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On the stability of radial solutions to an anisotropic Ginzburg-Landau equation. (English) Zbl 1483.35233

Summary: We study the linear stability of entire radial solutions \(u(re^{i\theta})=f(r)e^{i\theta}\), with positive increasing profile \(f(r)\), to the anisotropic Ginzburg-Landau equation \(-\Delta u -\delta (\partial_x+i\partial_y)^2\bar u =(1-|u|^2)u\), \(-1<\delta <1\), which arises in various liquid crystal models. In the isotropic case \(\delta=0\), Mironescu showed that such solution is nondegenerately stable. We prove stability of this radial solution in the range \(\delta\in (\delta_1,0]\) for some \(-1<\delta_1<0\) and instability outside this range. In strong contrast with the isotropic case, stability with respect to higher Fourier modes is not a direct consequence of stability with respect to lower Fourier modes. In particular, in the case where \(\delta\approx -1\), lower modes are stable and yet higher modes are unstable.

MSC:

35Q56 Ginzburg-Landau equations
35B35 Stability in context of PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
35R09 Integro-partial differential equations
76A15 Liquid crystals
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References:

[1] X. Chen, C. M. Elliott, and T. Qi, Shooting method for vortex solutions of a complex-valued Ginzburg-Landau equation, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), pp. 1075-1088, https://doi.org/10.1017/S0308210500030122. · Zbl 0816.34003
[2] M. G. Clerc, E. Vidal-Henriquez, J. D. Davila, and M. Kowalczyk, Symmetry breaking of nematic umbilical defects through an amplitude equation, Phys. Rev. E, 90 (2014), 012507, https://doi.org/10.1103/PhysRevE.90.012507.
[3] S. Colbert-Kelly, G. B. McFadden, D. Phillips, and J. Shen, Numerical analysis and simulation for a generalized planar Ginzburg-Landau equation in a circular geometry, Commun. Math. Sci., 15 (2017), pp. 329-357, https://doi.org/10.4310/CMS.2017.v15.n2.a3. · Zbl 1367.65142
[4] S. Colbert-Kelly and D. Phillips, Analysis of a Ginzburg-Landau type energy model for smectic \(C^\ast\) liquid crystals with defects, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), pp. 1009-1026, https://doi.org/10.1016/j.anihpc.2012.12.010. · Zbl 1288.35443
[5] M. del Pino, P. Felmer, and M. Kowalczyk, Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy’s type inequality, Int. Math. Res. Not., (2004), pp. 1511-1527, https://doi.org/10.1155/S1073792804133588. · Zbl 1112.35055
[6] P. Gravejat, E. Pacherie, and D. Smets, On the Stability of the Ginzburg-Landau Vortex, preprint, arXiv:2106.02511.
[7] R.-M. Hervé and M. Hervé, Étude qualitative des solutions réelles d’une équation différentielle liée à l’équation de Ginzburg-Landau, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), pp. 427-440, https://doi.org/10.1016/S0294-1449(16)30182-2. · Zbl 0836.34090
[8] 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), pp. 3390-3425, https://doi.org/10.1137/130948598. · Zbl 1321.34035
[9] 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), pp. 633-673, https://doi.org/10.1007/s00205-014-0791-4. · Zbl 1308.35213
[10] R. Ignat, L. Nguyen, V. Slastikov, and A. Zarnescu, Instability of point defects in a two-dimensional nematic liquid crystal model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), pp. 1131-1152, https://doi.org/10.1016/j.anihpc.2015.03.007. · Zbl 1351.82110
[11] R. Ignat, L. Nguyen, V. Slastikov, and A. Zarnescu, Stability of point defects of degree \(\pm\frac 12\) in a two-dimensional nematic liquid crystal model, Calc. Var. Partial Differential Equations, 55 (2016), 119, https://doi.org/10.1007/s00526-016-1051-2. · Zbl 1353.82076
[12] G. Kitavtsev, J. Robbins, V. Slastikov, and A. Zarnescu, Liquid crystal defects in the Landau-de-Gennes theory in two dimensions - beyond the one-constant approximation, Math. Models Methods Appl. Sci., 26 (2016), pp. 2769-2808. · Zbl 1357.82071
[13] P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. Anal., 130 (1995), pp. 334-344, https://doi.org/10.1006/jfan.1995.1073. · Zbl 0839.35011
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