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

Yangjun Ma

doi:10.3934/dcds.2020187

Article Contents

2020, Volume 40, Issue 7: 4479-4496. Doi: 10.3934/dcds.2020187

This issue Previous Article Entropy on regular trees Next Article Persistence of invariant tori for almost periodically forced reversible systems

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

Yangjun Ma^,

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

^* Corresponding author: Yangjun Ma
^* Corresponding author: Yangjun Ma

Received: August 31, 2019

Published: April 2020

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

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Abstract

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.

Keywords:

Incompressible non-inertial Qian-Sheng model,
$ Q $-tensor,
local and global well-posedness,
a priori estimates,
Parodi's relation.

Mathematics Subject Classification: Primary: 35A01, 35A09, 35B45, 35B65, 35Q35; Secondary: 76A15.

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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.
[2]	E. Feireisl, E. Rocca, G. 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.
[3]	S. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland Publishing Co., Amsterdam, 1962.
[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.
[5]	E. Kirr, M. 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.
[6]	F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.
[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.
[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.
[9]	N. J. Mottram and C. J. Newton, Introduction to Q-tensor theory, preprint, arXiv: 1409.3542, 2014.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	A. Zarnescu, Topics in the Q-tensor theory of liquid crystals, Topics Mathematical Modeling and Analysis, 7 (2012), 187-252.