**Mathematical Surveys and Monographs**

Volume: 188;
2013;
321 pp;
Hardcover

MSC: Primary 76; 35;

Print ISBN: 978-0-8218-9470-5

Product Code: SURV/188

List Price: $98.00

Individual Member Price: $78.40

**Electronic ISBN: 978-1-4704-0948-7
Product Code: SURV/188.E**

List Price: $98.00

Individual Member Price: $78.40

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#### Supplemental Materials

# The Water Waves Problem: Mathematical Analysis and Asymptotics

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*David Lannes*

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.

#### Table of Contents

# Table of Contents

## The Water Waves Problem: Mathematical Analysis and Asymptotics

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface xiii14 free
- Index of notations xvii18 free
- The water waves problem and its asymptotic regimes 122 free
- The Laplace equation 3758
- The Dirichlet-Neumann operator 6182
- Well-posedness of the water waves equations 91112
- Shallow water asymptotics: Systems. Part 1: Derivation 121142
- Shallow water asymptotics: Systems. Part 2: Justification 157178
- Shallow water asymptotics: Scalar equations 177198
- Deep water models and modulation equations 217238
- Water waves with surface tension 249270
- Appendix A. More on the Dirichlet-Neumann operator 267288
- Appendix B. Product and commutator estimates 283304
- Appendix C. Asymptotic models: A reader’s digest 297318
- Bibliography 305326
- Index 319340 free
- Back Cover Back Cover1346

#### Readership

Graduate students and research mathematicians interested in nonlinear PDEs and applications to oceanography.