ON SOME GEOMETRY OF PROPAGATION IN DIFFRACTIVE TIME SCALES

International audienceIn this article, we develop a non linear geometric optics which presents the two main following features. It is valid in diffractive times and it extends the classical approaches to the case of fast variable coefficients. In this context, we can show that the energy is transported along the rays associated with some non usual long-time hamiltonian. Our analysis needs structural assumptions and initial data suitably polrarized to be implemented. All the required conditions are met concerning a current model arising in fluid mechanics and which was the original motivation of our work. As a by product, we get results complementary to the litterature concerning the propagation of the Rossby waves which play a part in the description of large oceanic currents, like Gulf stream or Kuroshio

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final publication is available at http://www.springerlink.co

Fast Averaging for Long- and Short-wave Scaled Equatorial Shallow Water Equations with Coriolis Parameter Deviating from Linearity

The equatorial shallow water equations at low Froude number form a symmetric hyperbolic system with large terms containing a variable coefficient, the Coriolis parameter f, which depends on the latitude. The limiting behavior of the solutions as the Froude number tends to zero was investigated rigorously a few years ago, using the common approximation that the variations of f with latitude are linear. In that case, the large terms have a peculiar structure, due to special properties of the harmonic oscillator Hamiltonian, which can be exploited to prove strong uniform a priori estimates in adapted functional spaces. It is shown here that these estimates still hold when f deviates from linearity, even though the special properties on which the proofs were based have no obvious generalization. As in the linear case, existence, uniqueness and convergence properties of the solutions corresponding to general unbalanced data are deduced from the estimates.SCOPUS: ar.jinfo:eu-repo/semantics/publishe