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Generalized Fourier Integral Operator Methods for Hyperbolic Equations with Singularities

Published online by Cambridge University Press:  19 September 2013

Claudia Garetto  and
Michael Oberguggenberger
Claudia Garetto
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
Michael Oberguggenberger
Affiliation:
Institut für Grundlagen der Technischen Wissenschaften, Leopold-Franzens-Universität, Technikerstrasse 13, 6020 Innsbruck, Austria, (michael.oberguggenberger@uibk.ac.at)
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Abstract

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This paper addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalized functions. We employ the recently developed theory of generalized Fourier integral operators to construct parametrices for the solutions and to describe propagation of singularities in this setting. As required tools, the construction of generalized solutions to eikonal and transport equations is given and results on the microlocal regularity of the kernels of generalized Fourier integral operators are obtained.

Keywords

hyperbolic equationsColombeau generalized functionsmicrolocal analysisPrimary 35S3046F30Secondary 35B65
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 2 , June 2014 , pp. 426 - 463
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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