Conformal scattering for a nonlinear wave equation on a curved background

Abstract : The purpose of this paper is to establish a geometric scattering result for a conformally invariant nonlinear wave equation on an asymptotically simple spacetime. The scattering operator is obtained via trace operators at null infinities. The proof is achieved in three steps. A priori linear estimates are obtained via an adaptation of the Morawetz vector field in the Schwarzschild spacetime and a method used by Hörmander for the Goursat problem. A well-posedness result for the characteristic Cauchy problem on a light cone at infinity is then obtained. This requires a control of the nonlinearity uniform in time which comes from an estimates of the Sobolev constant and a decay assumption on the nonlinearity of the equation. Finally, the trace operators on conformal infinities are built and used to define the conformal scattering operator.
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http://hal.univ-brest.fr/hal-00470395
Contributeur : Jérémie Joudioux <>
Soumis le : jeudi 28 octobre 2010 - 11:14:23
Dernière modification le : jeudi 11 janvier 2018 - 06:16:45
Document(s) archivé(s) le : samedi 29 janvier 2011 - 02:38:24

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  • HAL Id : hal-00470395, version 2
  • ARXIV : 1004.1464

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Jérémie Joudioux. Conformal scattering for a nonlinear wave equation on a curved background. 2010. 〈hal-00470395v2〉

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