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 language | English |
|---|---|
| Pages (from-to) | 1471-1502 |
| Number of pages | 32 |
| Journal | Communications in Partial Differential Equations |
| Volume | 28 |
| Issue number | 7-8 |
| DOIs | |
| Publication status | Published - 01-01-2003 |
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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 journal › Article
TY - JOUR
T1 - Construction of Parametrix to Strictly Hyperbolic Cauchy Problems with Fast Oscillations in NonLipschitz Coefficients
AU - Kubo, Akisato
AU - Reissig, Michael
PY - 2003/1/1
Y1 - 2003/1/1
N2 - 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.
AB - 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.
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UR - http://www.scopus.com/inward/citedby.url?scp=0141707923&partnerID=8YFLogxK
U2 - 10.1081/PDE-120024375
DO - 10.1081/PDE-120024375
M3 - Article
AN - SCOPUS:0141707923
VL - 28
SP - 1471
EP - 1502
JO - Communications in Partial Differential Equations
JF - Communications in Partial Differential Equations
SN - 0360-5302
IS - 7-8
ER -