Global weak solutions to the Stochastic Ericksen--Leslie equations in dimension two

TL;DR

This paper proves the existence of global weak solutions for a stochastic liquid crystal flow model in two dimensions, using approximation and compactness methods.

Contribution

It introduces a new approach combining concentration-cancellation and Skorokhod theorems to establish solutions for the stochastic Ericksen--Leslie equations.

Findings

Existence of global weak martingale solutions in 2D.

Convergence of Ginzburg--Landau approximations.

Application of concentration-cancellation method.

Abstract

We establish the global existence of weak martingale solutions to the simplified stochastic Ericksen--Leslie system modeling the nematic liquid crystal flow driven by Wiener-type noises on the two-dimensional bounded domains. The construction of solutions is based on the convergence of Ginzburg--Landau approximations. To achieve such a convergence, we first utilize the concentration-cancellation method for the Ericksen stress tensor fields based on a Pohozaev type argument, and second the Skorokhod compactness theorem, which is built upon a uniform energy estimate.