American Institute of Mathematical Sciences

Advanced
December  2015, 8(4): 691-706. doi: 10.3934/krm.2015.8.691

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, China
2. 

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

Received  April 2015 Revised  May 2015 Published  July 2015

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: incompressible, energy dissipation, energy equation., liquid crystals, Weak solution.
Mathematics Subject Classification: 35D30, 76A15, 35Q3.
Citation: Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic & Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691
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.  doi: 10.1007/BF02099744.  Google Scholar

[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.  doi: 10.1002/cpa.3160350604.  Google Scholar

[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.  doi: 10.1016/j.jde.2013.03.009.  Google Scholar

[4]

J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations,, Nonlinearity, 13 (2000), 249.  doi: 10.1088/0951-7715/13/1/312.  Google Scholar

[5]

P. D. Gennes and J. Prost, The Physics of Liquid Crystals,, $2^{nd}$ edition, (1993).  doi: 10.1080/13583149408628646.  Google Scholar

[6]

S. Ding, J. Lin and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D,, Discrete Contin. Dyn. Syst., 32 (2012), 539.  doi: 10.3934/dcds.2012.32.539.  Google Scholar

[7]

J. L. Ericksen, Hydrostatic theory of liquid crystals,, Arch. Ration. Mech. Anal., 9 (1962), 371.   Google Scholar

[8]

G. L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer,, Physica, 78 (1994), 222.  doi: 10.1016/0167-2789(94)90117-1.  Google Scholar

[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.  doi: 10.1016/S0252-9602(13)60046-6.  Google Scholar

[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.  doi: 10.1007/s00526-010-0331-5.  Google Scholar

[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.  doi: 10.1016/j.aim.2012.06.009.  Google Scholar

[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.  doi: 10.1007/s00220-014-2079-9.  Google Scholar

[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.  doi: 10.1007/s00205-014-0768-3.  Google Scholar

[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.  doi: 10.1016/j.jfa.2013.07.026.  Google Scholar

[15]

F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system,, Math Methods Aool. Sci., 32 (2009), 2243.  doi: 10.1002/mma.1132.  Google Scholar

[16]

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

[17]

J. K. Li and Z. P. Xin, Global Weak Solutions to Non-isothermal Nematic Liquid Crystals in 2D,, , ().   Google Scholar

[18]

J. Leray, Etude de diverses équations intégrals nonlinéaires et de quelques problèmes que pose l'hydrodynamique,, J. Math. Pures. Appl., 28 (1933), 1.   Google Scholar

[19]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace. (French),, Acta. Math., 63 (1934), 193.  doi: 10.1007/BF02547354.  Google Scholar

[20]

F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena,, Commun Pure Appl. Math., 42 (1989), 789.  doi: 10.1002/cpa.3160420605.  Google Scholar

[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.   Google Scholar

[22]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, CPAM, 48 (1995), 501.  doi: 10.1002/cpa.3160480503.  Google Scholar

[23]

F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system,, Arch. Ration. Mech. Anal., 154 (2000), 135.  doi: 10.1007/s002050000102.  Google Scholar

[24]

C. Liu and J. Shen, On liquid crystal flows with free-slip boundary conditions,, Discrete Contin. Dynam. Systems, 7 (2001), 307.  doi: 10.3934/dcds.2001.7.307.  Google Scholar

[25]

F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions,, Arch. Ration. Mech. Anal., 197 (2010), 297.  doi: 10.1007/s00205-009-0278-x.  Google Scholar

[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, ().  doi: 10.1002/cpa.21583.  Google Scholar

[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.  doi: 10.1007/s11401-010-0612-5.  Google Scholar

[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.  doi: 10.1090/S0002-9939-2014-12057-0.  Google Scholar

[29]

L. Onsager, Statistical hydrodynamics,, Nuovo Cimento (Supplemento), 6 (1949), 279.   Google Scholar

[30]

H. Sun and C. Liu, On energetic variational approaches in modeling the nematic liquid crystal flows,, Discrete Contin. Dynam. Systems, 23 (2009), 455.  doi: 10.3934/dcds.2009.23.455.  Google Scholar

[31]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Studies in Mathematics and its Applications, 2 (1979).   Google Scholar

[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.  doi: 10.1007/s00205-011-0488-x.  Google Scholar

[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.  doi: 10.1007/s00526-013-0700-y.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.08.028.  Google Scholar

show all references

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.  doi: 10.1007/BF02099744.  Google Scholar

[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.  doi: 10.1002/cpa.3160350604.  Google Scholar

[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.  doi: 10.1016/j.jde.2013.03.009.  Google Scholar

[4]

J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations,, Nonlinearity, 13 (2000), 249.  doi: 10.1088/0951-7715/13/1/312.  Google Scholar

[5]

P. D. Gennes and J. Prost, The Physics of Liquid Crystals,, $2^{nd}$ edition, (1993).  doi: 10.1080/13583149408628646.  Google Scholar

[6]

