Spectral analysis of transport equations with bounce-back boundary conditions

TL;DR

This paper analyzes the spectral properties of the linear transport equation with bounce-back boundary conditions, deriving explicit semigroup expressions and showing the essential spectrum's invariance across different $L^p$ spaces.

Contribution

It provides a detailed spectral analysis of the transport operator and establishes the equivalence of the essential spectrum for the transport and streaming semigroups using compactness arguments.

Findings

Explicit expression of the streaming semigroup derived.

Spectral properties of the transport operator characterized.

Essential spectrum of transport and streaming semigroups shown to coincide.

Abstract

We investigate the spectral properties of the time-dependent linear transport equation with bounce-back boundary conditions. A fine analysis of the spectrum of the streaming operator is given and the explicit expression of the strongly continuous streaming semigroup is derived. Next, making use of a recent result from M. Sbihi \cite{Sbihi}, we prove, via a compactness argument, that the essential spectrum of the transport semigroup and that of the streaming semigroup coincide on all $L^p$-spaces with $1<p<\infty$.