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December  2017, 37(12): 6227-6242. doi: 10.3934/dcds.2017269

Stability of half-degree point defect profiles for 2-D nematic liquid crystal

Zhiyuan Geng 1, , Wei Wang 2, , Pingwen Zhang 3, and  Zhifei Zhang 3,

1. 

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012-1185, USA
2. 

Department of Mathematics, Zhejiang University, Hangzhou 310027, China
3. 

School of Mathematical Sciences, Peking University, Beijing 100871, China

Received  December 2016 Revised  July 2017 Published  August 2017

Fund Project: W. Wang is partly supported by NSF of China under Grant 11501502 and the Fundamental Research Funds for the Central Universities 2016QNA3004. P. Zhang is partly supported by NSF of China under Grant 11421101 and 11421110001. Z. Zhang is partly supported by NSF of China under Grant 11371039 and 11425103.

In this paper, we prove the stability of half-degree point defect profiles in $\mathbb{R}^2$ for the nematic liquid crystal within Landau-de Gennes model.

Keywords: Liquid crystal, point defect, Landau-de Gennes model, monotonicity, stability.
Mathematics Subject Classification: Primary: 76A15, 35A15; Secondary: 82D30, 35B38, 35J57.
Citation: Zhiyuan Geng, Wei Wang, Pingwen Zhang, Zhifei Zhang. Stability of half-degree point defect profiles for 2-D nematic liquid crystal. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6227-6242. doi: 10.3934/dcds.2017269
References:
[1]

P. Bauman, J. Park and D. Philips, Analysis of nematic liquid crystals with disclination lines, Arch. Ration. Mech. Anal., 205 (2012), 795-826.  doi: 10.1007/s00205-012-0530-7.

[2]

P. Biscari and G. G. Peroli, A hierarchy of defects in biaxial nematics, Commun. Math. Phys, 186 (1997), 381-392.  doi: 10.1007/s002200050113.

[3]

G. Canevari, Biaxiality in the asymptotic analysis of a 2-d Landau-de Gennes model for liquid crystals, ESAIM Control Optim. Calc. Var., 21 (2015), 101-137.  doi: 10.1051/cocv/2014025.

[4]

P. de Gennes and J. Prost, The Physics of Liquid Crystals, 2ndedition, Oxford University Press, Oxford, 1995.

[5]

G. Di Fratta, J. M. Robbins, V. Slastikov and A. Zarnescu, Half-integer point defects in the Q-tensor theory of nematic liquid crystals, Journal of Nonlinear Science, 26 (2016), 121-140.  doi: 10.1007/s00332-015-9271-8.

[6]

J. Ericksen, Liquid crystals with variable degree of orientation, Arch. Ration. Mech. Anal., 113 (1990), 97-120.  doi: 10.1007/BF00380413.

[7]

D. Golovaty and J. A. Montero, On minimizers of a Landau-de Gennes energy functional on planar domains, Arch. Ration. Mech. Anal., 213 (2014), 447-490.  doi: 10.1007/s00205-014-0731-3.

[8]

S. Gustafson and I. M. Sigal, The stability of magnetic vortices, Commun. Math. Phys., 212 (2000), 257-275.  doi: 10.1007/PL00005526.

[9]

R. Hardt, D. Kinderlehrer and F.-H. Lin, Existence and partial regularity of static liquid crystal configurations, Commun. Math. Phys., 105 (1986), 547-570.  doi: 10.1007/BF01238933.

[10]

F. Hélein, Minima de la fonctionelle energie libre des cristaux liquides, C. R. Acad. Sci. Paris, 305 (1987), 565-568. 

[11]

Y. Hu, Y. Qu and P. Zhang, On the disclination lines of nematic liquid crystals, Communications in Computational Physics, 19 (2016), 354-379.  doi: 10.4208/cicp.210115.180515a.

[12]

R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu, Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals, SIAM J. Math. Anal., 46 (2014), 3390-3425.  doi: 10.1137/130948598.

[13]

R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu, Stability of the melting hedgehog in the Landau-de Gennes theory of nematic liquid crystals, Arch. Ration. Mech. Anal., 215 (2015), 633-673.  doi: 10.1007/s00205-014-0791-4.

