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Journal d'Analyse Mathématique

, Volume 109, Issue 1, pp 81–162 | Cite as

Fourier-integral-operator product representation of solutions to first-order symmetrizable hyperbolic systems

  • Jérôme Le RousseauEmail author
Article
  • 69 Downloads

Abstract

We consider the first-order Cauchy problem
$$ \begin{gathered} \partial _z u + a(z,x,D_x )u = 0,0 < z \leqslant Z, \hfill \\ u|_{z = 0} = u_0 , \hfill \\ \end{gathered} $$
with Z > 0 and a(z, x,D x ) a k × k matrix of pseudodifferential operators of order one, whose principal part a 1 is assumed symmetrizable: there exists L(z, x, ξ) of order 0, invertible, such that
$$ a_1 (z,x,\xi ) = L(z,x,\xi )( - i\beta _1 (z,x,\xi ) + \gamma _1 (z,x,\xi ))(L(z,x,\xi ))^{ - 1} , $$
where β 1 and γ 1 are hermitian symmetric and γ 1 ≥ 0. An approximation Ansatz for the operator solution, U(z′, z), is constructed as the composition of global Fourier integral operators with complex matrix phases. In the symmetric case, an estimate of the Sobolev operator norm in L((H (s)(R n )) k , (H (s)(R n )) k ) of these operators is provided, which yields a convergence result for the Ansatz to U(z′, z) in some Sobolev space as the number of operators in the composition goes to ∞, in both the symmetric and symmetrizable cases. We thus obtain a representation of the solution operator U(z′, z) as an infinite product of Fourier integral operators with matrix phases.

Keywords

Cauchy Problem Phase Function Hyperbolic System Pseudodifferential Operator Scalar Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew University Magnes Press 2009

Authors and Affiliations

  1. 1.Laboratoire d’Analyse, Topologie Probabilités CNRS UMR 6632Universités d’Aix-Marseille Université de ProvenceMarseille cedex 13France

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