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We study the spatially homogeneous Boltzmann equation for Maxwell molecules, and its 1-dimensional model, the Kac equation. We prove propagation in time of stretched exponential moments of their weak solutions, both for the angular cutoff and the angular non-cutoff case. The order of the stretched exponential moments in question depends on the singularity rate of the angular kernel of the Boltzmann and the Kac equation. One of the main tools we use are Mittag-Leffler moments, which generalize the exponential ones.

Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules