A spatially homogeneous Boltzmann equation for elastic, inelastic and coalescing collisions

Abstract

Existence, uniqueness and qualitative behavior of the solution to a spatially homogeneous Boltzmann equation for particles undergoing elastic, inelastic and coalescing collisions are studied. Under general assumptions on the collision rates, we prove existence and uniqueness of a L 1 solution. This shows in particular that the cooling effect (due to inelastic collisions) does not occur in finite time. In the long time asymptotic, we prove that the solution converges to a mass-dependent Maxwellian function (when only elastic collisions are considered), to a velocity Dirac mass (when elastic and inelastic collisions are considered) and to 0 (when elastic, inelastic and coalescing collisions are taken into account). We thus show in the latter case that the effect of coalescence is dominating in large time. Our proofs gather deterministic and stochastic arguments.

Scalings for a Ballistic Aggregation Equation

“…We prove in this subsection an uniqueness and existence result for a general class of aggregation rates by adapting some arguments from [10,7], see also [13]. We then deduce the existence and uniqueness part in Theorem 3.1.…”

“…Existence of solutions under that condition has been proved in [14,2,7]. It has also been proved that f (t, ·) → 0 in L 1 (Y ), as t → +∞, (1.8) that is that the total number of particles tends to 0.…”

“…We refer to [10,7] where such kind of result is proved in a L 1 framework. The same result also holds for radially symmetric solutions when we assume that 11) and the second condition in (3.10) is replaced by…”

Mathematical Aspects of Coagulation-Fragmentation Equations

“…Still another model was studied by Fournier and Mischler [94,95], in which, although there is also no spacial dependence, there are, additionally to the binary collisions resulting in coagulating events, other binary elastic collisions (modelled by Boltzmann collision operator) and inelastic collisions (modelled by a granular collision operator).…”

A solution of the Boltzmann equation in the presence of inelastic collisions

“…Since the publication of this pioneering paper, a substantial number of papers have been published, in which various models of inelastic collision have been considered. Without making any attempt to present an extensive list of these papers we mention [33,14,9,4,5,3,8,10,21,30,17,15,11]. This list does not include papers where the models for collisions of specific atoms and molecules have been considered, such as H 2 -H 2 collisions [35], N-N 2 collisions [12] and H-N 2 collisions [31].…”