Hyperbolic Equations and Frequency Interactions
About this Title
Luis Caffarelli, New York University-Courant Institute of Mathematical Sciences, New York, NY and Weinan E, New York University-Courant Institute of Mathematical Sciences, New York, NY, Editors
Publication: IAS/Park City Mathematics Series
Publication Year: 1999; Volume 5
ISBNs: 978-0-8218-0592-3 (print); 978-1-4704-3904-0 (online)
MathSciNet review: MR1662834
MSC: Primary 35-06
The research topic for this IAS/PCMI Summer Session was nonlinear wave phenomena. Mathematicians from the more theoretical areas of PDEs were brought together with those involved in applications. The goal was to share ideas, knowledge, and perspectives.
How waves, or “frequencies”, interact in nonlinear phenomena has been a central issue in many of the recent developments in pure and applied analysis. It is believed that wavelet theory—with its simultaneous localization in both physical and frequency space and its lacunarity—is and will be a fundamental new tool in the treatment of the phenomena.
Included in this volume are write-ups of the “general methods and tools” courses held by Jeff Rauch and Ingrid Daubechies. Rauch's article discusses geometric optics as an asymptotic limit of high-frequency phenomena. He shows how nonlinear effects are reflected in the asymptotic theory. In the article “Harmonic Analysis, Wavelets and Applications” by Daubechies and Gilbert the main structure of the wavelet theory is presented.
Also included are articles on the more “specialized” courses that were presented, such as “Nonlinear Schrödinger Equations” by Jean Bourgain and “Waves and Transport” by George Papanicolaou and Leonid Ryzhik. Susan Friedlander provides a written version of her lecture series “Stability and Instability of an Ideal Fluid”, given at the Mentoring Program for Women in Mathematics, a preliminary program to the Summer Session.
This Summer Session brought together students, fellows, and established mathematicians from all over the globe to share ideas in a vibrant and exciting atmosphere. This book presents the compelling results.
Graduate students and research mathematicians working in partial differential equations.
Table of Contents
- Nonlinear Schrödinger equations
- Harmonic analysis, wavelets and applications
- Lectures on stability and instability of an ideal fluid
- Waves and transport
- Lectures on geometric optics