On Mittag-Leffler moments for the Boltzmann equation for hard potentials without cutoff

TL;DR

This paper introduces Mittag-Leffler moments as a new method to establish weighted propagation properties of solutions to the Boltzmann equation with hard potentials and non-integrable angular components, advancing understanding of solution behavior.

Contribution

It develops a novel approach using Mittag-Leffler moments to analyze and propagate weighted norms for the Boltzmann equation with angular singularities, extending previous exponential weight methods.

Findings

Established $L^1$ weighted propagation for solutions

Introduced Mittag-Leffler moments as a new analytical tool

Characterized weights based on angular singularity and potential rates

Abstract

We establish the $L^1$ weighted propagation properties for solutions of the Boltzmann equation with hard potentials and non-integrable angular components in the collision kernel. Our method identifies null forms by angular averaging and deploys moment estimates of solutions to the Boltzmann equation whose summability is achieved by introducing the new concept of Mittag-Leffler moments - extensions of $L^1$ exponentially weighted norms. Such $L^1$ weighted norms of solutions to the Boltzmann equation are, both, generated and propagated in time and the characterization of their corresponding Mittag-Leffler weights depends on the angular singularity and potential rates in the collision kernel. These estimates are a fundamental step in order to obtain $L^\infty$ exponentially weighted estimates for solutions of the Boltzmann equation being developed in a follow up work.