Instability of point defects in a two-dimensional nematic liquid crystal model

Ignat, Radu; Nguyen, Luc; Slastikov, Valeriy; Zarnescu, Arghir

doi:10.1016/j.anihpc.2015.03.007

Ignat, Radu ; Nguyen, Luc ; Slastikov, Valeriy ; Zarnescu, Arghir

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 4, pp. 1131-1152.

Résumé

We study a class of symmetric critical points in a variational 2D Landau–de Gennes model where the state of nematic liquid crystals is described by symmetric traceless $3 \times 3$ matrices. These critical points play the role of topological point defects carrying a degree $\frac{k}{2}$ for a nonzero integer k. We prove existence and study the qualitative behavior of these symmetric solutions. Our main result is the instability of critical points when $| k | \geq 2$ .

MR Zbl

DOI : 10.1016/j.anihpc.2015.03.007

Mots clés : Nonlinear elliptic PDE system, Singular ODE system, Stability, Vortex, Liquid crystal defects

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     title = {Instability of point defects in a two-dimensional nematic liquid crystal model},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1131--1152},
     publisher = {Elsevier},
     volume = {33},
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     zbl = {1351.82110},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2015.03.007/}
}

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Ignat, Radu; Nguyen, Luc; Slastikov, Valeriy; Zarnescu, Arghir. Instability of point defects in a two-dimensional nematic liquid crystal model. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 4, pp. 1131-1152. doi : 10.1016/j.anihpc.2015.03.007. http://www.numdam.org/articles/10.1016/j.anihpc.2015.03.007/

Bibliographie
Cité par

[1] Bauman, P.; Philips, D. Analysis of nematic liquid crystals with disclination lines, Arch. Ration. Mech. Anal., Volume 205 (2012), pp. 795–826 | DOI | MR | Zbl

[2] Bethuel, F.; Brezis, H.; Coleman, B.D.; Hélein, F. Bifurcation analysis of minimizing harmonic maps describing the equilibrium of nematic phases between cylinders, Arch. Ration. Mech. Anal., Volume 118 (1992) no. 2, pp. 149–168 | DOI | MR | Zbl

[3] Bethuel, F.; Brezis, H.; Hélein, F. Ginzburg–Landau Vortices, Prog. Nonlinear Differ. Equ. Appl., vol. 13, Birkhäuser Boston Inc., Boston, MA, 1994 | MR | Zbl

[4] Brezis, H.; Coron, J.-M.; Lieb, E.H. Harmonic maps with defects, Commun. Math. Phys., Volume 107 (1986) no. 4, pp. 649–705 | DOI | MR | Zbl

[5] Canevari, G. Biaxiality in the asymptotic analysis of a 2-d Landau–de Gennes model for liquid crystals, ESAIM Control Optim. Calc. Var., Volume 21 (2015) no. 1, pp. 101–137 | DOI | Numdam | MR | Zbl

[6] Chandrasekhar, S.; Ranganath, G. The structure and energetics of defects in liquid crystals, Adv. Phys., Volume 35 (1986), pp. 507–596 | DOI

[7] Cladis, P.; Kleman, M. Non-singular disclinations of strength $s = + 1$ in nematics, J. Phys., Volume 33 (1972), pp. 591–598 | DOI

[8] de Gennes, P.; Prost, J. The Physics of Liquid Crystals, Oxford University Press, Oxford, 1995

[9] del Pino, M.; Felmer, P.; Kowalczyk, M. Minimality and nondegeneracy of degree-one Ginzburg–Landau vortex as a Hardy's type inequality, Int. Math. Res. Not., Volume 30 (2004), pp. 1511–1527 | MR | Zbl

[10] di Fratta, G.; Robbins, J.; Slastikov, V.; Zarnescu, A. Profiles of point defects in two dimensions in Landau–de Gennes theory | arXiv

[11] Döring, L.; Ignat, R.; Otto, F. A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types, J. Eur. Math. Soc., Volume 16 (2014) no. 7, pp. 1377–1422 | DOI | MR | Zbl

[12] Ericksen, J. Liquid crystals with variable degree of orientation, Arch. Ration. Mech. Anal., Volume 113 (1990) no. 2, pp. 97–120 | DOI | MR | Zbl

[13] Fatkullin, I.; Slastikov, V. On spatial variations of nematic ordering, Phys. D, Volume 237 (2008) no. 20, pp. 2577–2586 | DOI | MR | Zbl

[14] Fatkullin, I.; Slastikov, V. Vortices in two-dimensional nematics, Commun. Math. Sci., Volume 7 (2009) no. 4, pp. 917–938 | DOI | MR | Zbl

[15] Frank, F. On the theory of liquid crystals, Discuss. Faraday Soc., Volume 25 (1958), pp. 19–28 | DOI

[16] Golovaty, D.; Montero, A. On minimizers of a Landau–de Gennes energy functional on planar domains, Arch. Ration. Mech. Anal., Volume 213 (2014), pp. 447–490 | DOI | MR | Zbl

[17] Hervé, R.-M.; Hervé, M. Étude qualitative des solutions réelles d'une équation différentielle liée à l'équation de Ginzburg–Landau, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 11 (1994) no. 4, pp. 427–440 | Numdam | MR | Zbl

[18] Hu, Y.; Qu, Y.; Zhang, P. On the disclination lines of nematic liquid crystals, 2014 | arXiv | MR | Zbl

[19] Ignat, R.; Nguyen, L.; Slastikov, V.; Zarnescu, A. Stability of the vortex defect in the Landau–de Gennes theory for nematic liquid crystals, C. R. Math. Acad. Sci. Paris, Volume 351 (2013) no. 13–14, pp. 533–537 | Zbl

[20] Ignat, R.; Nguyen, L.; Slastikov, V.; Zarnescu, A. Uniqueness results for an ODE related to a generalized Ginzburg–Landau model for liquid crystals, SIAM J. Math. Anal., Volume 46 (2014) no. 5, pp. 3390–3425 | DOI | Zbl

[21] Ignat, R.; Nguyen, L.; Slastikov, V.; Zarnescu, A. Stability of the melting hedgehog in the Landau–de Gennes theory of nematic liquid crystals, Arch. Ration. Mech. Anal., Volume 215 (2015) no. 2, pp. 633–673 | DOI | Zbl

[22] Ignat, R.; Otto, F. A compactness result for Landau state in thin-film micromagnetics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 28 (2011) no. 2, pp. 247–282 | DOI | Numdam | Zbl

[23] Kleman, M. Points, Lines and Walls in Liquid Crystals, Magnetic Systems and Various Ordered Media, John Wiley & Sons, New York, 1983

[24] Kleman, M.; Lavrentovich, O. Topological point defects in nematic liquid crystals, Philos. Mag., Volume 86 (2006), pp. 4117–4137 | DOI

[25] Kralj, S.; Virga, E.G. Universal fine structure of nematic hedgehogs, J. Phys. A, Math. Gen., Volume 34 (2001) no. 4, pp. 829–838 | DOI | Zbl

[26] Kralj, S.; Virga, E.G.; Zumer, S. Biaxial torus around nematic point defects, Phys. Rev. E, Volume 60 (1999) no. 2, pp. 1858–1866 | DOI

[27] Majumdar, A.; Zarnescu, A. Landau–de Gennes theory of nematic liquid crystals: the Oseen–Frank limit and beyond, Arch. Ration. Mech. Anal., Volume 196 (2010) no. 1, pp. 227–280 | DOI | Zbl

[28] Palffy-Muhoray, P. The diverse world of liquid crystals, Phys. Today, Volume 60 (2007), pp. 54–60 | DOI

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