# American Institute of Mathematical Sciences

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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [4] J. L. Ericksen, Conservation laws for liquid crystals,, \emph{Trans. Soc. Rheol.}, 5 (1961), 23.   Google Scholar [5] J. L. Ericksen, Hydrostatic theory of liquid crystals,, \emph{Arch. Rational Mech. Anal.}, 9 (1962), 371.   Google Scholar [6] E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004).   Google Scholar [7] E. Feireisl, Shape optimization in viscous compressible fluids,, \emph{Appl. Math. Optim.}, 47 (2003), 59.  doi: 10.1007/s00245-002-0737-3.  Google Scholar [8] F. C. Frank, On the theory of liquid crystals,, \emph{Discussions Faraday Soc.}, 25 (1958), 19.   Google Scholar [9] P. G. de Gennes, The Physics of Liquid Crystals,, Oxford University Press, (1974).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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).   Google Scholar [16] B. Kawohl, O. Pironneau, L. Tartar and J. P. Zolésio, Optimal Shape Design,, Lecture Notes in Mathematics 1740, (1740).  doi: 10.1007/BFb0106739.  Google Scholar [17] F. M. Leslie, Some constitutive equations for anisotropic fluids,, \emph{Quart. J. Mech. Appl. Math.}, 19 (1966), 357.   Google Scholar [18] F. M. Leslie, Some constitutive equations for liquid crystals,, \emph{Arch. Rational Mech. Anal.}, 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [23] F. H. Lin and C. Liu, Static and dynamic theories of liquid crystals,, \emph{Journal of Partial Differential Equations}, 14 (2001), 289.   Google Scholar [24] P. L. Lions, Mathematical Topics in Fluid Dynamics,, Vol. 1. Incmpressible models. Oxford Science Publication, (1996).   Google Scholar [25] P. L. Lions, Mathematical Topics in Fluid Dynamics,, Vol. 2. Compressible models. Oxford Science Publication, (1998).   Google Scholar [26] X. Liu, L. Liu and Y. Hao, Existence of strong solutions for the compressible Ericksen-Leslie model,, http://arxiv.org/abs/1106.6140, ().   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [32] W. C. Oseen, The theory of liquid crystals,, \emph{Trans. Faraday Soc.}, 29 (1933), 883.   Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [4] J. L. Ericksen, Conservation laws for liquid crystals,, \emph{Trans. Soc. Rheol.}, 5 (1961), 23.   Google Scholar [5] J. L. Ericksen, Hydrostatic theory of liquid crystals,, \emph{Arch. Rational Mech. Anal.}, 9 (1962), 371.   Google Scholar [6] E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004).   Google Scholar [7] E. Feireisl, Shape optimization in viscous compressible fluids,, \emph{Appl. Math. Optim.}, 47 (2003), 59.  doi: 10.1007/s00245-002-0737-3.  Google Scholar [8] F. C. Frank, On the theory of liquid crystals,, \emph{Discussions Faraday Soc.}, 25 (1958), 19.   Google Scholar [9] P. G. de Gennes, The Physics of Liquid Crystals,, Oxford University Press, (1974).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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).   Google Scholar [16] B. Kawohl, O. Pironneau, L. Tartar and J. P. Zolésio, Optimal Shape Design,, Lecture Notes in Mathematics 1740, (1740).  doi: 10.1007/BFb0106739.  Google Scholar [17] F. M. Leslie, Some constitutive equations for anisotropic fluids,, \emph{Quart. J. Mech. Appl. Math.}, 19 (1966), 357.   Google Scholar [18] F. M. Leslie, Some constitutive equations for liquid crystals,, \emph{Arch. Rational Mech. Anal.}, 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [23] F. H. Lin and C. Liu, Static and dynamic theories of liquid crystals,, \emph{Journal of Partial Differential Equations}, 14 (2001), 289.   Google Scholar [24] P. L. Lions, Mathematical Topics in Fluid Dynamics,, Vol. 1. Incmpressible models. Oxford Science Publication, (1996).   Google Scholar [25] P. L. Lions, Mathematical Topics in Fluid Dynamics,, Vol. 2. Compressible models. Oxford Science Publication, (1998).   Google Scholar [26] X. Liu, L. Liu and Y. Hao, Existence of strong solutions for the compressible Ericksen-Leslie model,, http://arxiv.org/abs/1106.6140, ().   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [32] W. C. Oseen, The theory of liquid crystals,, \emph{Trans. Faraday Soc.}, 29 (1933), 883.   Google Scholar [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.  Google Scholar
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