Abstract
In this paper, we prove several mathematical results related to a system of highly nonlinear stochastic partial differential equations (PDEs). These stochastic equations describe the dynamics of penalised nematic liquid crystals under the influence of stochastic external forces. Firstly, we prove the existence of a global weak solution (in the sense of both stochastic analysis and PDEs). Secondly, we show the pathwise uniqueness of the solution in a 2D domain. In contrast to several works in the deterministic setting we replace the Ginzburg–Landau function mathbb {1}_{|{mathbf {n}}|le 1}(|{mathbf {n}}|^2-1){mathbf {n}} by an appropriate polynomial f({mathbf {n}}) and we give sufficient conditions on the polynomial f for these two results to hold. Our third result is a maximum principle type theorem. More precisely, if we consider f({mathbf {n}})=mathbb {1}_{|d|le 1}(|{mathbf {n}}|^2-1){mathbf {n}} and if the initial condition {mathbf {n}}_0 satisfies |{mathbf {n}}_0|le 1, then the solution {mathbf {n}} also remains in the unit ball.
Highlights
Nematic liquid crystal is a state of matter that has properties which are between amorphous liquid and crystalline solid
To model the dynamics of nematic liquid crystals most scientists use the continuum theory developed by Ericksen [17] and Leslie [28]
From this theory Lin and Liu [29] derived the most basic and simplest form of the dynamical system describing the motion of nematic liquid crystals filling a bounded region O ⊂ Rd, d = 2, 3
Summary
Nematic liquid crystal is a state of matter that has properties which are between amorphous liquid and crystalline solid. The system (1.1)–(1.6) is the most basic and simplest form of equations from the Ericksen–Leslie continuum theory, it retains the most physical significance of the Nematic liquid crystals For a full understanding of the effect of fluctuating magnetic field on the behavior of the liquid crystals one needs to take into account the dynamics of n and v To initiate this kind of investigation we propose a mathematical study of (1.13)–(1.15) which basically describes an approximation of the system governing the nematic liquid crystals under the influence of fluctuating external forces. In “Appendix” section we recall or prove several crucial estimates about the nonlinear terms of the system (1.13)–(1.15)
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