Smooth self-similar blow-up profiles for the wave map equation

TL;DR

This paper investigates conditions for smooth self-similar blow-up profiles in the wave map equation from Minkowski space to symmetric manifolds, linking energy minimization to blow-up solutions.

Contribution

It establishes a precise criterion for the existence of smooth self-similar blow-up profiles in the 3D equivariant wave map equation, connecting elliptic energy minimization to hyperbolic blow-up.

Findings

Necessary and sufficient condition for smooth blow-up profiles in 3D

Relation between energy minimization and blow-up existence

Applications to wave map problems with symmetric targets

Abstract

Consider the equivariant wave map equation from Minkowski space to a rotationnally symmetric manifold which has an equator (example: the sphere). In dimension 3, this article gives a necessary and sufficient condition for the existence of a smooth self-similar blow up profile. More generally, we study the relation between 1. the minimizing properties of the equator map for the (elliptic) Dirichlet energy and 2. the existence of a smooth blow-up profile for the (hyperbolic) wave map problem. Several applications of this approach are described.