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May  2013, 12(3): 1445-1468. doi: 10.3934/cpaa.2013.12.1445

Energy conservative solutions to a nonlinear wave system of nematic liquid crystals

Geng Chen 1, , Ping Zhang 2, and  Yuxi Zheng 3,

1. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States
2. 

Academy of Mathematics & Systems Science, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, China
3. 

Department of Mathematical Sciences, Yeshiva University, New York, NY 10033

Received  March 2012 Revised  April 2012 Published  September 2012

We establish the global existence of solutions to the Cauchy problem for a system of hyperbolic partial differential equations in one space dimension modeling a type of nematic liquid crystals that has equal splay and twist coefficients. Our results have no restrictions on the angles of the director, as we use the director in its natural three-component form, rather than the two-component form of spherical angles.
Keywords: spherical angles, Camassa-Holm, three-component form., Hyperbolic.
Mathematics Subject Classification: Primary: 35L65, 35J70; Secondary: 35J6.
Citation: 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
References:
[1]

Giuseppe Alì and John Hunter, Orientation waves in a director field with rotational inertia, Kinet. Relat. Models, 2 (2009), 1-37.  Google Scholar

[2]

H. Berestycki, J. M. Coron and I. Ekeland (eds.), "Variational Methods,", in series, ().   Google Scholar

[3]

A. Bressan and Yuxi Zheng, Conservative solutions to a nonlinear variational wave equation, Comm. Math. Phys., 266 (2006), 471-497.  Google Scholar

[4]

D. Christodoulou and A. Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math., 46 (1993), 1041-1091.  Google Scholar

[5]

J. Coron, J. Ghidaglia and F. Hélein (eds.), "Nematics," Kluwer Academic Publishers, 1991. Google Scholar

[6]

J. L. Ericksen and D. Kinderlehrer (eds.), "Theory and Application of Liquid Crystals,", IMA Volumes in Mathematics and its Applications, ().   Google Scholar

[7]

James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697-715.  Google Scholar

[8]

R. Hardt, D. Kinderlehrer and Fanghua Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys., 105 (1986), 547-570.  Google Scholar

[9]

D. Kinderlehrer, Recent developments in liquid crystal theory, in "Frontiers in pure and applied mathematics" (R. Dautray ed.), Elsevier, New York, (1991), 151-178.  Google Scholar

[10]

R. A. Saxton, Dynamic instability of the liquid crystal director, in "Contemporary Mathematics, Vol. 100: Current Progress in Hyperbolic Systems" (W. B. Lindquist ed.), AMS, Providence, (1989), 325-330.  Google Scholar

[11]

J. Shatah, Weak solutions and development of singularities in the $SU(2)$ $\sigma$-model, Comm. Pure Appl. Math., 41 (1988), 459-469.  Google Scholar

[12]

J. Shatah and A. Tahvildar-Zadeh, Regularity of harmonic maps from Minkowski space into rotationally symmetric manifolds, Comm. Pure Appl. Math., 45 (1992), 947-971.  Google Scholar

[13]

J. Sterbenz and D. Tataru, Energy dispersed large data wave maps in $2+1$ dimensions, Comm. Math. Phys., 298 (2010), 139-230.  Google Scholar

[14]

J. Sterbenz and D. Tataru, Regularity of wave-maps in dimension $2+1$, Comm. Math. Phys., 298 (2010), 231-264.  Google Scholar

[15]

T. Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension, Internat. Math. Res. Notices, 6 (2001), 299-328.  Google Scholar

[16]

T. Tao, Global regularity of wave maps. II. Small energy in two dimensions, Comm. Math. Phys., 224 (2001), 443-544.  Google Scholar

[17]

E. Virga, "Variational Theories for Liquid Crystals," Chapman & Hall, New York, 1994. Google Scholar

[18]

Ping Zhang and Yuxi Zheng, Weak solutions to a nonlinear variational wave equation, Arch. Ration. Mech. Anal., 166 (2003), 303-319.  Google Scholar

[19]

Ping Zhang and Yuxi Zheng, Weak solutions to a nonlinear variational wave equation with general data, Ann. I. H. Poincaré, 22 (2005), 207-226.  Google Scholar

