American Institute of Mathematical Sciences

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

Energy dissipation for weak solutions of incompressible liquid crystal flows

 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 $D(u,d)=0$.
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

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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
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