Smooth self-similar blow-up profiles for the wave map equation
Analysis of PDEs
2008-06-26 v1 Differential Geometry
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.
Keywords
Cite
@article{arxiv.0806.4148,
title = {Smooth self-similar blow-up profiles for the wave map equation},
author = {Pierre Germain},
journal= {arXiv preprint arXiv:0806.4148},
year = {2008}
}