Asymptotic analysis of the linearized Boltzmann collision operator from angular cutoff to non-cutoff

TL;DR

This paper provides a detailed asymptotic analysis of the linearized Boltzmann collision operator, revealing the transition from cutoff to non-cutoff cases and clarifying spectral properties related to long-range interactions.

Contribution

It offers quantitative estimates on the asymptotics of the linearized Boltzmann operator, connecting hyperbolic and smoothing properties, and resolves a question about spectral gaps in cutoff versus non-cutoff cases.

Findings

No spectral gap in cutoff case; gap exists in non-cutoff case for moderate soft potentials.

Discovered new phenomena in asymptotic limit process using phase space localization.

Established link between hyperbolic and smoothing properties of the operator.

Abstract

We give quantitative estimates on the asymptotics of the linearized Boltzmann collision operator and its associated equation from angular cutoff to non cutoff. On one hand, the results disclose the link between the hyperbolic property resulting from the Grad's cutoff assumption and the smoothing property due to the long-range interaction. On the other hand, with the help of the localization techniques in the phase space, we observe some new phenomenon in the asymptotic limit process. As a consequence, we give the affirmative answer to the question that there is no jump for the property that the collision operator with cutoff does not have the spectrum gap but the operator without cutoff does have for the moderate soft potentials.