3D Asymptotics for Water-Waves

The asymptotical analysis of free surface waves has been at the origin of many models such as: shallow water equations, Korteweg-de Vries (KdV), Kadomtsev-Petviashvili (KP), Boussinesq, Camassa-Holm, Green-Naghdi, deep water, etc. In this talk, we will show how to derive rigorously all these models and explain what is their range of validity. We will then justify all these models in the following sense:

  1. We prove the existence of exact solutions to the water waves equations over a time scale large enough to see the dynamics specific to the asymptotical model
  2. We give a precise estimate of the error one makes by replacing the solution of the full waterwaves equations by a solution of the asymptotic model.

In order to provide such a justification at once for all the above asymptotical models, we work with a general nondimensiolized version of the water waves equations involving 5 physical parameters. Using a nonlocal energy, we are able to prove a well-posedness result, uniformly with respect to all these parameters. The particular asymptotical regimes mentioned above are then treated as particular cases.

David Lannes, CNRS at Bordeaux, Universite Bordeaux I, France

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