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Hyperbolic Equations and Frequency Interactions
Edited by: Luis Caffarelli and Weinan E, New York University-Courant Institute of Mathematical Sciences, NY
A co-publication of the AMS and IAS/Park City Mathematics Institute.
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IAS/Park City Mathematics Series
1999; 466 pp; hardcover
Volume: 5
ISBN-10: 0-8218-0592-4
ISBN-13: 978-0-8218-0592-3
List Price: US$84
Member Price: US$67.20
Order Code: PCMS/5
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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.

Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Readership

Graduate students and research mathematicians working in partial differential equations.

Table of Contents

Nonlinear Schrödinger equations
  • J. Bourgain -- Introduction
  • J. Bourgain -- Generalities and initial value problems
  • J. Bourgain -- The initial value problem (continued)
  • J. Bourgain -- A digressioin: The initial value problem for the KdV equation
  • J. Bourgain -- 1D invariant Gibbs measures
  • J. Bourgain -- Invariant measures (2D)
  • J. Bourgain -- Quasi-periodic solutions of Hamiltonian PDE
  • J. Bourgain -- Time periodic solutions
  • J. Bourgain -- Time quasi-periodic solutions
  • J. Bourgain -- Normal forms
  • J. Bourgain -- Applications of symplectic capacities to Hamiltonian PDE
  • J. Bourgain -- Remarks on longtime behaviour of the flow of Hamiltonian PDE
Harmonic analysis, wavelets and applications
  • I. C. Daubechies and A. C. Gilbert -- Introduction
  • I. C. Daubechies and A. C. Gilbert -- Constructing orthonormal wavelet bases: Multiresolution analysis
  • I. C. Daubechies and A. C. Gilbert -- Wavelet bases: Construction and algorithms
  • I. C. Daubechies and A. C. Gilbert -- More wavelet bases
  • I. C. Daubechies and A. C. Gilbert -- Wavelets in other functional spaces
  • I. C. Daubechies and A. C. Gilbert -- Pointwise convergence for wavelet expansions
  • I. C. Daubechies and A. C. Gilbert -- Two-dimensional wavelets and operators
  • I. C. Daubechies and A. C. Gilbert -- Wavelets and differential equations
  • I. C. Daubechies and A. C. Gilbert -- References
Lectures on stability and instability of an ideal fluid
  • S. Friedlander -- Introduction
  • S. Friedlander -- Equations of motion
  • S. Friedlander -- Initial-boundary value problem
  • S. Friedlander -- The type of the Euler equations
  • S. Friedlander -- Vorticity
  • S. Friedlander -- Steady flows
  • S. Friedlander -- Stability/instability of an equilibrium state
  • S. Friedlander -- Two-dimensional spectral problem
  • S. Friedlander -- "Arnold" criterion for nonlinear stability
  • S. Friedlander -- Plane parallel shear flow
  • S. Friedlander -- Instability in a vorticity norm
  • S. Friedlander -- Sufficient condition for instability
  • S. Friedlander -- Exponential stretching
  • S. Friedlander -- Integrable flows
  • S. Friedlander -- Baroclinic instability
  • S. Friedlander -- Nonlinear instability
  • S. Friedlander -- References
Waves and transport
  • G. Papanicolaou and L. Ryzhik -- Introduction
  • G. Papanicolaou and L. Ryzhik -- The Schrödinger equation
  • G. Papanicolaou and L. Ryzhik -- Symmetric hyperbolic systems
  • G. Papanicolaou and L. Ryzhik -- Waves in random media
  • G. Papanicolaou and L. Ryzhik -- The diffusion approximation
  • G. Papanicolaou and L. Ryzhik -- The geophysical applications
  • G. Papanicolaou and L. Ryzhik -- References
Lectures on geometric optics
  • J. Rauch and M. Keel -- Introduction
  • J. Rauch and M. Keel -- Basic linear existence theorems
  • J. Rauch and M. Keel -- Examples of propagation of singularities and of energy
  • J. Rauch and M. Keel -- Elliptic geometric optics
  • J. Rauch and M. Keel -- Linear hyperbolic geometric optics
  • J. Rauch and M. Keel -- Basic nonlinear existence theorems
  • J. Rauch and M. Keel -- One phase nonlinear geometric optics
  • J. Rauch and M. Keel -- Justification of one phase nonlinear geometric optics
  • J. Rauch and M. Keel -- References
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