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Stability of Spherically Symmetric Wave Maps
Joachim Krieger, Harvard University, Cambridge, MA

Memoirs of the American Mathematical Society
2006; 80 pp; softcover
Volume: 181
ISBN-10: 0-8218-3877-6
ISBN-13: 978-0-8218-3877-8
List Price: US$60
Individual Members: US$36
Institutional Members: US$48
Order Code: MEMO/181/853
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We study Wave Maps from \({\mathbf{R}}^{2+1}\) to the hyperbolic plane \({\mathbf{H}}^{2}\) with smooth compactly supported initial data which are close to smooth spherically symmetric initial data with respect to some \(H^{1+\mu}\), \(\mu>0\). We show that such Wave Maps don't develop singularities in finite time and stay close to the Wave Map extending the spherically symmetric data(whose existence is ensured by a theorem of Christodoulou-Tahvildar-Zadeh) with respect to all \(H^{1+\delta}, \delta < \mu_{0}\) for suitable \(\mu_{0}(\mu)>0\). We obtain a similar result for Wave Maps whose initial data are close to geodesic ones. This strengthens a theorem of Sideris for this context.

Table of Contents

  • Introduction, controlling spherically symmetric wave maps
  • Technical preliminaries. Proofs of main theorems
  • The proof of Proposition 2.2
  • Proof of theorem 2.3
  • Bibliography
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