Construction of Parametrix to Strictly Hyperbolic Cauchy Problems with Fast Oscillations in NonLipschitz Coefficients

Akisato Kubo, Michael Reissig

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

The goal of this article is to construct parametrix to strictly hyperbolic Cauchy problems with nonLipschitz coefficients depending on space and time. The nonLipschitz condition is described by the behavior of the time-derivative of coefficients. This leads to a classification of oscillations, where fast oscillations represent the critical case. To study this critical case we propose a refined perfect diagonalization procedure basing on suitable zones of the phase space and corresponding nonstandard symbol classes. After this diagonalization procedure we construct the parametrix in several steps. Here the perfect diagonalization helps to understand, that only a finite loss of derivatives appears for solutions valued in Sobolev spaces. We point out where this loss comes from. From construction of parametrix we conclude a result about C-well posedness of the Cauchy problem.

Original languageEnglish
Pages (from-to)1471-1502
Number of pages32
JournalCommunications in Partial Differential Equations
Volume28
Issue number7-8
DOIs
Publication statusPublished - 01-01-2003

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Hyperbolic Problems
Diagonalization
Cauchy Problem
Strictly
Critical Case
Oscillation
Derivatives
Sobolev spaces
Coefficient
Derivative
Well-posedness
Sobolev Spaces
Phase Space

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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Construction of Parametrix to Strictly Hyperbolic Cauchy Problems with Fast Oscillations in NonLipschitz Coefficients. / Kubo, Akisato; Reissig, Michael.

In: Communications in Partial Differential Equations, Vol. 28, No. 7-8, 01.01.2003, p. 1471-1502.

Research output: Contribution to journalArticle

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