July  2020, 40(7): 4479-4496. doi: 10.3934/dcds.2020187

Global well-posedness to incompressible non-inertial Qian-Sheng model

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

* Corresponding author: Yangjun Ma

Received  September 2019 Published  April 2020

Fund Project: This work is supported by the grants from the National Natural Foundation of China under contract No. 11971360

In this paper we study the incompressible non-inertial Qian-Sheng model, which describes the hydrodynamics of nematic liquid crystals without inertial effect in the $ Q $-tensor framework. Under some proper assumptions on the viscous coefficients, we prove the local well-posedness with large initial data and the global existence with small size of the initial data in the classical solutions regime.

Citation: Yangjun Ma. Global well-posedness to incompressible non-inertial Qian-Sheng model. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4479-4496. doi: 10.3934/dcds.2020187
References:
[1]

F. De Anna and A. Zarnescu, Global well-posedness and twist-wave solutions for the inertial Qian-Sheng model of liquid crystals, J. Differential Equations, 264 (2018), 1080-1118.  doi: 10.1016/j.jde.2017.09.031.  Google Scholar

[2]

E. FeireislE. RoccaG. Schimperna and A. Zarnescu, On a hyperbolic system arising in liquid crystals modeling, J. Hyperbolic Differ. Equ., 15 (2018), 15-35.  doi: 10.1142/S0219891618500029.  Google Scholar

[3]

S. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland Publishing Co., Amsterdam, 1962.  Google Scholar

[4]

N. Jiang and Y.-L. Luo, On well-posedness of Ericksen-Leslie's hyperbolic incompressible liquid crystal model, SIAM J. Math. Anal., 51 (2019), 403-434.  doi: 10.1137/18M1167310.  Google Scholar

[5]

E. KirrM. Wilkinson and A. Zarnescu, Dynamic statistical scaling in the Landau-de Gennes theory of nematic liquid crystals, J. Stat. Phys., 155 (2014), 625-657.  doi: 10.1007/s10955-014-0970-6.  Google Scholar

[6]

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

[7]

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

[8]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Oxford Lecture Series in Mathematics and its Applications, Vol. 3, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[9]

N. J. Mottram and C. J. Newton, Introduction to Q-tensor theory, preprint, arXiv: 1409.3542, 2014. Google Scholar

[10]

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

[11]

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

[12]

V. Popa-Nita and P. Oswald, Waves at the nematic-isotropic interface: The role of surface tension anisotropy, curvature elasticity, and backflow effects, Phys. Rev. E., 68 (2003), 061707. doi: 10.1103/PhysRevE.68.061707.  Google Scholar

[13]

V. Popa-Nita, T. J. Sluckin and S. Kralj, Waves at the nematic-isotropic interface: Thermotropic nematogen-non-nematogen mixtures, Phys. Rev. E., 71 (2005), 061706. doi: 10.1103/PhysRevE.71.061706.  Google Scholar

[14]

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

[15]

Y. Xiao, Global strong solution to the three-dimensional liquid crystal flows of Q-tensor model, J. Differential Equations, 262 (2017), 1291-1316.  doi: 10.1016/j.jde.2016.10.011.  Google Scholar

[16]

A. Zarnescu, Topics in the Q-tensor theory of liquid crystals, Topics Mathematical Modeling and Analysis, 7 (2012), 187-252.   Google Scholar

show all references

References:
[1]

F. De Anna and A. Zarnescu, Global well-posedness and twist-wave solutions for the inertial Qian-Sheng model of liquid crystals, J. Differential Equations, 264 (2018), 1080-1118.  doi: 10.1016/j.jde.2017.09.031.  Google Scholar

[2]

E. FeireislE. RoccaG. Schimperna and A. Zarnescu, On a hyperbolic system arising in liquid crystals modeling, J. Hyperbolic Differ. Equ., 15 (2018), 15-35.  doi: 10.1142/S0219891618500029.  Google Scholar

[3]

S. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland Publishing Co., Amsterdam, 1962.  Google Scholar

[4]

N. Jiang and Y.-L. Luo, On well-posedness of Ericksen-Leslie's hyperbolic incompressible liquid crystal model, SIAM J. Math. Anal., 51 (2019), 403-434.  doi: 10.1137/18M1167310.  Google Scholar

[5]

E. KirrM. Wilkinson and A. Zarnescu, Dynamic statistical scaling in the Landau-de Gennes theory of nematic liquid crystals, J. Stat. Phys., 155 (2014), 625-657.  doi: 10.1007/s10955-014-0970-6.  Google Scholar

[6]

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

[7]

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

[8]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Oxford Lecture Series in Mathematics and its Applications, Vol. 3, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[9]

N. J. Mottram and C. J. Newton, Introduction to Q-tensor theory, preprint, arXiv: 1409.3542, 2014. Google Scholar

[10]

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

[11]

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

[12]

V. Popa-Nita and P. Oswald, Waves at the nematic-isotropic interface: The role of surface tension anisotropy, curvature elasticity, and backflow effects, Phys. Rev. E., 68 (2003), 061707. doi: 10.1103/PhysRevE.68.061707.  Google Scholar

[13]

V. Popa-Nita, T. J. Sluckin and S. Kralj, Waves at the nematic-isotropic interface: Thermotropic nematogen-non-nematogen mixtures, Phys. Rev. E., 71 (2005), 061706. doi: 10.1103/PhysRevE.71.061706.  Google Scholar

[14]

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

[15]

Y. Xiao, Global strong solution to the three-dimensional liquid crystal flows of Q-tensor model, J. Differential Equations, 262 (2017), 1291-1316.  doi: 10.1016/j.jde.2016.10.011.  Google Scholar

[16]

A. Zarnescu, Topics in the Q-tensor theory of liquid crystals, Topics Mathematical Modeling and Analysis, 7 (2012), 187-252.   Google Scholar

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