Exponential tails for solutions to the homogeneous Boltzmann equation
We present recent results on the tail behavior for solutions to the homogeneous Boltzmann equation for variable hard potentials and their differences depending on the singular behavior of the angular cross section. More precisely, we discuss the generation and propagation of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. In the case of non-integrable angular cross section (i.e. without Grad's cutoff condition), the tails that are propagated depend on the the singularity rate of the angular cross-section. For some of those rates the corresponding functional weights are stretched exponential functions above Gaussians, and, for other singular rates the weights are Mittag-Leffler functions which can be viewed as fractional power series behaving asymptotically as such stretched exponentials as well.
This is work in collaboration with Ricardo J. Alonso, Irene M. Gamba and Natasa Pavlovic.