280
Views
0
CrossRef citations to date
0
Altmetric
Research Article

Stochastic effects on solution landscapes for nematic liquid crystals

, , &
Pages 276-296 | Received 15 Jul 2023, Accepted 11 Dec 2023, Published online: 18 Jan 2024

References

  • de Gennes PG, Prost J. The physics of liquid crystals. 2nd ed. Oxford (UK): Clarendon Press; 1993.
  • Canevari G, Majumdar A, Spicer A. Order reconstruction for nematics on squares and hexagons: a landau-de gennes study. SIAM J Appl Math. 2017;77(1):267–293. doi: 10.1137/16M1087990
  • Robinson M, Luo C, Farrell PE, et al. From molecular to continuum modelling of bistable liquid crystal devices. Liq Cryst. 2017;44(14–15):2267–2284. doi: 10.1080/02678292.2017.1290284
  • Gartland EC, Mkaddem S. Instability of radial hedgehog configurations in nematic liquid crystals under Landau–de Gennes free-energy models. Phy Rev E. 1999;59(1):563–567. doi: 10.1103/PhysRevE.59.563
  • Kloeden PE, Platen E. Stochastic differential equations. Heidelberg: Springer Berlin; 1992.
  • Lord GJ, Powell CE, Shardlow T. An introduction to computational stochastic PDEs. Cambridge (UK): Cambridge University Press; 2014.
  • Øksendal B. Stochastic differential equations: an introduction with applications. Heidelberg: Springer Berlin; 2003.
  • Brzeźniak Z, Hausenblas E, Razafimandimby PA. Some results on the penalised nematic liquid crystals driven by multiplicative noise: weak solution and maximum principle. Stoch Partial Differ Equ. 2019;7(3):417–475. doi: 10.1007/s40072-018-0131-z
  • Brzeźniak Z, Manna U, Panda AA. Large deviations for stochastic nematic liquid crystals driven by multiplicative gaussian noise. Potential Anal. 2020;53(3):799–838. doi: 10.1007/s11118-019-09788-6
  • Wang Y, Canevari G, Majumdar A. Order reconstruction for nematics on squares with isotropic inclusions: a Landau-de Gennes study. SIAM J Appl Math. 2019;79(4):1314–1340. doi: 10.1137/17M1179820
  • Majumdar A. The radial-hedgehog solution in Landau–de Gennes’ theory for nematic liquid crystals. Eur J Appl Math. 2012;23(1):61–97. doi: 10.1017/S0956792511000295
  • Majumdar A. Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory. Eur J Appl Math. 2010;21(2):181–203. doi: 10.1017/S0956792509990210
  • Majumdar A, Zarnescu A. Landau-de Gennes theory of nematic liquid crystals: the Oseen–Frank limit and beyond. Arch Ration Mech Anal. 2010;196(1):227––280. doi: 10.1007/s00205-009-0249-2
  • Golovaty D, Montero JA, Sternberg P. Dimension reduction for the Landau-de Gennes model in planar nematic thin films. J Nonlinear Sci. 2015;25(6):1431–1451. doi: 10.1007/s00332-015-9264-7
  • Mottram NJ, Newton CJ. Introduction to Q-tensor theory. arXiv:14093542 [cond-matsoft]. 2014.
  • Tsakonas C, Davidson AJ, Brown CV, et al. Multistable alignment states in nematic liquid crystal filled wells. Appl Phys Lett. 2007;90(11):111913. doi: 10.1063/1.2713140
  • Luo C, Majumdar A, Erban R. Multistability in planar liquid crystal wells. Phys Rev E. 2012;85(6):061702. doi: 10.1103/PhysRevE.85.061702
  • Kralj S, Majumdar A. Order reconstruction patterns in nematic liquid crystal wells. Proc Math Phys Eng Sci. 2014;470(2169):20140276. doi: 10.1098/rspa.2014.0276
  • Canevari G, Harris J, Majumdar A, et al. The well order reconstruction solution for three-dimensional wells, in the Landau-de Gennes theory. Int J Non Linear Mech. 2020;119:103342. doi: 10.1016/j.ijnonlinmec.2019.103342
  • Butcher JC. The numerical analysis of ordinary differential equations : Runge-Kutta and general linear methods. Chichester: Wiley; 1987.
  • Hairer M, Ryser M, Weber H. Triviality of the 2d stochastic Allen-Cahn equation. Electron J Probab. 2012;17(39):1–14. doi: 10.1214/EJP.v17-1731
  • Ryser MD, Nigam N, Tupper PF. On the well-posedness of the stochastic Allen–Cahn equation in two dimensions. J Comput Phys. 2012;231(6):2537–2550. doi: 10.1016/j.jcp.2011.12.002
  • Gard TC. Introduction to stochastic differential equations. 1988.
  • Arnold L. Random dynamical systems. Heidelberg: Springer Berlin; 1998.
  • Chunrong F, Wu Y, Zhao H. Anticipating random periodic solutions–ii. SPDES with multiplicative linear noise. arXiv preprint arXiv:180300503. 2018.
  • Liu W, Mao X, Wu Y. The backward Euler-maruyama method for invariant measures of stochastic differential equations with super-linear coefficients. Appl Numer Math. 2023;184:137–150. doi: 10.1016/j.apnum.2022.09.017
  • Lewis AH, Garlea I, Alvarado J, et al. Colloidal liquid crystals in rectangular confinement: theory and experiment. Soft Matter. 2014;10(39):7865–7873. doi: 10.1039/C4SM01123F
  • Kralj S, Rosso R, Virga EG. Finite-size effects on order reconstruction around nematic defects. Phy Rev E. 2010;81(2):021702. doi: 10.1103/PhysRevE.81.021702
  • Lamy X. Some properties of the nematic radial hedgehog in the Landau–de gennes theory. J Math Anal Appl. 2013;397(2):586–594. doi: 10.1016/j.jmaa.2012.08.011
  • Sofi JA, Dhara S. Stability of liquid crystal micro-droplets based optical microresonators. Liq Cryst. 2019;46(4):629–639. doi: 10.1080/02678292.2018.1515373