On the existence of smooth self-similar blowup profiles for the wave map equation

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Abstract

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 map problem. This has several applications to questions of regularity and uniqueness for the wave map equation.

Original languageEnglish (US)
Pages (from-to)706-728
Number of pages23
JournalCommunications on Pure and Applied Mathematics
Volume62
Issue number5
DOIs
StatePublished - May 2009

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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