Stéphane Mischler

On the spatially homogeneous Boltzmann equation

Abstract We consider the question of existence and uniqueness of solutions to the spatially homogeneous Boltzmann equation. The main result is that to any initial data with finite mass and energy, there exists a unique solution for which the same two quantities are conserved. We also prove that any solution which satisfies certain bounds on moments of order s A second part of the paper is devoted to the time discretization of the Boltzmann equation, the main results being estimates of the rate of convergence for the explicit and implicit Euler schemes. Two auxiliary results are of independent interest: a sharpened form of the so called Povzner inequality, and a regularity result for an iterated gain term.

Gelation and mass conservation in coagulation-fragmentation models

Abstract The occurrence of gelation and the existence of mass-conserving solutions to the continuous coagulation–fragmentation equation are investigated under various assumptions on the coagulation and fragmentation rates, thereby completing the already known results. A non-uniqueness result is also established and a connection to the modified coagulation model of Flory is made.

From the discrete to the continuous coagulation–fragmentation equations

The connection between the discrete and the continuous coagulation–fragmentation models is investigated. A weak stability principle relying on a priori estimates and weak compactness in L 1 is developed for the continuous model. We approximate the continuous model by a sequence of discrete models and, writing the discrete models as modified continuous ones, we prove the convergence of the latter towards the former with the help of the above-mentioned stability principle. Another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation-fragmentation model.

Fractional Diffusion Limit for Collisional Kinetic Equations

This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation.

On Kac's chaos and related problems

This paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by M. Kac \cite{Kac1956} in his study of mean-field limit for systems of

On coalescence equations and related models

N

STABILITY IN A NONLINEAR POPULATION MATURATION MODEL

undistinguishable particles as

On a quantum Boltzmann equation for a gas of photons

N\to\infty

Cooling Process for Inelastic Boltzmann Equations for Hard Spheres, Part II: Self-Similar Solutions and Tail Behavior

. First, we quantitatively liken three usual measures of {\it Kacs chaos}, some involving the all

Stability, Convergence to Self-Similarity and Elastic Limit for the Boltzmann Equation for Inelastic Hard Spheres

N