Asymptotic models for surface and internal waves. Paper from the 29th Brazilian mathematics colloquium -- \(29^\circ\) Colóquio Brasileiro de Matemática, Rio de Janeiro, Brazil, July 22 -- August 2, 2013 (Q2848114)

In this book, the author introduces and justifies the water waves system as well as many related asymptotic models. A special emphasis is put on the limit of validity of these models, consequence of the nature of the corresponding approximation. The book offers a panorama on various mathematical questions related to these models.NEWLINENEWLINEChapters 2 and 3 recall general tools concerning mainly the Cauchy problem: compactness techniques, oscillatory integrals, interpolation.NEWLINENEWLINEChapter 4 is devoted to models for surface waves. It introduces the water waves system (Euler equations with free surface) and several related asymptotic models: Boussinesq regime, KP regime, Camassa-Holm regime, the Saint-Venant regime, the full dispersion regime and finally the so-called modulation regime.NEWLINENEWLINEChapter 5 is devoted to models for internal waves, for which a similar presentation is done, with various relevant asymptotic regimes.NEWLINENEWLINEIn Chapter 6, the author discusses some rigorous justifications of the asymptotic models from the literature.NEWLINENEWLINEChapters 7, 8 and 9 are devoted to up-to-date well-posedness and ill-posedness results for some classical nonlinear dispersive equations (among them : the KdV equation, the BBM equation, the KP equation).NEWLINENEWLINEChapter 10 is devoted to several aspects of blow up: hyperbolic blow up, nonlinear blow up, dispersive blow up, with a selection of typical results.NEWLINENEWLINEChapter 11 presents an introduction to long-time existence issues.NEWLINENEWLINESeveral existence/non existence results of solitary waves are given in Chapter 12.NEWLINENEWLINEFinally, Chapters 13 and 14 discuss more specific and complicated models.NEWLINENEWLINEAn extensive bibliography (more than 250 references) completes this book and reinforces its interest as an wide introduction to the subject.