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Wave Maps and Ill-posedness of their Cauchy Problem

  • Piero D’Ancona
  • Vladimir Georgiev
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 159)

Abstract

In this review article we present an introduction to the theory of wave maps, give a short overview of some recent methods used in this field and finally we prove some recent results on ill-posedness of the corresponding Cauchy problem for the wave maps in critical Sobolev spaces. The low regularity solutions for the wave map problem are studied by the aid of appropriate bilinear estimates in the spirit of ones introduced by Klainerman and Bourgain. Our approach to obtain ill-posedness in critical Sobolev norms uses suitable family of wave maps constructed via geodesic flow on the target manifold. We give two alternative proofs of the ill-posedness: the first approach is based on the application of Fourier analysis tools, while the second proof is based on the application of the classical fundamental solution representation for the free wave equation. For the case of subcritical Sobolev norms we establish non-uniqueness of the corresponding Cauchy problem.

Keywords

Equivariant wave maps Hs-spaces blow-up of solution 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Piero D’Ancona
    • 1
  • Vladimir Georgiev
    • 2
  1. 1.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.Dipartimento di MatematicaUniversità di PisaPisaItaly

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