Etienne Bernard

The Diffusion Approximation for the Linear Boltzmann Equation with Vanishing Scattering Coefficient

The present paper discusses the diffusion approximation of the linear Boltzmann equation in cases where the collision frequency is not uniformly large in the spatial domain. Our results apply for instance to the case of radiative transfer in a composite medium with optically thin inclusions in an optically thick background medium. The equation governing the evolution of the approximate particle density coincides with the limit of the diffusion equation with infinite diffusion coefficient in the optically thin inclusions.

A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures

In this short paper, we formally derive the thin spray equation for a steady Stokes gas, i.e. the equation consists in a coupling between a kinetic (Vlasov type) equation for the dispersed phase and a (steady) Stokes equation for the gas. Our starting point is a system of Boltzmann equations for a binary gas mixture. The derivation follows the procedure already outlined in [Bernard-Desvillettes-Golse-Ricci, arXiv:1608.00422 [math.AP]] where the evolution of the gas is governed by the Navier-Stokes equation.

A Derivation of the Vlasov-Navier-Stokes Model for Aerosol Flows from Kinetic Theory

This article proposes a derivation of the Vlasov-Navier-Stokes system for spray/aerosol flows. The distribution function of the dispersed phase is governed by a Vlasov-equation, while the velocity field of the propellant satisfies the Navier-Stokes equations for incompressible fluids. The dynamics of the dispersed phase and of the propellant are coupled through the drag force exerted by the propellant on the dispersed phase. We present a formal derivation of this model from a multiphase Boltzmann system for a binary gaseous mixture, involving the droplets/dust particles in the dispersed phase as one species, and the gas molecules as the other species. Under suitable assumptions on the collision kernels, we prove that the sequences of solutions to the multiphase Boltzmann system converge to distributional solutions to the Vlasov-Navier-Stokes equation in some appropriate distinguished scaling limit. Specifically, we assume (a) that the mass ratio of the gas molecules to the dust particles/droplets is small, (b) that the thermal speed of the dust particles/droplets is much smaller than that of the gas molecules and (c) that the mass density of the gas and of the dispersed phase are of the same order of magnitude.

Homogenization of the Linear Boltzmann Equation in a Domain with a Periodic Distribution of Holes

Consider a linear Boltzmann equation posed on the Euclidian plane with a periodic system of circular holes and for particles moving at speed 1. Assuming that the holes are absorbing, i.e., that particles falling in a hole remain trapped there forever, we discuss the homogenization limit of that equation in the case where the reciprocal number of holes per unit surface and the length of the circumference of each hole are asymptotically equivalent small quantities. We show that the mass loss rate due to particles falling into the holes is governed by a renewal equation that involves the distribution of free path lengths for the periodic Lorentz gas. In particular, it is proved that the total mass of the particle system at time t decays exponentially quickly as

Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate

t\to+\infty

On the Convergence to Equilibrium for Degenerate Transport Problems

. This is at variance with the collisionless case discussed in [E. Caglioti and F. Golse, Comm. Math. Phys., 236 (2003), pp. 199–221], where the total mass decays as

On the exponential decay to equilibrium of the degenerate linear Boltzmann equation

C/t

Optimal Estimate of the Spectral Gap for the Degenerate Goldstein-Taylor Model

as

HOMOGENIZATION OF TRANSPORT PROBLEMS AND SEMIGROUPS

t\to+\infty

Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate

.