Type II blow-up mechanism for supercritical harmonic map heat flow

2016/01/08 by Biernat, Paweł, Seki, Yukihiro · 1 citation
#35J62 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1601.01831

Abstract

The harmonic map heat flow is a geometric flow well known to produce solutions whose gradient blows up in finite time. A popular model for investigating the blow-up is the heat flow for maps \mathbb R^d→ S^d, restricted to equivariant maps. This model displays a variety of possible blow-up mechanisms, examples include self-similar solutions for 3≤ d≤ 6 and a so-called Type II blow-up in the critical dimension d=2. Here we present the first constructive example of Type II blow-up in higher dimensions: for each d≥7 we construct a countable family of Type II solutions, each characterized by a different blow-up rate. We study the mechanism behind the formation of these singular solutions and we relate the blow-up to eigenvalues associated to linearization of the harmonic map heat flow around the equatorial map. Some of the solutions constructed by us were already observed numerically.

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