Day 5 Titles and Abstracts



The three-wave coupling system. Application to Laser-plasma interaction

R. Sentis
CEA-Bruyeres, France
remi.sentis -at- cea.fr

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In the framework of Laser-Plasma interaction problems, one have to address the coupling between two electromagnetic fields - the transmitted and backscattered laser fields - and an ion hydrodynamic field. The characteristic speed of this last one is of course very small compared to the speed of light. The aim of this talk is to analyse this three-wave coupling system. This non-linear hyperbolic system has been used for fourty years by physisics but we give here new mathematical results on this system. Moreover we propose a numerical scheme which may be used in large 3D simulation paraxial propagation code.

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Incompressible fluids in porous media

Rafael Orive
Universidad Autonama de Madrid, Spain
rafael.orive -at- uam.es

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We study the fluid interface problem through porous media given by two incompressible 2D fluids of different densities. This problem is mathematically analogous to the dynamics interface for convection in porous media, where the free boundary evolves between fluids with different temperatures. We present the last works done with the collaboration of A. Castro, D. Córdoba, F. Gancedo and R. Granero.

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Strong solutions of systems coupling elastic structures with the Navier-Stokes equations

J.-P. Raymond
Institut de Mathématiques,
Université Paul Sabatier, France
raymond -at- math.univ-toulouse.fr

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In this talk, we are interested in models coupling the incompressible Navier-Stokes equations with equations of elastic structures. When the structure is located at the boundary of the domain occupied by the fluid it will be described by a plate or a beam equation. We also consider the case of an elastic solid, described by the Lamé system, inside an incompressible viscous fluid. The solutions are found by using a fixed point method that requires the analysis of a linearized system. We prove new existence results and we recover some other ones. In both cases, the analysis of the linearized system is new.

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Semiclassical and spectral analysis of a model arising in oceanography.

Isabelle Gallagher
University Paris Diderot, Paris, France
isabelle.gallagher -at- math.jussieu.fr

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In this talk we shall discuss a model of rotating shallow water equations arising in the study of oceanic waves. Under some appropriate scaling assumptions we exhibit Poincaré waves and Rossby waves and prove that the former have a dispersive behaviour, including for very long times, whereas the latter may exhibit a rapping phenomenon. This is a joint work with C. Cheverry, T. Paul and L. Saint-Raymond.

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A review of results concerning quasilinear parabolic and elliptic singular problems

Jacques Giacomoni
University of Pau, France
jacques.giacomoni -at- univ-pau.fr

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In this talk, I will present recent and joint contributions about quasilinear and singular elliptic and parabolic equations (existence, multiplicity of solutions, regularity). In particular, I will focus on the following problem:

$\displaystyle {{\rm P_t}} \left\{ \begin{aligned}u_t-\Delta_p u=\frac{1}{u^\de... ...imes\partial\Omega,\\ u(0,x) &=u_0(x)\;\text{ in }\Omega \end{aligned} \right.$

where $ \Omega$ is an open bounded domain with smooth boundary, $ 1 < p< \infty$ and $ 0<\delta$, $ f$ a locally lipschtiz function in $ {I\!\!R}^+$ and $ u_0\in L^2(\Omega)\cap W^{1,p}_0(\Omega)$.

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Some shape optimization problems for the $ p$-Laplacian operator

A. Chorwadwala & S. Kesavan &R. Mahadevan
University BioBio, Concepcion, Chile
rmahadevan -at- udec.cl

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Let $ B_1$ be a ball in $ {I\!\!R}^N$ and let $ B_0$ be a smaller ball inside it. It is shown that if $ u$ is the solution of the problem $ -\Delta_p u = 1$ in $ B_1 \setminus \bar{B_0}$ vanishing on the boundary, then the Dirichlet-energy of $ u$ on $ B_1 \setminus \bar{B_0}$ is minimal if and only if the two balls are concentric. It is also shown that the principal Dirichlet eigenvalue of the $ p$-Laplacian is maximal if and only if the two balls are concentric. The shape derivative calculus for these nonlinear problems is developed. A Weak Comparison Principle for the $ p$-Laplacian (with non-vanishing boundary condition) necessary for analyzing the shape derivatives is proved.

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Gamma-lower limit for energy convergence for weak data.

Muthukumar
Indian Institute of Technology, Kanpur, India
tmk -at- iitk.ac.in

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The talk will focus on problems that motivate the study of $ H$-convergence corresponding to weakly converging data from $ H^{-1}$. We shall conjecture on the $ \Gamma$- lower limit of energy functional and obtain the result for the periodic case.

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