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The water waves problem. Mathematical analysis and asymptotics. (English) Zbl 1410.35003

Mathematical Surveys and Monographs 188. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9470-5/hbk). xx, 321 p. (2013).
This monograph is devoted to the study of water waves from the point of view of applied analysis. The main emphasis is made on the rigorous justification of various asymptotic models such as the Korteweg-de Vries and nonlinear Schrödinger equations, which have been used formally for description of water waves for many years. The author collects together outcomes of years of his research, but also uses the recent advances of various authors to simplify and improve some of the proofs.
According to the author, “the rationale of this book is to propose a simple and robust framework allowing one to address some important issues raised by the water wave equations”. Each chapter is well structured around a specific point. Thanks to the clear organization of the book, it is easy to follow for individual results in each chapter and to combine them all together in a joint picture.
Chapter 1 includes a survey of various formulations of the water wave equations as well as their asymptotic reductions. Chapters 2, 3, and 4 are devoted to the well-posedness of the water wave equations and include a survey of solutions of the Laplace equation in appropriate function spaces, a survey of properties of the Dirichlet-Neumann operator, and the state of the art in the questions of well-posedness.
Chapters 5, 6 and 7 collect together derivation and justification of the shallow water asymptotic models used in coastal oceanography. These include the Green-Naghdi equations, Boussinesq models and, finally, the Korteweg-de Vries and Camassa-Holm equations.
Chapter 8 is devoted to the deep water models, which include the Benney-Roskes model, the Davey-Stewartson model, and the celebrated nonlinear Schrödinger equation. This chapter is open-ended, as there still remain many open problems in the full justification of these models.
Chapter 9 analyzes the influence of surface tension in the water wave models. Finally, Appendices A, B and C include supplementary material, which may be useful on its own. In particular, the appendices cover the additional results on the Dirichlet-Neumann operator, product and commutator estimates, and the full list of asymptotic models used in the book.
All together, the monograph is written by an international expert on the subject and should be a valuable resource both for newcomers and for specialists in water wave equations.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q35 PDEs in connection with fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B25 Singular perturbations in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35C20 Asymptotic expansions of solutions to PDEs
76D33 Waves for incompressible viscous fluids
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