S. Ding, J. Lin and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D,, Discrete Contin. Dyn. Syst., 32 (2012), 539.  doi: 10.3934/dcds.2012.32.539.  Google Scholar

[7]

J. L. Ericksen, Hydrostatic theory of liquid crystals,, Arch. Ration. Mech. Anal., 9 (1962), 371.   Google Scholar

[8]

G. L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer,, Physica, 78 (1994), 222.  doi: 10.1016/0167-2789(94)90117-1.  Google Scholar

[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.  doi: 10.1016/S0252-9602(13)60046-6.  Google Scholar

[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.  doi: 10.1007/s00526-010-0331-5.  Google Scholar

[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.  doi: 10.1016/j.aim.2012.06.009.  Google Scholar

[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.  doi: 10.1007/s00220-014-2079-9.  Google Scholar

[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.  doi: 10.1007/s00205-014-0768-3.  Google Scholar

[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.  doi: 10.1016/j.jfa.2013.07.026.  Google Scholar

[15]

F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system,, Math Methods Aool. Sci., 32 (2009), 2243.  doi: 10.1002/mma.1132.  Google Scholar

[16]

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

[17]

J. K. Li and Z. P. Xin, Global Weak Solutions to Non-isothermal Nematic Liquid Crystals in 2D,, , ().   Google Scholar

[18]

J. Leray, Etude de diverses équations intégrals nonlinéaires et de quelques problèmes que pose l'hydrodynamique,, J. Math. Pures. Appl., 28 (1933), 1.   Google Scholar

[19]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace. (French),, Acta. Math., 63 (1934), 193.  doi: 10.1007/BF02547354.  Google Scholar

[20]

F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena,, Commun Pure Appl. Math., 42 (1989), 789.  doi: 10.1002/cpa.3160420605.  Google Scholar

[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.   Google Scholar

[22]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, CPAM, 48 (1995), 501.  doi: 10.1002/cpa.3160480503.  Google Scholar

[23]

F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system,, Arch. Ration. Mech. Anal., 154 (2000), 135.  doi: 10.1007/s002050000102.  Google Scholar

[24]

C. Liu and J. Shen, On liquid crystal flows with free-slip boundary conditions,, Discrete Contin. Dynam. Systems, 7 (2001), 307.  doi: 10.3934/dcds.2001.7.307.  Google Scholar

[25]

F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions,, Arch. Ration. Mech. Anal., 197 (2010), 297.  doi: 10.1007/s00205-009-0278-x.  Google Scholar

[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, ().  doi: 10.1002/cpa.21583.  Google Scholar

[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.  doi: 10.1007/s11401-010-0612-5.  Google Scholar

[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.  doi: 10.1090/S0002-9939-2014-12057-0.  Google Scholar

[29]

L. Onsager, Statistical hydrodynamics,, Nuovo Cimento (Supplemento), 6 (1949), 279.   Google Scholar

[30]

H. Sun and C. Liu, On energetic variational approaches in modeling the nematic liquid crystal flows,, Discrete Contin. Dynam. Systems, 23 (2009), 455.  doi: 10.3934/dcds.2009.23.455.  Google Scholar

[31]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Studies in Mathematics and its Applications, 2 (1979).   Google Scholar

[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.  doi: 10.1007/s00205-011-0488-x.  Google Scholar

[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.  doi: 10.1007/s00526-013-0700-y.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.08.028.  Google Scholar

[1]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[2]

Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1445-1468. doi: 10.3934/cpaa.2013.12.1445
[3]

Kyungkeun Kang, Jinhae Park. Partial regularity of minimum energy configurations in ferroelectric liquid crystals. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1499-1511. doi: 10.3934/dcds.2013.33.1499
[4]

Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357
[5]

Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681
[6]

Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591
[7]

Xian-Gao Liu, Jianzhong Min, Kui Wang, Xiaotao Zhang. Serrin's regularity results for the incompressible liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5579-5594. doi: 10.3934/dcds.2016045
[8]

Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055
[9]

Moez Daoulatli, Irena Lasiecka, Daniel Toundykov. Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 67-94. doi: 10.3934/dcdss.2009.2.67
[10]

Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757
[11]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064
[12]

Henri Schurz. Dissipation of mean energy of discretized linear oscillators under random perturbations. Conference Publications, 2005, 2005 (Special) : 778-783. doi: 10.3934/proc.2005.2005.778
[13]

Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124
[14]

Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237
[15]

Kazumasa Fujiwara, Shuji Machihara, Tohru Ozawa. Remark on a semirelativistic equation in the energy space. Conference Publications, 2015, 2015 (special) : 473-478. doi: 10.3934/proc.2015.0473
[16]

Daomin Cao, Hang Li. High energy solutions of the Choquard equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3023-3032. doi: 10.3934/dcds.2018129
[17]

Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857
[18]

Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483
[19]

Michele Coti Zelati, Piotr Kalita. Smooth attractors for weak solutions of the SQG equation with critical dissipation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1857-1873. doi: 10.3934/dcdsb.2017110
[20]

Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences