Local well-posedness and small Deborah limit of a molecule-based $Q$-tensor system

TL;DR

This paper establishes local well-posedness for a molecular $Q$-tensor model of nematic liquid crystals and rigorously derives the Ericksen-Leslie theory as a small Deborah number limit.

Contribution

It proves local existence and uniqueness for the molecular $Q$-tensor system and derives the Ericksen-Leslie equations via a rigorous asymptotic analysis.

Findings

Existence and uniqueness of local strong solutions.

Rigorous derivation of Ericksen-Leslie theory from molecular model.

Validation of the small Deborah number limit approach.

Abstract

In this paper, we consider a hydrodynamic $Q$-tensor system for nematic liquid crystal flow, which is derived from Doi-Onsager molecular theory by the Bingham closure. We first prove the existence and uniqueness of local strong solution. Furthermore, by taking Deborah number goes to zero and using the Hilbert expansion method, we present a rigorous derivation from the molecule-based $Q$-tensor theory to the Ericksen-Leslie theory.