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On the existence of smooth self‐similar blowup profiles for the wave map equation
Author(s) -
Germain Pierre
Publication year - 2009
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20266
Subject(s) - harmonic map , mathematics , equivariant map , uniqueness , mathematical analysis , minkowski space , wave equation , equator , dirichlet problem , elliptic curve , space (punctuation) , manifold (fluid mechanics) , pure mathematics , geometry , boundary value problem , physics , mechanical engineering , linguistics , philosophy , astronomy , engineering , latitude
Consider the equivariant wave map equation from Minkowski space to a rotationally symmetric manifold N that has an equator (e.g., the sphere). In dimension 3, this paper presents a necessary and sufficient condition on N for the existence of a smooth self‐similar blowup profile. More generally, we study the relation between the minimizing properties of the equator map for the Dirichlet energy corresponding to the (elliptic) harmonic map problem and the existence of a smooth blowup profile for the (hyperbolic) wave mapproblem.This has several applications to questions of regularity and uniqueness for the wave map equation. © 2008 Wiley Periodicals, Inc.