A Global 2D Well-Posedness Result on the Order Tensor Liquid Crystal Theory

TL;DR

This paper extends the mathematical understanding of liquid crystal flow models by establishing well-posedness results for lower regularity solutions and providing an alternative proof for the existence of weak solutions in two dimensions.

Contribution

It proves well-posedness and uniqueness of weak solutions for the order tensor system at lower regularities, complementing prior higher regularity results, and offers an alternative existence proof.

Findings

Proved uniqueness of weak solutions for $H^s$ with $0<s extless=1$

Established propagation of lower regularity solutions

Provided an alternative existence proof for weak solutions

Abstract

Paicu and Zarnescu have studied an order tensor system which describes the flow of a liquid crystal. They have proven the existence of weak solutions, the propagation of higher regularities, namely $H^s$ with $s>1$ and the weak-strong uniqueness in dimension two. This paper is devoted to the propagation of lower regularities, namely $H^s$ for $0<s\leq 1$ and to prove the uniqueness of the weak solutions. For the completeness of this research, we also propose an alternative approach in order to prove the existence of weak solutions.