A multi-scale approach to hyperbolic evolution equations with limited smoothness

Research output: Contribution to journalArticle

Abstract

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.

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Subject classification (UKÄ) – MANDATORY

  • Mathematics

Keywords

  • pseudodifferential evolution equations, decomposition, paradifferential, curvelets, dyadic parabolic decomposition
Original languageEnglish
Pages (from-to)988-1017
JournalCommunications in Partial Differential Equations
Volume33
Issue number6
StatePublished - 2008
Peer-reviewedYes