September  2015, 14(5): 1623-1639. doi: 10.3934/cpaa.2015.14.1623

Shape optimization in compressible liquid crystals

1. 

College of Information and Management Science, Henan Agricultural University, Zhengzhou, China

2. 

School of Mathematical Sciences, Fudan University, Shanghai

Received  October 2012 Revised  August 2013 Published  June 2015

The shape optimization problem for the profile in compressible liquid crystals is considered in this paper. We prove that the optimal shape with minimal volume is attainable in an appropriate class of admissible profiles which subjects to a constraint on the thickness of the boundary. Such consequence is mainly obtained from the well-known weak sequential compactness method (see [25]).
Citation: 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
References:
[1]

D. Bucur and J. P. Zolésio, Free boundary problems and density perimeter,, \emph{J. Differential Equations}, 126 (1996), 224. doi: 10.1006/jdeq.1996.0050.

[2]

Y. Chu, W. Ma and X. Liu, Long-time behaviour of solutions to the compressible liquid crystals,, \emph{Sci. Sin. Math.}, 42 (2012), 107.

[3]

S. J. Ding, C. Y. Wang and H. Y. Wen, Weak solution to compressible hydrodynamic flow of liquid crystal in dimension one,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 15 (2011), 357. doi: 10.3934/dcdsb.2011.15.357.

[4]

J. L. Ericksen, Conservation laws for liquid crystals,, \emph{Trans. Soc. Rheol.}, 5 (1961), 23.

[5]

J. L. Ericksen, Hydrostatic theory of liquid crystals,, \emph{Arch. Rational Mech. Anal.}, 9 (1962), 371.

[6]

E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004).

[7]

E. Feireisl, Shape optimization in viscous compressible fluids,, \emph{Appl. Math. Optim.}, 47 (2003), 59. doi: 10.1007/s00245-002-0737-3.

[8]

F. C. Frank, On the theory of liquid crystals,, \emph{Discussions Faraday Soc.}, 25 (1958), 19.

[9]

P. G. de Gennes, The Physics of Liquid Crystals,, Oxford University Press, (1974).

[10]

D. Hoff, Strong convergence to global solutions for multidimensonal flows of compressible, isothermal flow with large, discontinuous initial data,, \emph{Arch. Rational Mech. Anal.}, 132 (1995), 1. doi: 10.1007/BF00390346.

[11]

M. Hong, J. Li and Z. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in $\mathbb R^3$,, Comm. Partial Differential Equations, 39 (2014), 1284. doi: 10.1080/03605302.2013.871026.

[12]

M. Hong, Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbb R^2,$, Adv. Math., 231 (2012), 1364.

[13]

T. Huang, C. Y. Wang and H. Y. Wen, Strong solutions of the compressible nematic liquid crystal flow,, \emph{J. Differential Equations}, 252 (2012), 2222. doi: 10.1016/j.jde.2011.07.036.

[14]

T. Huang, C. Y. Wang and H. Y. Wen, Blow up criteridon for compressible nematic liquid crystal flows in dimension three,, \emph{Arch. Ration. Mech. Anal.}, 204 (2012), 285.

[15]

S. Kaur, S. P. Singh and A. M. Biradar, Enhanced electro-optical properties in gold nanoparticles doped ferroelectric liquid crystals,, \emph{Appl. Phys. Lett.}, 91 (2007).

[16]

B. Kawohl, O. Pironneau, L. Tartar and J. P. Zolésio, Optimal Shape Design,, Lecture Notes in Mathematics 1740, (1740). doi: 10.1007/BFb0106739.

[17]

F. M. Leslie, Some constitutive equations for anisotropic fluids,, \emph{Quart. J. Mech. Appl. Math.}, 19 (1966), 357.

[18]

F. M. Leslie, Some constitutive equations for liquid crystals,, \emph{Arch. Rational Mech. Anal.}, 28 (1968), 265. doi: 10.1007/BF00251810.

[19]

F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena,, \emph{Comm. Pure. Appl. Math.}, 42 (1989), 789. doi: 10.1002/cpa.3160420605.

