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

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 an $L^1$ 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 (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 considered). We thus show in the latter case that the effect of coalescence is dominating in large time. Our proofs gather deterministic and stochastic arguments.

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