There is a vast literature devoted to the study of water waves ranging from
coastal engineering preoccupations to a very theoretical mathematical analysis of
the equations. Since all the scientific communities involved have their own approach
and terminology, it is sometimes quite a challenge to check that they actually speak
about the same things. The rationale of this book is to propose a simple and robust
framework allowing one to address some important issues raised by the water waves
equations. Of course, such a global approach is sometimes not compatible with
sharpness; it has been a deliberate choice to sacrifice the latter when a choice was
necessary (on minimal regularity assumptions, for instance). Hopefully, experts on
the well-posedness of the water waves equations or on the mathematical properties
of some asymptotic system, or on any other aspect, may, however, discover here
other related subjects of interest and some open problems.
Since this book is addressed to various audiences, there has been an effort to
make each chapter as thematically focused as possible. For instance, physical as-
pects are not often present in the chapters devoted to the well-posedness of the
equations, and, vice versa, the chapter devoted to the derivation of the asymp-
totic models is rather oriented by physical considerations and requires only basic
mathematical tools. More precisely,
- Chapter 1 is a general and basic introduction. It also includes a review of
various approaches developed in recent years for the mathematical anal-
ysis of the water waves equations. Some extensions raising several new
problems are also presented (such as moving bottoms and rough topog-
raphy), and the physical assumptions made to derive the water waves
equations are discussed.
- Chapters 2, 3, and 4 are devoted to the well-posedness of the water waves
equations. They are addressed to mathematicians looking for a basic intro-
duction to the Cauchy problem for water waves, as well as to researchers
more familiar with these equations but not necessarily with their behavior
in shallow water. Some aspects of the proof (e.g., the study of the Laplace
equation, properties of the Dirichlet-Neumann operator) can also be of
interest by themselves and we therefore gave sharper results than those
actually needed to prove the well-posedness of the water waves equations.
- In Chapter 5, we derive many shallow water asymptotic models used in
coastal oceanography. This chapter is addressed to oceanographers and
people working on such models and those who want to know precisely what
their range of validity is. This chapter uses only basic mathematical tools
and should be readable independently of Chapters 2, 3, and 4. Note that
the derivation of various models presented here (the so-called models with
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