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

## TL;DR

This paper proves the propagation of stretched exponential moments over time for solutions to the Boltzmann and Kac equations with Maxwell molecules, using Mittag-Leffler moments to handle angular singularities.

## Contribution

It establishes the propagation of stretched exponential moments for Maxwell molecules' Boltzmann and Kac equations, including non-cutoff cases, using Mittag-Leffler moments.

## Key findings

- Propagation of moments depends on angular kernel singularity.
- Results apply to both cutoff and non-cutoff cases.
- Uses Mittag-Leffler moments to handle singularities.

## Abstract

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.