On 3-D vortex patches in bounded domains

This article concerns the equations of motion of perfect incompressible fluids in a 3-D, smooth, bounded, simply connected domain. We suppose that the curl of the initial velocity field is a vortex patch, and examine the classical problems of the existence of a solution, either locally or globally in time, and of the persistence of the initial regularity.Comment: 34 page

Mathematical study of the β\beta-plane model for rotating fluids in a thin layer

This article is concerned with an oceanographic model describing the asymptotic behaviour of a rapidly rotating and incompressible fluid with an inhomogeneous rotation vector; the motion takes place in a thin layer. We first exhibit a stationary solution of the system which consists of an interior part and a boundary layer part. The spatial variations of the rotation vector generate strong singularities within the boundary layer, which have repercussions on the interior part of the solution. The second part of the article is devoted to the analysis of two-dimensional and three-dimensional waves. It is shown that the thin layer effect modifies the propagation of three-dimensional Poincar\'e waves by creating small scales. Using tools of semi-classical analysis, we prove that the energy propagates at speeds of order one, i.e. much slower than in traditional rotating fluid models.Comment: 46 page

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

On the 2D Isentropic Euler System with Unbounded Initial vorticity

37 pagesThis paper is devoted to the study of the low Mach number limit for the 2D isentropic Euler system associated to ill-prepared initial data with slow blow up rate on $\log\varepsilon^{-1}$. We prove in particular the strong convergence to the solution of the incompressible Euler system when the vorticity belongs to some weighted $BMO$ spaces allowing unbounded functions. The proof is based on the extension of the result of \cite{B-K} to a compressible transport model

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

Slow Convergence to Vortex Patches in Quasigeostrophic Balance

Taking advantage of dispersive effects - which supposes a flow extended in the whole space - a sluggish convergence of the solutions of the Boussinesq equations to a solution of the quasigeostrophic system, when the Rossby number tends to zero, can be proved under weaker assumptions on the initial data than usual. In particular, no assumption of well-preparedness is needed. Two examples are given, both involving fields with striated potential vorticity. A result of convergence to a vortex patch is deduced from one of them.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Examples of dispersive effects in non-viscous rotating fluids

info:eu-repo/semantics/publishe

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