Global Existence and Regularity for the Active Liquid Crystal System

TL;DR

This paper establishes the global existence and regularity of solutions for the active liquid crystal system within the Q-tensor framework, advancing understanding of its mathematical properties in multiple dimensions.

Contribution

It proves global weak solutions exist in 2D and 3D, and demonstrates higher regularity and uniqueness in 2D using Littlewood-Paley decomposition.

Findings

Global weak solutions exist in 2D and 3D.

Higher regularity of solutions in 2D.

Uniqueness of weak-strong solutions in 2D.

Abstract

In this paper, we study the active hydrodynamics, described in the Q-tensor liquid crystal framework. We prove the existence of global weak solutions in dimension two and three, with suitable initial datas. By using Littlewood-Paley decomposition, we also get the higher regularity of the weak solutions and the uniqueness of weak-strong solutions in dimension two.