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Influence of bottom topography on long water waves. (English) Zbl 1144.76005
Summary: We focus here on the water waves problem for uneven bottoms in the long-wave regime, on an unbounded two or three-dimensional domain. In order to derive asymptotic models for this problem, we consider two different regimes of bottom topography, one for small variations in amplitude, and one for strong variations. Starting from the Zakharov formulation of this problem, we rigorously compute the asymptotic expansion of the involved Dirichlet-Neumann operator. Then, following the global strategy introduced by J. L. Bona et al. [Arch. Ration. Mech. Anal. 178, No. 3, 373–410 (2005; Zbl 1108.76012)], we derive new symetric asymptotic models for each regime. The solutions of these systems are proved to give good approximations of solutions of the water wave problem. These results hold for solutions that evanesce at infinity as well as for spatially periodic solutions.

MSC:
 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics 35Q35 PDEs in connection with fluid mechanics
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