[20]

Ping Zhang and Yuxi Zheng, Conservative solutions to a system of variational wave equations of nematic liquid crystals, Arch. Ration. Mech. Anal., 195 (2010),701-727.  Google Scholar

[21]

Ping Zhang and Yuxi Zheng, Energy conservative solutions to a one-dimensional full variational wave system, Comm. Pure Appl. Math., 65 (2012), 683-726. Google Scholar

show all references

References:
[1]

Giuseppe Alì and John Hunter, Orientation waves in a director field with rotational inertia, Kinet. Relat. Models, 2 (2009), 1-37.  Google Scholar

[2]

H. Berestycki, J. M. Coron and I. Ekeland (eds.), "Variational Methods,", in series, ().   Google Scholar

[3]

A. Bressan and Yuxi Zheng, Conservative solutions to a nonlinear variational wave equation, Comm. Math. Phys., 266 (2006), 471-497.  Google Scholar

[4]

D. Christodoulou and A. Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math., 46 (1993), 1041-1091.  Google Scholar

[5]

J. Coron, J. Ghidaglia and F. Hélein (eds.), "Nematics," Kluwer Academic Publishers, 1991. Google Scholar

[6]

J. L. Ericksen and D. Kinderlehrer (eds.), "Theory and Application of Liquid Crystals,", IMA Volumes in Mathematics and its Applications, ().   Google Scholar

[7]

James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697-715.  Google Scholar

[8]

R. Hardt, D. Kinderlehrer and Fanghua Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys., 105 (1986), 547-570.  Google Scholar

[9]

D. Kinderlehrer, Recent developments in liquid crystal theory, in "Frontiers in pure and applied mathematics" (R. Dautray ed.), Elsevier, New York, (1991), 151-178.  Google Scholar

[10]

R. A. Saxton, Dynamic instability of the liquid crystal director, in "Contemporary Mathematics, Vol. 100: Current Progress in Hyperbolic Systems" (W. B. Lindquist ed.), AMS, Providence, (1989), 325-330.  Google Scholar

[11]

J. Shatah, Weak solutions and development of singularities in the $SU(2)$ $\sigma$-model, Comm. Pure Appl. Math., 41 (1988), 459-469.  Google Scholar

[12]

J. Shatah and A. Tahvildar-Zadeh, Regularity of harmonic maps from Minkowski space into rotationally symmetric manifolds, Comm. Pure Appl. Math., 45 (1992), 947-971.  Google Scholar

[13]

J. Sterbenz and D. Tataru, Energy dispersed large data wave maps in $2+1$ dimensions, Comm. Math. Phys., 298 (2010), 139-230.  Google Scholar

[14]

J. Sterbenz and D. Tataru, Regularity of wave-maps in dimension $2+1$, Comm. Math. Phys., 298 (2010), 231-264.  Google Scholar

[15]

T. Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension, Internat. Math. Res. Notices, 6 (2001), 299-328.  Google Scholar

[16]

T. Tao, Global regularity of wave maps. II. Small energy in two dimensions, Comm. Math. Phys., 224 (2001), 443-544.  Google Scholar

[17]

E. Virga, "Variational Theories for Liquid Crystals," Chapman & Hall, New York, 1994. Google Scholar

[18]

Ping Zhang and Yuxi Zheng, Weak solutions to a nonlinear variational wave equation, Arch. Ration. Mech. Anal., 166 (2003), 303-319.  Google Scholar

[19]

Ping Zhang and Yuxi Zheng, Weak solutions to a nonlinear variational wave equation with general data, Ann. I. H. Poincaré, 22 (2005), 207-226.  Google Scholar

[20]

Ping Zhang and Yuxi Zheng, Conservative solutions to a system of variational wave equations of nematic liquid crystals, Arch. Ration. Mech. Anal., 195 (2010),701-727.  Google Scholar

[21]

Ping Zhang and Yuxi Zheng, Energy conservative solutions to a one-dimensional full variational wave system, Comm. Pure Appl. Math., 65 (2012), 683-726. Google Scholar

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