The Water Waves Problem: Mathematical Analysis and Asymptotics
About this Title
David Lannes, Ecole Normale Supérieure et CNRS, Paris, France
Publication: Mathematical Surveys and Monographs
Publication Year 2013: Volume 188
ISBNs: 978-0-8218-9470-5 (print); 978-1-4704-0948-7 (online)
MathSciNet review: 3060183
MSC: Primary 35Q53; Secondary 35B25, 35C20, 76B15, 76D33
This monograph provides a comprehensive and self-contained study on the theory of water waves equations, a research area that has been very active in recent years. The vast literature devoted to the study of water waves offers numerous asymptotic models. Which model provides the best description of waves such as tsunamis or tidal waves? How can water waves equations be transformed into simpler asymptotic models for applications in, for example, coastal oceanography? This book proposes a simple and robust framework for studying these questions.
The book should be of interest to graduate students and researchers looking for an introduction to water waves equations or for simple asymptotic models to describe the propagation of waves. Researchers working on the mathematical analysis of nonlinear dispersive equations may also find inspiration in the many (and sometimes new) models derived here, as well as precise information on their physical relevance.
Graduate students and research mathematicians interested in nonlinear PDEs and applications to oceanography.
Table of Contents
- 1. The water waves problem and its asymptotic regimes
- 2. The Laplace equation
- 3. The Dirichlet-Neumann operator
- 4. Well-posedness of the water waves equations
- 5. Shallow water asymptotics: Systems. Part 1: Derivation
- 6. Shallow water asymptotics: Systems. Part 2: Justification
- 7. Shallow water asymptotics: Scalar equations
- 8. Deep water models and modulation equations
- 9. Water waves with surface tension
- Appendix A. More on the Dirichlet-Neumann operator
- Appendix B. Product and commutator estimates
- Appendix C. Asymptotic models: A reader’s digest