Ebook: Propagation and Interaction of Singularities in Nonlinear Hyperbolic Problems
Author: Michael Beals (auth.)
- Tags: Partial Differential Equations
- Series: Progress in Nonlinear Differential Equations and Their Applications 130
- Year: 1989
- Publisher: Birkhäuser Basel
- Edition: 1
- Language: English
- pdf
This book developed from a series of lectures I gave at the Symposium on Nonlinear Microlocal Analysis held at Nanjing University in October. 1988. Its purpose is to give an overview of the use of microlocal analysis and commutators in the study of solutions to nonlinear wave equations. The weak singularities in the solutions to such equations behave up to a certain extent like those present in the linear case: they propagate along the null bicharacteristics of the operator. On the other hand. examples exhibiting singularities not present in the linear case can also be constructed. I have tried to present a crossection of both the regularity results and the singular examples. for problems on the interior of a domain and on domains with boundary. The main emphasis is on the case of more than one space dimen sion. since that case is treated in great detail in the paper of Rauch-Reed 159]. The results presented here have for the most part appeared elsewhere. and are the work of many authors. but a few new examples and proofs are given. I have attempted to indicate the essential ideas behind the arguments. so that only some of the results are proved in full detail. It is hoped that the central notions of the more technical proofs appearing in research papers will be illuminated by these simpler cases.
Content:
Front Matter....Pages i-ix
Introduction....Pages 1-3
Nonlinear Microlocal Analysis....Pages 4-29
Appearance of Nonlinear Singularities....Pages 30-51
Conormal Singularities....Pages 52-73
Conormal Regularity after Nonlinear Interaction....Pages 74-95
Regularity and Singularities in Problems on Domains With Boundary....Pages 96-116
Conormal Waves on Domains with Boundary....Pages 117-135
Back Matter....Pages 136-145
Content:
Front Matter....Pages i-ix
Introduction....Pages 1-3
Nonlinear Microlocal Analysis....Pages 4-29
Appearance of Nonlinear Singularities....Pages 30-51
Conormal Singularities....Pages 52-73
Conormal Regularity after Nonlinear Interaction....Pages 74-95
Regularity and Singularities in Problems on Domains With Boundary....Pages 96-116
Conormal Waves on Domains with Boundary....Pages 117-135
Back Matter....Pages 136-145
....