[14]

R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu, Instability of point defects in a two-dimensional nematic liquid crystal model, Ann. I. H. Poincare-AN, 33 (2016), 1131-1152.  doi: 10.1016/j.anihpc.2015.03.007.

[15]

R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu, Stability of point defects of degree $±1/2$ in a two-dimensional nematic liquid crystal model Calculus of Variations and Partial Differential Equations, 55 (2016), 33pp. doi: 10.1007/s00526-016-1051-2.

[16]

M. Kleman and O. D. Lavrentovich, Topological point defects in nematic liquid crystals, Philosophical Magazine, 86 (2006), 4117-4137.  doi: 10.1080/14786430600593016.

[17]

X. Lamy, Some properties of the nematic radial hedgehog in the Landau-de Gennes theory, J. Math. Anal. Appl., 397 (2013), 586-594.  doi: 10.1016/j.jmaa.2012.08.011.

[18]

E. H. Lieb and M. Loss, Symmetry of the Ginzburg-Landau mimimizer in a disc, Math. Res. Lett., 1 (1994), 701-715.  doi: 10.4310/MRL.1994.v1.n6.a7.

[19]

F.-H. Lin and C. Liu, Static and dynamic theories of liquid crystals, J. Partial Differ. Equ., 14 (2001), 289-330. 

[20]

T.-C. Lin, The stability of the radial solution to the Ginzburg-Landau equation, Commun. PDE, 22 (1997), 619-632.  doi: 10.1080/03605309708821276.

[21]

A. Majumdar, The radial-hedgehog solution in Landau-de Gennes' theory for nematic liquid crystals, Euro. J. Appl. Math., 23 (2012), 61-97.  doi: 10.1017/S0956792511000295.

[22]

A. Majumdar and A. Zarnescu, Landau-de Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196 (2010), 227-280.  doi: 10.1007/s00205-009-0249-2.

[23]

N. D. Mermin, The topological theory of defects in ordered media, Rev. Modern Phys., 51 (1979), 591-648.  doi: 10.1103/RevModPhys.51.591.

[24]

P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. Anal., 130 (1995), 334-344.  doi: 10.1006/jfan.1995.1073.

[25]

Manuel de Pino, P. Felmer and M. Kowalczyk, Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's inequality, IMRN, 30 (2004), 1511-1527.  doi: 10.1155/S1073792804133588.

[26]

R. Rosso and E. G. Virga, Metastable nematic hedgehogs, J. Phys. A, 29 (1996), 4247-4264.  doi: 10.1088/0305-4470/29/14/041.

[27]

G. Toulouse and M. Kleman, Principles of a classification of defects in ordered media, Journal de Physique Lettres, 37 (1976), 149-151.  doi: 10.1051/jphyslet:01976003706014900.

show all references

References:
[1]

P. Bauman, J. Park and D. Philips, Analysis of nematic liquid crystals with disclination lines, Arch. Ration. Mech. Anal., 205 (2012), 795-826.  doi: 10.1007/s00205-012-0530-7.

[2]

P. Biscari and G. G. Peroli, A hierarchy of defects in biaxial nematics, Commun. Math. Phys, 186 (1997), 381-392.  doi: 10.1007/s002200050113.

[3]

G. Canevari, Biaxiality in the asymptotic analysis of a 2-d Landau-de Gennes model for liquid crystals, ESAIM Control Optim. Calc. Var., 21 (2015), 101-137.  doi: 10.1051/cocv/2014025.

[4]

P. de Gennes and J. Prost, The Physics of Liquid Crystals, 2ndedition, Oxford University Press, Oxford, 1995.

[5]

G. Di Fratta, J. M. Robbins, V. Slastikov and A. Zarnescu, Half-integer point defects in the Q-tensor theory of nematic liquid crystals, Journal of Nonlinear Science, 26 (2016), 121-140.  doi: 10.1007/s00332-015-9271-8.

[6]

J. Ericksen, Liquid crystals with variable degree of orientation, Arch. Ration. Mech. Anal., 113 (1990), 97-120.  doi: 10.1007/BF00380413.

