, Introduction to $Q$-tensor theory,, preprint, ().
[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.
[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.
[23]	O. Parodi, Stress tensor for a nematic liquid crystal, J. Phys. France, 31 (1970), 581-584. doi: 10.1051/jphys:01970003107058100.
[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.
[25]	H. Triebel, Theory of Function Spaces, Monographs in Mathematics, Birkhäuser Verlag, Basel, Boston, 1983. doi: 10.1007/978-3-0346-0416-1.
[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.
[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.
[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.
[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.

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.
[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, ().
[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.
[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.
[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.
[6]	P. G. De Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974.
[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.
[8]	J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 23-34. doi: 10.1122/1.548883.
[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.
[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.
[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.
[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.
[13]	J. Huang and S. Ding, Global well-posedness for a coupled incompressible Navier-Stokes and Q-tensor system,, preprint, ().
[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.
[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.
[16]	F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.
[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.
[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.
[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.
[20]	N. J. Mottram and C. J. P. Newton, Introduction to $Q$-tensor theory,, preprint, ().
[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.
[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.
[23]	O. Parodi, Stress tensor for a nematic liquid crystal, J. Phys. France, 31 (1970), 581-584. doi: 10.1051/jphys:01970003107058100.
[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.
[25]	H. Triebel, Theory of Function Spaces, Monographs in Mathematics, Birkhäuser Verlag, Basel, Boston, 1983. doi: 10.1007/978-3-0346-0416-1.
[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.
[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.
[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.
[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.

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