Energy dissipation for weak solutions of incompressible liquid crystal flows

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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

Abstract
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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 and 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), 207-209. 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 $\mathbb{R}^2$, 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 $\mathbb{R}^2$, 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.

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), 207-209. 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 $\mathbb{R}^2$, 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 $\mathbb{R}^2$, 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.

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