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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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https://hal.univ-brest.fr/hal-00470395
Contributor : Jérémie Joudioux <>
Submitted on : Thursday, October 28, 2010 - 11:14:23 AM
Last modification on : Wednesday, April 1, 2020 - 1:57:10 AM
Long-term archiving on: : Saturday, January 29, 2011 - 2:38:24 AM

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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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