Nematic Liquid Crystals in Lipschitz domains

TL;DR

This paper establishes local and global well-posedness for the simplified Ericksen-Leslie model of nematic liquid crystals in three-dimensional Lipschitz domains, extending analysis beyond smooth-boundary cases using a semilinear approach.

Contribution

It proves well-posedness results for nematic liquid crystal models in Lipschitz domains, a significant extension from previous smooth-boundary analyses.

Findings

Well-posedness in critical spaces for initial data in $L^3_{\sigma}$ and $W^{1,3}$

Extension of analysis to Lipschitz domains with nonsmooth boundaries

Application of a semilinear approach to the Ericksen-Leslie model

Abstract

We consider the simplified Ericksen-Leslie model in three dimensional bounded Lipschitz domains. Applying a semilinear approach, we prove local and global well-posedness (assuming a smallness condition on the initial data) in critical spaces for initial data in $L^3_{\sigma}$ for the fluid and $W^{1,3}$ for the director field. The analysis of such models, so far, has been restricted to domains with smooth boundaries.