Some results on the penalised nematic liquid crystals driven by multiplicative noise: weak solution and maximum principle

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 by an appropriate polynomial 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 and if the initial condition satisfies , then the solution also remains in the unit ball.

Stochastic system for generalized polytropic filtration

“…Therefore, fixing > 0 and letting → ∞ and arguing similarly as in Brzeźniak et al, 55 we infer that for any Φ ∈ L 2 ( Ω,  , P; L 2 (D) ) ,…”

“…The stochastic process W = {W(t) ∶ t ∈ [0, T]} is a K-cylindrical Wiener process evolving on L 2 (D), and for any , t ∈ [0, T], the increments W(t) − W( ) is independent of the -algebra generated by (u( ), W( )) for any∈ [0, ].Proof. The proof of this result is a verbatim repetition of the proof in Brzeźniak et al,8 , lemma 5.2, p25 the proofs of Proposition 4.11 in Breit and Hofmanova,54 , p. 1211 and Proposition 3.23 in Brzeźniak et al55 …”

“…Indeed, this filtration satisfies the usual conditions. Following, for instance, previous studies, 8,54,55 we can easily check that for each , the sequences of stochastic processes (W m (t); t ∈ [0, T]) m∈N form K-valued -Wiener processes defined on the probability space (Ω,  ; P) and that for each…”

Large Deviations for Stochastic Nematic Liquid Crystals Driven by Multiplicative Gaussian Noise

“…He is very grateful for the financial support he received from the International Centre for Mathematical Sciences (ICMS) Edinburgh. Last, but not the least, the authors wish to thank Professor Guoli Zhou for pointing out some gaps in the previous version of this paper [8].…”

“…However, it is pointed out in [23,Chapter 5] that the fluid flow disturbs the alignment and conversely a change in the alignment will induce a flow in the nematic liquid crystal. It is this gap in knowledge that is the motivation for our mathematical study which was initiated in the old unpublished preprints [7] and [8], see also the recent papers [6] and [5].…”