A scaling limit from the wave map to the heat flow into $\mathbb{S}^2$

TL;DR

This paper investigates the mathematical connection between wave maps and heat flows into the sphere, providing a quantitative limit analysis that aids understanding of the zero inertia limit in liquid crystal models.

Contribution

It establishes a rigorous limit from scaled wave maps to heat flows into , including initial layer corrections, advancing the theoretical understanding of these geometric PDEs.

Findings

Quantitative connection between wave map and heat flow equations

Introduction of initial layer correction for the limit process

Foundation for analyzing zero inertia limit in liquid crystal models

Abstract

In this paper we study a limit connecting a scaled wave map with the heat flow into the unit sphere $\mathbb{S}^2$. We show quantitatively how that the two equations are connected by means of an initial layer correction. This limit is motivated as a first step into understanding the limit of zero inertia for the hyperbolic-parabolic Ericksen-Leslie's liquid crystal model.