Asymptotic analysis of a Vlasov-Boltzmann equation with anomalous scaling

TL;DR

This paper investigates how an external acceleration field affects the asymptotic behavior of solutions to a Vlasov-Boltzmann equation, showing that it leads to a fractional diffusion with an added advection term under certain scalings.

Contribution

It extends existing fractional diffusion approximations of the Boltzmann equation by incorporating the effects of an external acceleration field, analyzing critical and supercritical cases.

Findings

Fractional diffusion with advection term arises under specific scalings.

Both critical and supercritical acceleration field cases are analyzed.

The results generalize previous models lacking external fields.

Abstract

This paper is devoted to the approximation of the linear Boltzmann equation by fractional diffusion equations. Most existing results address this question when there is no external acceleration field. The goal of this paper is to investigate the case where a given acceleration field is present. The main result of this paper shows that for an appropriate scaling of the acceleration field, the usual fractional diffusion equation is supplemented by an advection term. Both the critical and supercritical case are considered.