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Surface water waves and Friedrichs expansion in the Lagrange system of coordinates. (Sur les ondes superficieles de l’eau et le dévelopement de Friedrichs dans le système de coordonnées de Lagrange.) (French) Zbl 1050.35085

Ancona, Vincenzo (ed.) et al., Hyperbolic differential operators and related problems. New York, NY: Marcel Dekker (ISBN 0-8247-0963-2/pbk). Lect. Notes Pure Appl. Math. 233, 199-231 (2003).
The authors are investigating various equations or systems of partial differential equations describing the flow of a liquid, under various conditions. Airy’s equations, Boussinesq’s equation as well as Korteweg-de Vries equation are considered. A rather detailed history of the problem is sketched, with adequate references, including such names as Stokes, Lord Raleigh, Levi-Civita, Struik, Lavrentiev, Friedrichs-Hyers, Ovsjannikov, Courant, W. Craig and other contributors with significant clarifying aspects to the problem of water waves. The following sections of the paper are related to various items concerning the topic; surface water waves; the problem in Lagrange coordinates; a priori estimates; a priori estimates for the pressure; the convergence of the power series (representing the solution); approximate equations (deep water case, Boussinesq and Korteweg-de Vries for long waves are dealt with). The references cover a period starting in 1815 and up to recent years.
For the entire collection see [Zbl 1027.00009].

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q53 KdV equations (Korteweg-de Vries equations)
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
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