Nima Moini | Searcharxiv

arXiv2021

Kinetic interaction Morawetz and concentration estimates

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

math.AP↗

arXiv2020

Evolution of mesoscopic interactions and scattering solutions of the Boltzmann equation

The purpose of this paper is to study the evolution of moving interacting particles on the mesoscopic scale. We will introduce an uncertainty principle and a new priori bound for the evolution of particles subject to a general mesoscopic interaction and use this setting to demonstrate evidence for the universal dispersion of particles in the absence of the conventional methods like velocity averaging lemmas or Strichartz-type estimates. We will develop a scattering theory in generality and use its frame work to show the existence and uniqueness of a class scattering solutions to the Boltzmann equation with asymptotic stability and completeness in the $L^{\infty}$ setting.

math.AP↗