Kamel Hamdache

Global Existence and Large Time Behaviour of Solutions for the Vlasov-Stokes Equations

One considers the Vlasov-Stokes equations to model the motion of a solid particles suspension in a Stokes flow. The dispersed phase, is modelled by a transport kinetic equation with acceleration corresponding to the Stokes drag and gravity field. The viscous fluid is assumed to be incompressible and its velocity satisfies the Stokes equations with an external force. This force is due to the relative velocity of the dispersed phase in the fluid. Denoting byN the space dimension, we prove global existence results of solutions forN ≥ 2. We also obtain the large time asymptotic behaviour of the solutions forN = 2 andN = 3.

MATHEMATICAL ANALYSIS FOR COMPRESSIBLE MISCIBLE DISPLACEMENT MODELS IN POROUS MEDIA

We discuss a three-dimensional displacement model of one miscible compressible fluid by another in a porous medium. The motion is modeled by a nonlinear system of parabolic type coupling the pressure and the concentration. We give an existence result of weak solutions for a model with diffusion and dispersion, using the Schauder fixed point theorem. We also study a model in the absence of diffusion and dispersion. The system becomes of parabolic-hyperbolic type, the existence of global weak solutions is then obtained through a compensated compactness argument.

Homogénéisation d'équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux

Resume On s’interesse a l’homogeneisation de l’equation hyperbolique modele ∂ t u e + a e ( t , y ) ∂ x u e = 0 , t > 0 , x ∈ ℝ , y ∈ Ω ⊂ ℝ N , munie d’une condition initiale (et d’une condition aux limites lorsque x ∈]0, 1[). Pour cela, nous caracterisons la limite L∞(Ω) faible * de fonctions du type φ x e ( λ ) = ( λ − A e ( y ) ) − 1 definies pour , λ ∉ [m, M] et verifiant 0

Homogenisation of transport kinetic equations with oscillating potentials

We consider the homogenisation of transport kinetic equations with a highly periodic oscillating external field. The external field, acting on the particles, consists of a sum of a field deriving from a periodic potential and a bounded periodic perturbation. For the profile function generated by the oscillating solution of the problem, we derive a kinetic model with transmission boundary conditions in the energy variable. In some cases, for example when the field is not perturbed, we deduce a transport kinetic equation with memory effect satisfied by the weak-* limit of the sequence of solutions.

Chirality in the Maxwell equations by the dipole approximation

In this paper, we show how chiral materials are realized by embedding chiral objects (helices) in an isotropic medium. More precisely, we derive the Drude--Born--Fedorov constitutive relations, governing the propagation of electromagnetic waves in chiral media, from the standard constitutive relations for a homogeneous, isotropic medium by embedding in the medium a large number of regularly spaced, ramdomly oriented helical conductors, each modeled as a dipole.

Global existence and regularity of solutions to a system of nonlinear Maxwell equations

Abstract We consider the model that has been suggested by Greenberg et al. (Physica D 134 (1999) 362–383) for the ferroelectric behavior of materials. In this model, the usual (linear) Maxwells equations are supplemented with a constitutive relation in which the electric displacement equals a constant times the electric field plus an internal polarization variable which evolves according to an internal set of nonlinear Maxwells equations. For such model we provide rigorous proofs of global existence, uniqueness, and regularity of solutions. We also provide some preliminary results on the long-time behavior of solutions. The main difficulties in this study are due to the loss of compactness in the system of Maxwells equations. These results generalize those of Greenberg et al., where only solutions with TM (transverse magnetic) symmetry were considered.

Homogénéisation d'un modèle d'écoulements miscibles en milieu poreux

We consider a 1-D model for incompressible miscible displacements in porous media without any dispersion term. Existence and uniqueness results for nonsmooth data are proved. We study the homogenization of the model. The limit problem is of the same type. The result is obtained thanks to compactness properties of the corresponding characteristic curves.

INTERLAYER EXCHANGE COUPLING FOR FERROMAGNETS THROUGH SPACERS

We consider the interlayer exchange coupling between two ferromagnetic layers mediated by a nonmagnetic spacer layer. We adopt the theoretical model which is based on a macroscopic description using the Landau--Lifshitz equations and the Hoffmann boundary conditions characterizing the interlayer exchange coupling between the interfaces of the ferromagnetic films. An asymptotic study of such a layered system for either small or large spacer thicknesses is presented. The asymptotic problem and the boundary conditions at interfaces both for the magnetization and the magnetic field are characterized.

Homogenization of degenerate wave equations with periodic coefficients

In this paper the authors discuss homogenization of hyperbolic equations involving periodic coefficients which are degenerate relative to a certain direction. The general scheme by which effective equations are obtained is as the reiterated homogenization. The first step of the process leads to equations describing the oscillatory behavior in the direction of the propagation. Next, space averaging in the degenerate direction gives the result. The process in the second step produces nonlocal effects.

Problemes aux limites pour l'equation de boltzmann: existence globale de solutions

This work is concerned by the initial and boundary value problem for the Boltzmann equation. Existence in the large and asymptotic behaviour results are proved for small data and general reemission laws. The colisional kernel satisfies the cut—off angular hypothesesand, the intermolecular potential is such that 7/3 <s≤∞ for the Dirichlet case and 7/3 <s≤5for the general case Mot—cles: Equation de Boltzmann— Problemes aux limites interieurs et exterieurs — Gaz rarefies —Condition aux limites de Dirichlet— Conditions aux limites non linaeaires— Existence globale de petites solutions