[7]

D. Golovaty and J. A. Montero, On minimizers of a Landau-de Gennes energy functional on planar domains, Arch. Ration. Mech. Anal., 213 (2014), 447-490.  doi: 10.1007/s00205-014-0731-3.

[8]

S. Gustafson and I. M. Sigal, The stability of magnetic vortices, Commun. Math. Phys., 212 (2000), 257-275.  doi: 10.1007/PL00005526.

[9]

R. Hardt, D. Kinderlehrer and F.-H. Lin, Existence and partial regularity of static liquid crystal configurations, Commun. Math. Phys., 105 (1986), 547-570.  doi: 10.1007/BF01238933.

[10]

F. Hélein, Minima de la fonctionelle energie libre des cristaux liquides, C. R. Acad. Sci. Paris, 305 (1987), 565-568. 

[11]

Y. Hu, Y. Qu and P. Zhang, On the disclination lines of nematic liquid crystals, Communications in Computational Physics, 19 (2016), 354-379.  doi: 10.4208/cicp.210115.180515a.

[12]

R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu, Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals, SIAM J. Math. Anal., 46 (2014), 3390-3425.  doi: 10.1137/130948598.

[13]

R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu, Stability of the melting hedgehog in the Landau-de Gennes theory of nematic liquid crystals, Arch. Ration. Mech. Anal., 215 (2015), 633-673.  doi: 10.1007/s00205-014-0791-4.

[14]

R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu, Instability of point defects in a two-dimensional nematic liquid crystal model, Ann. I. H. Poincare-AN, 33 (2016), 1131-1152.  doi: 10.1016/j.anihpc.2015.03.007.

[15]

R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu, Stability of point defects of degree $±1/2$ in a two-dimensional nematic liquid crystal model Calculus of Variations and Partial Differential Equations, 55 (2016), 33pp. doi: 10.1007/s00526-016-1051-2.

[16]

M. Kleman and O. D. Lavrentovich, Topological point defects in nematic liquid crystals, Philosophical Magazine, 86 (2006), 4117-4137.  doi: 10.1080/14786430600593016.

[17]

X. Lamy, Some properties of the nematic radial hedgehog in the Landau-de Gennes theory, J. Math. Anal. Appl., 397 (2013), 586-594.  doi: 10.1016/j.jmaa.2012.08.011.

[18]

E. H. Lieb and M. Loss, Symmetry of the Ginzburg-Landau mimimizer in a disc, Math. Res. Lett., 1 (1994), 701-715.  doi: 10.4310/MRL.1994.v1.n6.a7.

[19]

F.-H. Lin and C. Liu, Static and dynamic theories of liquid crystals, J. Partial Differ. Equ., 14 (2001), 289-330. 

[20]

T.-C. Lin, The stability of the radial solution to the Ginzburg-Landau equation, Commun. PDE, 22 (1997), 619-632.  doi: 10.1080/03605309708821276.

[21]

A. Majumdar, The radial-hedgehog solution in Landau-de Gennes' theory for nematic liquid crystals, Euro. J. Appl. Math., 23 (2012), 61-97.  doi: 10.1017/S0956792511000295.

[22]

A. Majumdar and A. Zarnescu, Landau-de Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196 (2010), 227-280.  doi: 10.1007/s00205-009-0249-2.

[23]

N. D. Mermin, The topological theory of defects in ordered media, Rev. Modern Phys., 51 (1979), 591-648.  doi: 10.1103/RevModPhys.51.591.

[24]

P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. Anal., 130 (1995), 334-344.  doi: 10.1006/jfan.1995.1073.

[25]

Manuel de Pino, P. Felmer and M. Kowalczyk, Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's inequality, IMRN, 30 (2004), 1511-1527.  doi: 10.1155/S1073792804133588.

[26]

R. Rosso and E. G. Virga, Metastable nematic hedgehogs, J. Phys. A, 29 (1996), 4247-4264.  doi: 10.1088/0305-4470/29/14/041.

[27]

G. Toulouse and M. Kleman, Principles of a classification of defects in ordered media, Journal de Physique Lettres, 37 (1976), 149-151.  doi: 10.1051/jphyslet:01976003706014900.

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