[20]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, \emph{Comm. Pure Appl. Math.}, 48 (1995), 501. doi: 10.1002/cpa.3160480503.

[21]

F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals,, \emph{Discrete and Continuous Dynamic Systems}, 2 (1996), 1.

[22]

F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system,, \emph{Arch. Rational Mech. Anal.}, 154 (2000), 135. doi: 10.1007/s002050000102.

[23]

F. H. Lin and C. Liu, Static and dynamic theories of liquid crystals,, \emph{Journal of Partial Differential Equations}, 14 (2001), 289.

[24]

P. L. Lions, Mathematical Topics in Fluid Dynamics,, Vol. 1. Incmpressible models. Oxford Science Publication, (1996).

[25]

P. L. Lions, Mathematical Topics in Fluid Dynamics,, Vol. 2. Compressible models. Oxford Science Publication, (1998).

[26]

X. Liu, L. Liu and Y. Hao, Existence of strong solutions for the compressible Ericksen-Leslie model,, http://arxiv.org/abs/1106.6140, ().

[27]

L. Liu and X. Liu, A blow-up ctiterion of strong solutions to the compressible liquid crystals system,, \emph{Chinese Journal of Contemporary Mathematics}, 32 (2011), 211.

[28]

X. Liu and J. Qing, Globally weak solutions to the flow of compressible liquid crystals system,, \emph{Discrete and Continuous Dynamical System A}, 33 (2013), 757. doi: 10.3934/dcds.2013.33.757.

[29]

X. Liu and Z. Zhang, Global existence of weak solutions for the incompressible liquid crystals,, \emph{Chinese Ann. Math. Ser. A}, 30 (2009), 1.

[30]

W. Ma, H. Gong, J. Li, Global strong solutions to incompressible Ericksen-Leslie system in $\mathbb R^3$,, \emph{Nonlinear Anal.}, 109 (2014), 230.

[31]

W. Ma and X. Liu, The boundedness of energy for the compressible liqud crystals system,, \emph{Chinese Ann. Math. Ser. A}, 32 (2011), 1.

[32]

W. C. Oseen, The theory of liquid crystals,, \emph{Trans. Faraday Soc.}, 29 (1933), 883.

[33]

D. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals,, \emph{Arch. Rational Mech. Anal.}, 204 (2012), 881. doi: 10.1007/s00205-011-0488-x.

show all references

References:
[1]

D. Bucur and J. P. Zolésio, Free boundary problems and density perimeter,, \emph{J. Differential Equations}, 126 (1996), 224. doi: 10.1006/jdeq.1996.0050.

[2]

Y. Chu, W. Ma and X. Liu, Long-time behaviour of solutions to the compressible liquid crystals,, \emph{Sci. Sin. Math.}, 42 (2012), 107.

[3]

S. J. Ding, C. Y. Wang and H. Y. Wen, Weak solution to compressible hydrodynamic flow of liquid crystal in dimension one,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 15 (2011), 357. doi: 10.3934/dcdsb.2011.15.357.

[4]

J. L. Ericksen, Conservation laws for liquid crystals,, \emph{Trans. Soc. Rheol.}, 5 (1961), 23.

[5]

J. L. Ericksen, Hydrostatic theory of liquid crystals,, \emph{Arch. Rational Mech. Anal.}, 9 (1962), 371.

[6]

E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004).

[7]

E. Feireisl, Shape optimization in viscous compressible fluids,, \emph{Appl. Math. Optim.}, 47 (2003), 59. doi: 10.1007/s00245-002-0737-3.

[8]

F. C. Frank, On the theory of liquid crystals,, \emph{Discussions Faraday Soc.}, 25 (1958), 19.

[9]

P. G. de Gennes, The Physics of Liquid Crystals,, Oxford University Press, (1974).

[10]

D. Hoff, Strong convergence to global solutions for multidimensonal flows of compressible, isothermal flow with large, discontinuous initial data,, \emph{Arch. Rational Mech. Anal.}, 132 (1995), 1. doi: 10.1007/BF00390346.

