Stability of the melting hedgehog in the Landau-de Gennes theory of nematic liquid crystals. (English) Zbl 1308.35213
Summary: We investigate stability properties of the radially symmetric solution corresponding to the vortex defect (the so called “melting hedgehog”) in the framework of the Landau-de Gennes model of nematic liquid crystals. We prove local stability of the melting hedgehog under arbitrary \(Q\)-tensor valued perturbations in the temperature regime near the critical supercooling temperature. As a consequence of our method, we also rediscover the loss of stability of the vortex defect in the deep nematic regime.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76A15 | Liquid crystals |
| 35A15 | Variational methods applied to PDEs |
| 35B35 | Stability in context of PDEs |
References:
| [1] | Ball J.M., Zarnescu A.: Orientability and energy minimization for liquid crystal models. Arch. Ration. Mech. Anal., 202, 493-535 (2011) · Zbl 1263.76010 · doi:10.1007/s00205-011-0421-3 |
| [2] | Béthuel F., Brezis H., Hélein F.: Asymptotics for the minimization of a Ginzburg-Landau functional. Calc. Var. Partial Diff. Equ. 1, 123-148 (1993) · Zbl 0834.35014 · doi:10.1007/BF01191614 |
| [3] | Brezis H., Coron J.-M., Lieb E.H.: Harmonic maps with defects. Commun. Math. Phys. 107, 649-705 (1986) · Zbl 0608.58016 · doi:10.1007/BF01205490 |
| [4] | Chandrasekhar , Ranganath G.: The structure and energetics of defects in liquid crystals. Adv. Phys. 35, 507-596 (1986) · doi:10.1080/00018738600101941 |
| [5] | Cladis, P., Kleman, M.: Non-singular disclinations of strength s = + 1 in nematics. J. Phys. 33, 591-598 (1972) · Zbl 0611.35077 |
| [6] | Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Interscience Publishers, Inc., New York, 1953 · Zbl 0051.28802 |
| [7] | de Gennes, P., Prost, J.: The Physics of Liquid Crystals, 2nd edn. Oxford University Press, Oxford, 1995 |
| [8] | de Gennes P.G.: Types of singularities permitted in the ordered phase. C. R. Acad. Sci. Paris Ser. B 275, 319-321 (1972) |
| [9] | Ericksen J.: Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113, 97-120 (1990) · Zbl 0729.76008 · doi:10.1007/BF00380413 |
| [10] | Fatkullin I., Slastikov V.: On spatial variations of nematic ordering. Physica D 237, 2577-2586 (2008) · Zbl 1149.76005 · doi:10.1016/j.physd.2008.03.048 |
| [11] | Fatkullin I., Slastikov V.: Vortices in two-dimensional nematics. Commun. Math. Sci. 7, 917-938 (2009) · Zbl 1187.82035 · doi:10.4310/CMS.2009.v7.n4.a6 |
| [12] | Frank, F.: On the theory of liquid crystals. Discuss. Faraday Soc. 25, 1 (1958) |
| [13] | Gartland E.C., Mkaddem S.: Instability of radial hedgehog configurations in nematic liquid crystals under Landau-de Gennes free-energy models. Phys. Rev. E 59, 563-567 (1999) · doi:10.1103/PhysRevE.59.563 |
| [14] | Gustafson S.: Symmetric solutions of the Ginzburg-Landau equation in all dimensions. Int. Math. Res. Not., 16, 807-816 (1997) · Zbl 0883.35041 · doi:10.1155/S1073792897000524 |
| [15] | Hardt R., Kinderlehrer D., Lin F.-H.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105, 547-570 (1986) · Zbl 0611.35077 · doi:10.1007/BF01238933 |
| [16] | 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. doi:10.1137/130948598 · Zbl 1321.34035 |
| [17] | 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 351, 533-537 (2013) · Zbl 1276.35030 |
| [18] | Kleman M., Lavrentovich O.: Topological point defects in nematic liquid crystals. Philos. Mag. 86, 4117-4137 (2006) · doi:10.1080/14786430600593016 |
| [19] | Kralj S., Virga E.G.: Universal fine structure of nematic hedgehogs. J. Phys. A Gen. 34, 829-838 (2001) · Zbl 1015.82036 · doi:10.1088/0305-4470/34/4/309 |
| [20] | Lin F.-H., Liu C.: Static and dynamic theories of liquid crystals. J. Partial Differ. Equ. 14, 289-330 (2001) · Zbl 1433.82014 |
| [21] | Majumdar A., Zarnescu A.: Landau-de Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond. Arch. Ration. Mech. Anal. 196, 227-280 (2010) · Zbl 1304.76007 · doi:10.1007/s00205-009-0249-2 |
| [22] | Millot V., Pisante A.: Symmetry of local minimizers for the three-dimensional Ginzburg-Landau functional. J. Eur. Math. Soc. 12, 1069-1096 (2010) · Zbl 1204.35156 · doi:10.4171/JEMS/223 |
| [23] | Mironescu P.: On the stability of radial solutions of the Ginzburg-Landau equation. J. Funct. Anal. 130, 334-344 (1995) · Zbl 0839.35011 · doi:10.1006/jfan.1995.1073 |
| [24] | Mkaddem S., Gartland E.C.: Fine structure of defects in radial nematic droplets. Phys. Rev. E 62, 6694-6705 (2000) · doi:10.1103/PhysRevE.62.6694 |
| [25] | Nehring J., Saupe A.: Schlieren texture in nematic and smectic liquid-crystals. J. Chem. Soc. Faraday Trans. II 68, 1-15 (1972) · doi:10.1039/f29726800001 |
| [26] | Nguyen L., Zarnescu A.: Refined approximation for minimizers of a Landau-de Gennes energy functional. Calc. Var. Partial Differ. Equ. 47, 383-432 (2013) · Zbl 1273.35260 · doi:10.1007/s00526-012-0522-3 |
| [27] | Rosso R., Virga E.G.: Metastable nematic hedgehogs. J. Phys. A 29, 4247-4264 (1996) · Zbl 0902.34045 · doi:10.1088/0305-4470/29/14/041 |
| [28] | Virga, E.G.: Variational Theories for Liquid Crystals. Applied Mathematics and Mathematical Computation, vol. 8. Chapman & Hall, London, 1994 · Zbl 0814.49002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
