Energy dissipation for weak solutions of incompressible liquid crystal flows

Shanshan  Guo; Zhong Tan

doi:10.3934/krm.2015.8.691

Article Contents

2015, Volume 8, Issue 4: 691-706. Doi: 10.3934/krm.2015.8.691

This issue Previous Article Strong continuity for the 2D Euler equations Next Article An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach

Energy dissipation for weak solutions of incompressible liquid crystal flows

Shanshan Guo^1, and
Zhong Tan^2,

1.
School of Mathematical Sciences, Xiamen University, Xiamen, 361005
2.
School of Mathematical Sciences and Fujian Provincial Key Laboratory, on Mathematical Modeling and Scientific Computing, Xiamen University, Xiamen, 361005

Received: March 31, 2015

Revised: April 30, 2015

Published: November 2015

Abstract Related Papers Cited by

Abstract

In this paper, we are concerned with the simplified Ericksen-Leslie system （1）--（3）, modeling the flow of nematic liquid crystals for any initial and boundary (or Cauchy) data $(u_0, d_0)\in {\bf H}\times H^1(\Omega, \mathbb{S}^2)$, with $d_0(\Omega)\subset\mathbb{S}^2_+$. We define a dissipation term $D(u,d)$ that stems from an eventual lack of smoothness in the solutions, and then obtain a local equation of energy for weak solutions of liquid crystals in dimensions three. As a consequence, we consider the 2D case and obtain $D(u,d)=0$.

Keywords:

Mathematics Subject Classification: 35D30, 76A15, 35Q35.

Citation:

Related Papers

Cited by

References

[1]	P. Constain, E. Weinan and E. S. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Commun. Math. Phys., 165 (1994), 207-209.doi: 10.1007/BF02099744.
[2]	L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the navier-stokes equations, Comm. Pure. Appl. Math., 35 (1982), 771-831.doi: 10.1002/cpa.3160350604.
[3]	C. Cavaterra, E. Rocca and H. Wu, Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows, J. Differential Equations, 255 (2013), 24-57, arXiv:1212.0043.doi: 10.1016/j.jde.2013.03.009.
[4]	J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, 13 (2000), 249-255.doi: 10.1088/0951-7715/13/1/312.
[5]	P. D. Gennes and J. Prost, The Physics of Liquid Crystals, $2^{nd}$ edition, International Series of Monographs on Physics No 83, (1993)doi: 10.1080/13583149408628646.
[6]	S. Ding, J. Lin and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dyn. Syst., 32 (2012), 539-563.doi: 10.3934/dcds.2012.32.539.
[7]	J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378.
[8]	G. L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer, Physica, 78 (1994), 222-240.doi: 10.1016/0167-2789(94)90117-1.
[9]	Z. S. Gao, Z. Tan and G. C. Wu, Energy dissipation for weak solutions of incompressible MHD equations, Acta Mathematica Scientia, 33 (2013), 865-671.doi: 10.1016/S0252-9602(13)60046-6.
[10]	M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc Var Partial Differential Equations, 40 (2011), 15-36.doi: 10.1007/s00526-010-0331-5.
[11]	M. C. Hong and Z. P. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbbR^2$, Adv. Math., 231 (2012), 1364-1400.doi: 10.1016/j.aim.2012.06.009.
[12]	J. R. Huang, F. H. Lin and C. Y. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbbR^2$, Adv. Math., 331 (2012), 805-850.doi: 10.1007/s00220-014-2079-9.
[13]	F. Jiang, S. Jiang and D. H. Wang, Global weak solutions to the equations of compressible flow of nematic liquid crystals in Two Dimensions, Arch. Ration. Mech. Anal., 214 (2014), 403-451.doi: 10.1007/s00205-014-0768-3.
[14]	F. Jiang, S. Jiang and D. H. Wang, On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain, J. Funct. Anal., 265 (2013), 3369-3397.doi: 10.1016/j.jfa.2013.07.026.
[15]	F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system, Math Methods Aool. Sci., 32 (2009), 2243-2266.doi: 10.1002/mma.1132.
[16]	F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.doi: 10.1007/BF00251810.
[17]	J. K. Li and Z. P. Xin, Global Weak Solutions to Non-isothermal Nematic Liquid Crystals in 2D, arXiv:1307.2065.
[18]	J. Leray, Etude de diverses équations intégrals nonlinéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures. Appl., 28 (1933), 1-82.
[19]	J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace. (French), Acta. Math., 63 (1934), 193-248.doi: 10.1007/BF02547354.
[20]	F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Commun Pure Appl. Math., 42 (1989), 789-814.doi: 10.1002/cpa.3160420605.
[21]	F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 2 (1996), 1-22.
[22]	F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, CPAM, 48 (1995), 501-537.doi: 10.1002/cpa.3160480503.
[23]	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.
[24]	C. Liu and J. Shen, On liquid crystal flows with free-slip boundary conditions, Discrete Contin. Dynam. Systems, 7 (2001), 307-318.doi: 10.3934/dcds.2001.7.307.
[25]	F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.doi: 10.1007/s00205-009-0278-x.
[26]	F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three, Communications on Pure and Applied Mathematics, arXiv:1408.4146. doi: 10.1002/cpa.21583.
[27]	F. H. Lin and C. Y. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math, 31 (2010), 921-938.doi: 10.1007/s11401-010-0612-5.
[28]	Z. Lei, D. Li and X. Y. Zhang, Remarks of global wellposedness of liquid crystal flows and heat flows of harmonic maps in two dimensions, Proc. Amer. Math. Soc., 142 (2014), 3801-3810.doi: 10.1090/S0002-9939-2014-12057-0.
[29]	L. Onsager, Statistical hydrodynamics, Nuovo Cimento (Supplemento), 6 (1949), 279-287.
[30]	H. Sun and C. Liu, On energetic variational approaches in modeling the nematic liquid crystal flows, Discrete Contin. Dynam. Systems, 23 (2009), 455-475.doi: 10.3934/dcds.2009.23.455.
[31]	R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North Holland, Amsterdam, 1979.
[32]	D. H. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Ration. Mech. Anal., 204 (2012), 881-915.doi: 10.1007/s00205-011-0488-x.
[33]	W. D. Wang and M. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Calc. Var. Partial Differential Equations, 51 (2014), 915-962.doi: 10.1007/s00526-013-0700-y.
[34]	X. Xu and Z. F. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows, J. Differ. Equ., 252 (2012), 1169-1181.doi: 10.1016/j.jde.2011.08.028.