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

TL;DR

This paper constructs and analyzes new Type II blow-up solutions for the harmonic map heat flow in higher dimensions, revealing diverse blow-up rates and mechanisms related to linearized eigenvalues, extending understanding beyond previously known cases.

Contribution

It provides the first explicit construction of Type II blow-up solutions in dimensions greater than six, with a detailed analysis of their formation and underlying linearized eigenvalue problem.

Findings

Constructed countable family of Type II solutions for d≥7

Linked blow-up behavior to eigenvalues of linearized flow

Identified solutions previously observed numerically

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 $\mathbb R^{d}\to S^{d}$, restricted to equivariant maps. This model displays a variety of possible blow-up mechanisms, examples include self-similar solutions for $3\le d\le 6$ 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 $d\ge7$ 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.