A multi-scale approach to hyperbolic evolution equations with limited smoothness
Författare
Summary, in English
We discuss how techniques from multiresolution analysis and phase space transforms can be exploited in solving a general class of evolution equations with limited smoothness. We have wave propagation in media of limited smoothness in mind. The frame that appears naturally in this context belongs to the family of frames of curvelets. The construction considered here implies a full-wave description on the one hand but reveals the geometrical properties derived from the propagation of singularities on the other hand. The approach and analysis we present (i) aids in the understanding of the notion of scale in the wavefield and how this interacts with the configuration or medium, (ii) admits media of limited smoothness, viz. with Holder regularity s >= 2, and (iii) suggests a novel computational algorithm that requires solving for the mentioned geometry on the one hand and solving a matrix Volterra integral equation of the second kind on the other hand. The Volterra equation can be solved by recursionas in the computation of certain multiple scattering seriesrevealing a curvelet-curvelet interaction. We give precise estimates expressing the degree of concentration of curvelets following the propagation of singularities.
Avdelning/ar
Publiceringsår
2008
Språk
Engelska
Sidor
988-1017
Publikation/Tidskrift/Serie
Communications in Partial Differential Equations
Volym
33
Issue
6
Dokumenttyp
Artikel i tidskrift
Förlag
Taylor & Francis
Ämne
- Mathematics
Nyckelord
- pseudodifferential evolution equations
- decomposition
- paradifferential
- curvelets
- dyadic parabolic decomposition
Status
Published
Forskningsgrupp
- Harmonic Analysis and Applications
ISBN/ISSN/Övrigt
- ISSN: 0360-5302