Kinetic interaction Morawetz and concentration estimates

TL;DR

This paper develops kinetic interaction Morawetz estimates for mesoscopic evolutions, introduces interaction uncertainty, and shows how interactions concentrate over time within small cones, based solely on conservation laws.

Contribution

It introduces the notion of interaction uncertainty and demonstrates its divergence over time, providing new a-priori bounds for mesoscopic kinetic equations.

Findings

Interaction uncertainty diverges with time.

Interactions concentrate within small cones over time.

Results apply to Boltzmann equation and variants.

Abstract

In this paper we will develop the kinetic interaction Morawetz estimate for mesoscopic evolutions. Furthermore, we will find a new a-priori bound and introduce the notion of interaction uncertainty. We will demonstrate that interaction uncertainty goes to infinity with time and that the interactions, averaged over time, concentrate within a specific collection of blind cones with an arbitrarily small apex angle. These results are solely based on conservation laws and thereby are all true for the Boltzmann equation and its different variants