Carlos E. Kenig – författare

Alberto P. Calderon (1920-1998) was one of the 20th century's leading mathematical analysts. His contributions have changed the way researchers approach and think about a variety of topics in mathematics and its applications, including harmonic analysis, partial differential equations and complex analysis, as well as in more applied fields such as signal processing, geophysics and tomography. In addition, he helped define the "Chicago school" of analysis, which remains influential to this day. In 1995, more than 300 mathematicians from around the world gathered in Chicago for a conference on harmonic analysis and partial differential equations held in Calderon's honour. This volume originated in papers given there and presents syntheses of several major fields of mathematics as well as original research articles. The book should be of use to researchers in these and other related fields.

This collection of new and original papers on mathematical aspects of nonlinear dispersive equations includes both expository and technical papers that reflect a number of recent advances in the field. The expository papers describe the state of the art and research directions. The technical papers concentrate on a specific problem and the related analysis and are addressed to active researchers. The book deals with many topics that have been the focus of intensive research and, in several cases, significant progress in recent years, including hyperbolic conservation laws, Schrodinger operators, nonlinear Schrodinger and wave equations, and the Euler and Navier-Stokes equations.

This monograph deals with recent advances in the study of the long-time asymptotics of large solutions to critical nonlinear dispersive equations. The first part of the monograph describes, in the context of the energy critical wave equation, the ``concentration-compactness/rigidity theorem method'' introduced by C. Kenig and F. Merle. This approach has become the canonical method for the study of the ``global regularity and well-posedness'' conjecture (defocusing case) and the ``ground-state'' conjecture (focusing case) in critical dispersive problems.The second part of the monograph describes the ``channel of energy'' method, introduced by T. Duyckaerts, C. Kenig, and F. Merle, to study soliton resolution for nonlinear wave equations. This culminates in a presentation of the proof of the soliton resolution conjecture, for the three-dimensional radial focusing energy critical wave equation.It is the intent that the results described in this book will be a model for what to strive for in the study of other nonlinear dispersive equations.

The origins of the harmonic analysis go back to an ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. Today's harmonic analysis incorporates the elements of geometric measure theory, number theory, probability, and has countless applications from data analysis to image recognition and from the study of sound and vibrations to the cutting edge of contemporary physics. The present volume is based on lectures presented at the summer school on Harmonic Analysis. These notes give fresh, concise, and high-level introductions to recent developments in the field, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field and to senior researchers wishing to keep up with current developments.