Large Deviations for Stochastic Nematic Liquid Crystals Driven by Multiplicative Gaussian Noise
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Large Deviations for Stochastic Nematic Liquid Crystals Driven by Multiplicative Gaussian Noise Zdzisław Brze´zniak1 · Utpal Manna2
· Akash Ashirbad Panda2
Received: 16 September 2018 / Accepted: 27 May 2019 / © Springer Nature B.V. 2019
Abstract We study a stochastic two-dimensional nematic liquid crystal model with multiplicative Gaussian noise. We prove the Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions using weak convergence method. Keywords Nematic liquid crystal · Weak martingale solution · Large deviations principle Mathematics Subject Classification (2010) 60F10 · 60H15 · 60J75 · 76A15 · 76B03
1 Introduction The dissimilarity between the states of matter arises due to the degree and type of ordering, the molecules of the matter display with respect to their neighbours. In between solid and liquid states, there lives an intermediate state, which displays long-range orientational order. Liquid crystals are such phases containing molecules with high shape-anisotropy. At elevated temperatures, the axes of the liquid crystal molecules orient in a random manner. On cooling, the phase to evolve first is the nematic phase. The molecules in this phase exhibit orientational order but have no positional order. On average, the molecules in the nematic phase align parallel to a well-defined spatial direction which is denoted by the unit vector d, known as the director. To model the dynamics of the nematic liquid crystals, most of the scientists bank on the continuum theory developed by Ericksen [16] and Leslie [19] in the 1960’s. Stimulated by Utpal Manna
[email protected] Zdzisław Brze´zniak [email protected] Akash Ashirbad Panda [email protected] 1
Department of Mathematics, The University of York, Heslington, York YO10 5DD, UK
2
School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, Vithura, Thiruvananthapuram 695551, India
Z. Brze´zniak et al.
this theory, Lin and Liu [20] established the most elementary form of dynamical system representing the motion of nematic liquid crystals. This system can be derived as ∂u + (u · ∇)u − μu + ∇p = −λ∇ · (∇d ∇d) , (1.1) ∂t ∇ · u = 0, (1.2) ∂d + (u · ∇)d = γ d + |∇d|2 d , (1.3) ∂t (1.4) |d|2 = 1. This holds in OT := (0, T ] × O, where O ⊂ R2 and 0 < T < ∞. The vector field u : [0, T ) × O → R2 denotes the velocity of the fluid, d : [0, T ) × O → R3 is the director field that represents the macroscopic molecular orientation of the liquid crystal material, p : [0, T ) × O → R denotes the pressure function. The constants μ, λ and γ are positive constants that represent viscosity, the competition between kinetic energy and potential energy, and microscopic elastic relaxation time for the molecular orientation field. ∇· denotes the divergence operator. The symbol ∇d∇d is the 2×2-matrix with the entries [∇d ∇d]i,j =
3
∂xi d(k) ∂xj d(k) ,
i, j = 1, 2.
k=1
We equip the system with the initial and boundary conditions respectively as follows u(0) = u0 with ∇
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