Lectures on the Energy Critical Nonlinear Wave Equation
About this Title
Carlos E. Kenig, University of Chicago, Chicago, IL
Publication: CBMS Regional Conference Series in Mathematics
Publication Year 2015: Volume 122
ISBNs: 978-1-4704-2014-7 (print); 978-1-4704-2273-8 (online)
MathSciNet review: MR3328916
MSC: Primary 35L71; Secondary 35B44, 35C08, 35L15
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.
Graduate students and research mathematicians interested in nonlinear wave equations.
Table of Contents
- 1. The local theory of the Cauchy problem
- 2. The “road map”: The concentration compactness/rigidity theorem method for critical problems I
- 3. The “road map”: The concentration compactness/rigidity theorem method for critical problems II
- 4. Properties of compact solutions and some more rigidity theorems, with applications to an extension of Theorem 2.6
- 5. Proof of the rigidity theorems
- 6. Type II blow-up solutions
- 7. Channels of energy and outer energy lower bounds
- 8. Universal type II blow-up profiles
- 9. Soliton resolution for radial solutions to (NLW), I
- 10. Soliton resolution for radial solutions to (NLW), II
- 11. Soliton resolution for radial solutions to (NLW), III