Infinite energy solutions to inelastic homogeneous boltzmann equations

This paper is concerned with the existence, shape and dynamical stability of infiniteenergy equilibria for a class of spatially homogeneous kinetic equations in space dimensions d ≥ 2. Our results cover in particular Bobylev’s model for inelastic Maxwell molecules. First, we show under certain conditions on the collision kernel, that there exists an index α ∈ (0, 2) such that the equation possesses a nontrivial stationary solution, which is a scale mixture of radially symmetric stable laws. We also characterize the mixing distribution as the fixed point of a smoothing transformation. Second, we prove that any transient solution that emerges from the NDA of some (not necessarily radial symmetric) α-stable distribution converges to an equilibrium. The key element of the convergence proof is an application of the central limit theorem to a representation of the transient solution as a weighted sum of projections of randomly rotated i.i.d. random vectors.