[11]

M. Hong, J. Li and Z. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in $\mathbb R^3$,, Comm. Partial Differential Equations, 39 (2014), 1284. doi: 10.1080/03605302.2013.871026.

[12]

M. Hong, Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbb R^2,$, Adv. Math., 231 (2012), 1364.

[13]

T. Huang, C. Y. Wang and H. Y. Wen, Strong solutions of the compressible nematic liquid crystal flow,, \emph{J. Differential Equations}, 252 (2012), 2222. doi: 10.1016/j.jde.2011.07.036.

[14]

T. Huang, C. Y. Wang and H. Y. Wen, Blow up criteridon for compressible nematic liquid crystal flows in dimension three,, \emph{Arch. Ration. Mech. Anal.}, 204 (2012), 285.

[15]

S. Kaur, S. P. Singh and A. M. Biradar, Enhanced electro-optical properties in gold nanoparticles doped ferroelectric liquid crystals,, \emph{Appl. Phys. Lett.}, 91 (2007).

[16]

B. Kawohl, O. Pironneau, L. Tartar and J. P. Zolésio, Optimal Shape Design,, Lecture Notes in Mathematics 1740, (1740). doi: 10.1007/BFb0106739.

[17]

F. M. Leslie, Some constitutive equations for anisotropic fluids,, \emph{Quart. J. Mech. Appl. Math.}, 19 (1966), 357.

[18]

F. M. Leslie, Some constitutive equations for liquid crystals,, \emph{Arch. Rational Mech. Anal.}, 28 (1968), 265. doi: 10.1007/BF00251810.

[19]

F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena,, \emph{Comm. Pure. Appl. Math.}, 42 (1989), 789. doi: 10.1002/cpa.3160420605.

[20]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, \emph{Comm. Pure Appl. Math.}, 48 (1995), 501. doi: 10.1002/cpa.3160480503.

[21]

F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals,, \emph{Discrete and Continuous Dynamic Systems}, 2 (1996), 1.

[22]

F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system,, \emph{Arch. Rational Mech. Anal.}, 154 (2000), 135. doi: 10.1007/s002050000102.

[23]

F. H. Lin and C. Liu, Static and dynamic theories of liquid crystals,, \emph{Journal of Partial Differential Equations}, 14 (2001), 289.

[24]

P. L. Lions, Mathematical Topics in Fluid Dynamics,, Vol. 1. Incmpressible models. Oxford Science Publication, (1996).

[25]

P. L. Lions, Mathematical Topics in Fluid Dynamics,, Vol. 2. Compressible models. Oxford Science Publication, (1998).

[26]

X. Liu, L. Liu and Y. Hao, Existence of strong solutions for the compressible Ericksen-Leslie model,, http://arxiv.org/abs/1106.6140, ().

[27]

L. Liu and X. Liu, A blow-up ctiterion of strong solutions to the compressible liquid crystals system,, \emph{Chinese Journal of Contemporary Mathematics}, 32 (2011), 211.

[28]

X. Liu and J. Qing, Globally weak solutions to the flow of compressible liquid crystals system,, \emph{Discrete and Continuous Dynamical System A}, 33 (2013), 757. doi: 10.3934/dcds.2013.33.757.

[29]

X. Liu and Z. Zhang, Global existence of weak solutions for the incompressible liquid crystals,, \emph{Chinese Ann. Math. Ser. A}, 30 (2009), 1.

[30]

W. Ma, H. Gong, J. Li, Global strong solutions to incompressible Ericksen-Leslie system in $\mathbb R^3$,, \emph{Nonlinear Anal.}, 109 (2014), 230.

[31]

W. Ma and X. Liu, The boundedness of energy for the compressible liqud crystals system,, \emph{Chinese Ann. Math. Ser. A}, 32 (2011), 1.

[32]

W. C. Oseen, The theory of liquid crystals,, \emph{Trans. Faraday Soc.}, 29 (1933), 883.

[33]

D. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals,, \emph{Arch. Rational Mech. Anal.}, 204 (2012), 881. doi: 10.1007/s00205-011-0488-x.

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