Abstract
In this note we prove the existence and uniqueness of local maximal smooth solution of the stochastic simplified Ericksen-Leslie systems modelling the dynamics of nematic liquid crystals under stochastic perturbations.
Highlights
Liquid crystal, which is a state of matter that has properties between amorphous liquid and crystalline solid, can be classified into two groups according to the form of their molecules
Liquid crystals with rod-shaped molecules are called calamitics while those with disc-like molecules are referred to discotics
Razafimandimby was at the University of Pretoria; he is grateful to the funding he received from the National Research Foundation South Africa (Grant Numbers 109355 and 112084). He is grateful to the European Mathematical Society for the EMS-Simons for Africa-Collaborative research grant which enables him to visit Montanuniversitat Leoben, Austria. ̊ Corresponding author: Paul Razafimandimby
Summary
Liquid crystal, which is a state of matter that has properties between amorphous liquid and crystalline solid, can be classified into two groups according to the form of their molecules. Liu [34] derived the most basic and simplest form of the dynamical system describing the motion of nematic liquid crystals flowing in Rdpd “ 2, 3q This system is given by vtpv ∇qv ∆v ∇p “ ́λ∇ ̈ p∇d d ∇dq, ∇ ̈ v “ 0, dtpv ∇qd “ γp∆d |∇d|2dq, |d|2 “ 1. Valued random variable pv0, d0q we can find a stopping time τ8 which can be approximated by an increasing sequence of stopping times pτmqmPN and a unique local stochastic process pv, dq “ pvptq, dptqq, 0 ď t ă τ8 satisfying the following conditions. If Assumption 1 is satisfied, for all F0-measurable and square integrable Hαsol Hα1-valued random variables y0 “ pv0, d0q the problem (2.11) for the stochastic liquid crystals has a unique local maximal strong solution ppv; dq, τ8q satisfying the following properties: 1.
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