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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

 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.
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

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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
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