Radiation fields

Piotr T. Chruściel; Olivier Lengard

Bulletin de la Société Mathématique de France (2005)

  • Volume: 133, Issue: 1, page 1-72
  • ISSN: 0037-9484

Abstract

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We study the “hyperboloidal Cauchy problem” for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behavior at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a λ φ p nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal completions at null infinity, as well to a large class of equations with a similar non-linearity structure. We prove existence of solutions with controlled asymptotic behavior, and asymptotic expansions for solutions when the initial data have such expansions. In particular we prove that polyhomogeneous initial data (satisfying compatibility conditions) lead to solutions which are polyhomogeneous at the conformal boundary ℐ + of the Minkowski space-time.

How to cite

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Chruściel, Piotr T., and Lengard, Olivier. "Radiation fields." Bulletin de la Société Mathématique de France 133.1 (2005): 1-72. <http://eudml.org/doc/272421>.

@article{Chruściel2005,
abstract = {We study the “hyperboloidal Cauchy problem” for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behavior at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a $\lambda \phi ^p$ nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal completions at null infinity, as well to a large class of equations with a similar non-linearity structure. We prove existence of solutions with controlled asymptotic behavior, and asymptotic expansions for solutions when the initial data have such expansions. In particular we prove that polyhomogeneous initial data (satisfying compatibility conditions) lead to solutions which are polyhomogeneous at the conformal boundary $\mathcal \{I\}^+$ of the Minkowski space-time.},
author = {Chruściel, Piotr T., Lengard, Olivier},
journal = {Bulletin de la Société Mathématique de France},
keywords = {wave equations; asymptotic behavior; conformal infinity; polyhomogeneous expansions; singularity propagation; symmetric hyperbolic systems},
language = {eng},
number = {1},
pages = {1-72},
publisher = {Société mathématique de France},
title = {Radiation fields},
url = {http://eudml.org/doc/272421},
volume = {133},
year = {2005},
}

TY - JOUR
AU - Chruściel, Piotr T.
AU - Lengard, Olivier
TI - Radiation fields
JO - Bulletin de la Société Mathématique de France
PY - 2005
PB - Société mathématique de France
VL - 133
IS - 1
SP - 1
EP - 72
AB - We study the “hyperboloidal Cauchy problem” for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behavior at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a $\lambda \phi ^p$ nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal completions at null infinity, as well to a large class of equations with a similar non-linearity structure. We prove existence of solutions with controlled asymptotic behavior, and asymptotic expansions for solutions when the initial data have such expansions. In particular we prove that polyhomogeneous initial data (satisfying compatibility conditions) lead to solutions which are polyhomogeneous at the conformal boundary $\mathcal {I}^+$ of the Minkowski space-time.
LA - eng
KW - wave equations; asymptotic behavior; conformal infinity; polyhomogeneous expansions; singularity propagation; symmetric hyperbolic systems
UR - http://eudml.org/doc/272421
ER -

References

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