This paper deals with explicit spectral gap estimates for the linearized Boltzmann operator with hard potentials (and hard spheres). We prove that it can be reduced to the Maxwellian case, for which explicit estimates are already known. Such a method is constructive, does not rely on Weyl's Theorem and thus does not require Grad's splitting. The more physical idea of the proof is to use geometrical properties of the whole collision operator. In a second part, we use the fact that the Landau operator can be expressed as the limit of the Boltzmann operator as collisions become grazing in order to deduce explicit spectral gap estimates for the linearized Landau operator with hard potentials.

Sprays are complex flows made of liquid droplets surrounded by a gas. They can be modeled by introducing a system coupling a kinetic equation (for the droplets) of Vlasov type and a (Euler-like) fluid equation for the gas. In this paper, we prove that, for the so-called thin sprays, this coupled model is well-posed, in the sense that existence and uniqueness of classical solutions holds for small time, provided the initial data are sufficiently smooth and their support have suitable properties.

We here deal with a model of therapeutic sprays for the upper airways. We aim to model both inhaled and injected sprays. We propose a numerical solver for the kinetic equation which underlies our model, using a particle method. Eventually, we present two numerical tests for simple geometries of the airways. Résumé. Nous proposons dans ce travail une modélisation du comportement d'un brouillard de gouttelettesà but thérapeutique dans les voies respiratoires supérieures. Notre objectif est de mettre en place un modle pouvant représenterà la fois des sprays inhalés et des sprays injectés. Ce modèle est porté par uneéquation cinétique, pour laquelle nous présentons une résolution numérique par méthode particulaire. Enfin, nous donnons deux cas-tests pour des conduits respiratoiresà géométrie simple.

Revised version (new material: a section on the numerical boundary conditions, some minor modifications)International audienceMost of deterministic solvers for rarefied gas dynamics use discrete velocity (or discrete ordinate) approximations of the distribution function on a Cartesian grid. This grid must be sufficiently large and fine to describe the distribution functions at every space position in the computational domain. For 3-dimensional hypersonic flows, like in re-entry problems, this induces much too dense velocity grids that cannot be practically used, for memory storage requirements. In this article, we present an approach to generate automatically a locally refined velocity grid adapted to a given simulation. This grid contains much less points than a standard Cartesian grid and allows us to make realistic 3-dimensional simulations at a reduced cost, with a comparable accuracy