The Boltzmann equation can be derived rigorously from a system of elastic hard spheres (Lanford?s theorem, [11], [8]). Kinetic theory may also be fruitfully used to model large systems of particles that interact inelastically (sand, snow, interstellar dust, see [3], [10], [4]). Such materials are known as granular media. The theory enables for instance to explain the onset of inhomogeneities, as well as to quantify the decay of the temperature (Haff?s law, [9], [3]). In this case, the derivation of the inelastic Boltzmann equation is still open, mainly due to the complicated dynamics of the particles. In particular, it is still unknown if the dynamics of such particle systems is well-posed. One major difficulty comes from the phenomenon of inelastic collapse. A system of particles is said to experience an inelastic collapse when infinitely many collisions take place in finite time. It is known that inelastic collapse may take place for systems of only three particles [12].
We studied systems of three particles, in dimension d ? 2. Assuming that the restitution coefficient r is constant, we obtained general results of convergence and asymptotics concerning the variables of the dynamical system describing a collapsing system of particles. We prove a complete classification of the singularities when a collapse of three particles takes place, obtaining only two possible orders of collisions between the particles. In the first case we recover that the particles arrange in a nearly-linear chain, already studied by Zhou and Kadanoff [13], and in the second case we obtain that the particles arrange in a triangle, and we show that, after sufficiently many collisions, the particles collide according to a unique order of collisions, which is periodic. Finally, we construct an initial configuration leading to a nearly-linear collapse, stable under perturbations, and such that the angle between the particles at the time of collapse can be chosen a priori, with an arbitrary precision. Another important question is the following: since inelastic collapse can take place, is it possible to continue the dynamics of the particles anyway? We report also partial results in this direction.
Considering on the other hand another law of collision, prescribing that a fixed quantity of kinetic energy is lost during each collision, we obtained results on systems of an arbitrary number of particles interacting according to this law, that look a priori contradictory. Namely, we proved that the flow of such a system of particles conserves the measure in the phase space, whereas the kinetic energy is not conserved. From these results, we deduce an Alexander?s theorem [1] for such systems of particles: for almost every initial datum, the dynamics of such systems is globally well-posed. To the best of our knowledge, this is the first result of global well-posedness concerning the dynamics of systems of inelastic particles.
The results are taken from [5], [7], [6], obtained in collaboration with Juan J. L. Velázquez (Universität Bonn).
