From the Q-tensor flow for the liquid crystal to the Harmonic map flow

TL;DR

This paper investigates the behavior of the relaxed Q-tensor flow in three dimensions, demonstrating its convergence to harmonic map flow and providing a new proof for the global existence of weak solutions using Ginzburg-Landau approximation.

Contribution

It establishes the limiting behavior of the Q-tensor flow and offers a novel proof for the global existence of harmonic map flow solutions in three dimensions.

Findings

Limiting map is the harmonic map flow.

Convergence of Q-tensor flow to harmonic map flow.

New proof for global existence of weak solutions.

Abstract

In this paper, we consider the solutions of the relaxed Q-tensor flow in $\R^3$ with small parameter $\epsilon$. Firstly, we show that the limiting map is the so called harmonic map flow; Secondly, we also present a new proof for the global existence of weak solution for the harmonic map flow in three dimensions as in \cite{struwe88} and \cite{keller}, where Ginzburg-Landau approximation approach was used.