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Wave Maps and Ill-posedness of their Cauchy Problem

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New Trends in the Theory of Hyperbolic Equations

Part of the book series: Operator Theory: Advances and Applications ((APDE,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.

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazzale A. Moro 2, I-00185, Roma, Italy

    Piero D’Ancona

  2. Dipartimento di Matematica, Università di Pisa, Via Buonarroti No. 2, 56127, Pisa, Italy

    Vladimir Georgiev

Authors
  1. Piero D’Ancona
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  2. Vladimir Georgiev
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Editors and Affiliations

  1. Fakultät für Mathematik und Informatik Institut für Angewandte Analysis, TU Bergakademie Freiberg, 09596, Freiberg, Germany

    Michael Reissig

  2. Institut für Mathematik, Universität Potsdam, 14415, Potsdam, Germany

    Bert-Wolfgang Schulze

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D’Ancona, P., Georgiev, V. (2005). Wave Maps and Ill-posedness of their Cauchy Problem. In: Reissig, M., Schulze, BW. (eds) New Trends in the Theory of Hyperbolic Equations. Operator Theory: Advances and Applications, vol 159. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7386-5_1

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