On strong solution to the 2D stochastic Ericksen-Leslie system: A Ginzburg-Landau approximation approach

TL;DR

This paper establishes the existence of local strong solutions for the 2D stochastic Ericksen-Leslie equations modeling nematic liquid crystals, using a Ginzburg-Landau approximation approach, advancing understanding of stochastic PDEs in liquid crystal dynamics.

Contribution

It proves the existence of local strong solutions for the stochastic Ericksen-Leslie equations via a Ginzburg-Landau approximation, a novel approach in this context.

Findings

Existence of local strong solutions for SELEs.

Convergence of the stochastic Ginzburg-Landau approximation.

Solutions are martingale and strong in PDEs sense.

Abstract

In this manuscript, we consider a highly nonlinear and constrained stochastic PDEs modelling the dynamics of 2-dimensional nematic liquid crystals under random perturbation. This system of SPDEs is also known as the stochastic Ericksen-Leslie equations (SELEs). We discuss the existence of local strong solution to the stochastic Ericksen-Leslie equations. In particular, we study the convergence the stochastic Ginzburg-Landau approximation of SELEs, and prove that the SELEs with initial data in $\sh^1\times \sh^2$ has at least a martingale, local solution which is strong in PDEs sense.