TY - JOUR
T1 - Construction of Parametrix to Strictly Hyperbolic Cauchy Problems with Fast Oscillations in NonLipschitz Coefficients
AU - Kubo, Akisato
AU - Reissig, Michael
N1 - Funding Information:
Authors would like to express many thanks to DFG for financial support for the first author from August to October 2001 and to Faculty of Mathematics and Computer Science of TU Bergakademie Freiberg for hospitality. During this stay the case of slow oscillations (γ 2 ½0,1Þ in Eq. (I.4)) was studied. The refined diagonalization procedure which allows to study the case γ ¼ 1 was suggested by the second author during his work as a Foreign Professor at the University of Tsukuba in February, March 2002. He is grateful to the staff of Institute of Mathematics, especially the research groups of Prof. Kajitani, Prof. Taira, and Prof. Wakabayashi for their hospitality. The authors thank the referee for valuable comments.
PY - 2003
Y1 - 2003
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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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 -