Renormalization and blow up for charge one equivariant critical wave maps

2006/10/07 by J. Krieger, Joachim Krieger, Wilhelm Schlag +3 · 148 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Charge (physics) #Energy (signal processing) #Equivariant map #Ground state #Harmonic map #Nonlinear Partial Differential Equations #Renormalization #Spectral Theory in Mathematical Physics #State (computer science) #Zero (linguistics) #math-ph #math.AP #math.MP #msc:35L05 #msc:35P25 #msc:35Q75

paper · pdf · doi:10.1007/s00222-007-0089-3

published in Inventiones mathematicae 171(3), 543-615 (Springer Science+Business Media)

arxiv created 2006/10/07 · openalex publication_date 2007/11/14 · arxiv updated 2015/06/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We prove the existence of equivariant finite time blow up solutions for the wave map problem from 2+1 dimensions into the 2-sphere. These solutions are the sum of a dynamically rescaled ground-state harmonic map plus a radiation term. The local energy of the latter tends to zero as time approaches blow up time. This is accomplished by first "renormalizing" the rescaled ground state harmonic map profile by solving an elliptic equation, followed by a perturbative analysis.

Citations

• Weak solutions and development of singularities of the SU(2) σ‐model

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