Smoothness and Liouville type theorem for a system with cubic nonlinearities from biaxial nematic liquid crystals in two dimensions. (English) Zbl 1504.35314
Summary: In recent paper, we will consider some properties of weak solutions to the Dirichlet problem of a system with cubic nonlinearities from biaxial nematic liquid crystals in two dimensions. The smoothness of weak solution is obtained firstly. Then we study the constant boundary value problems of this system and obtain a Liouville-type theorem.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76A15 | Liquid crystals |
| 35D30 | Weak solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
References:
| [1] | De Gennes, P. G., The Physics of Liquid Crystals (1974), Clarendon Press: Clarendon Press Oxford |
| [2] | Govers, E.; Vertogen, G., Elastic continuum theory of biaxial nematics, Phys. Rev. A, 30 (1984) |
| [3] | Govers, E.; Vertogen, G., Erratum: Elastic continuum theory of biaxial nematics [phys. Rev. a, 1984, 30], Phys. Rev. A, 31 (1985) |
| [4] | Lin, J.; Li, Y.; Wang, C., On static and hydrodynamic biaxial nematic liquid crystals (2020), arXiv:2006.04207 |
| [5] | Huang, J.; Lin, J., Orientability and asymptotic convergence of Q-tensor flow of biaxial nematic liquid crystals, Cal. Var. PDE, 61, 173 (2022) · Zbl 1503.35164 |
| [6] | Lemaire, L., Applications harmoniques de surfaces Riemanniennes, J. Differential Geom., 13, 51-78 (1978) · Zbl 0388.58003 |
| [7] | Karcher, H.; Wood, J., Non-existence results and growth properties for harmonic maps and forms, J. Reine Angew. Math., 353, 165-180 (1984) · Zbl 0544.58008 |
| [8] | Héleinre, F., Regularity of weakly harmonic maps between a surface and an \(n\)-sphere, C. R. Acad. Sci. Paris Sér. I Math., 311, 519-524 (1990) · Zbl 0728.35014 |
| [9] | Lin, F.; Wang, C., The Analysis of Harmonic Maps and their Heat Flows (2008), World Scientific · Zbl 1203.58004 |
| [10] | Riviére, T., Conservation laws for conformally invariant variational problems, Invent. Math., 168, 1-22 (2007) · Zbl 1128.58010 |
| [11] | Muller, F.; Schikorra, A., Boundary regularity via Uhlenbeck-Rivière decomposition, Analysis, 29, 199-220 (2009) · Zbl 1181.35102 |
| [12] | Schoen, R.; Yau, S., Lectures on Harmonic Maps (1997), International Press of Boston · Zbl 0886.53004 |
| [13] | Moser, R., Partial Regularity for Harmonic Maps and Related Problems (2005), World Scientific · Zbl 1246.58012 |
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.
