Abstract. Geophysical fluids all exhibit a common feature: their aspect ratio (depth to horizontal width) is very small. This leads to an asymptotic model widely used in meteorology, oceanography, and limnology, namely the hydrostatic approximation of the time-dependent incompressible Navier-Stokes equations. It relies on the hypothesis that pressure increases linearly in the vertical direction. In the following, we prove a convergence and existence theorem for this model by means of anisotropic estimates and a new time-compactness criterium.

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We study the existence of travelling-waves and local well-posedness in a subspace of C 1 b (R) for a nonlinear evolution equation recently proposed by Andrew C. Fowler to describe the dynamics of dunes.2000 Mathematics Subject Classification. 47J35; 35G25; 76B25.

In this paper, we prove existence, uniqueness and regularity for a class of stochastic partial differential equations with a fractional Laplacian driven by a spacetime white noise in dimension one. The equation we consider may also include a reaction term.

The paper describes how to tackle new challenging coastal engineering problems related to beach erosion with a shape optimization approach. The method modifies the shape of the sea bottom in order to reduce beach erosion effects. Global optimization is shown to be necessary as the related functionals have several local minima. We describe the physical model used, the proposed protection devices against beach erosion and real case applications.