Global well-posedness of stochastic nematic liquid crystals with random initial and random boundary conditions driven by multiplicative noise

TL;DR

This paper proves the global well-posedness of 2D stochastic nematic liquid crystal models with random initial and boundary conditions influenced by multiplicative noise, using Malliavin calculus techniques.

Contribution

It establishes the first rigorous proof of global solutions for stochastic nematic liquid crystals with random initial and boundary data under Malliavin regularity.

Findings

Global existence of solutions proven under certain regularity conditions

Solutions remain well-posed even with boundary noise perturbations

Malliavin calculus is effectively used for stochastic PDE analysis

Abstract

The flow of nematic liquid crystals can be described by a highly nonlinear stochastic hydrodynamical model, thus is often influenced by random fluctuations, such as uncertainty in specifying initial conditions and boundary conditions. In this article, we consider the $2$-D stochastic nematic liquid crystals with the velocity field perturbed by affine-linear multiplicative white noise, with random initial data and random boundary conditions. Our main objective is to establish the global well-posedness of the stochastic equations under certain sufficient Malliavin regularity of the initial conditions and the boundary conditions. The Malliavin calculus techniques play important roles in proving the global existence of the solutions to the stochastic nematic liquid crystal models with random initial and random boundary conditions. It should be pointed out that the global well-posedness is also true when the stochastic system is perturbed by the noise on the boundary.