Remarks of Global Wellposedness of Liquid Crystal Flows and Heat Flows of Harmonic Maps in Two Dimensions

TL;DR

This paper proves the global well-posedness of smooth solutions for 2D liquid crystal and harmonic map heat flow equations with large initial data, using a new approach based on a rigidity theorem and frequency localization.

Contribution

It provides a new proof of global well-posedness under a geometric angle condition, extending previous results with a novel technical approach.

Findings

Global well-posedness for large initial data

Rigidity theorem ensures coercivity of harmonic energy

Frequency localization and concentration-compactness techniques used

Abstract

We consider the Cauchy problem to the two-dimensional incompressible liquid crystal equation and the heat flows of harmonic maps equation. Under a natural geometric angle condition, we give a new proof of the global well-posedness of smooth solutions for a class of large initial data in energy space. This result was originally obtained by Ding-Lin in \cite{DingLin} and Lin-Lin-Wang in \cite{LinLinWang}. Our main technical tool is a rigidity theorem which gives the coercivity of the harmonic energy under certain angle condition. Our proof is based on a frequency localization argument combined with the concentration-compactness approach which can be of independent interest.