Pointwise Behavior of the Linearized Boltzmann Equation on Torus

TL;DR

This paper analyzes the pointwise behavior of the linearized Boltzmann equation on a torus, revealing fluid and kinetic aspects, with decay rates influenced by domain size and new proof techniques for key lemmas.

Contribution

It introduces a novel approach to construct fluid and kinetic-like waves, providing insights into their decay and regularity properties on the torus.

Findings

Fluid-like waves depend on domain size for decay rates

Kinetic-like waves exhibit exponential decay

New proof of the mixture lemma avoids explicit solutions

Abstract

We study the pointwise behavior of the linearized Boltzmann equation on torus for non-smooth initial perturbation. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as part of the long-wave expansion in the spectrum of the Fourier mode for the space variable, the time decay rate of the fluid-like waves depends on the size of the domain. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. Moreover, the mixture lemma plays an important role in constructing the kinetic-like waves, we supply a new proof of this lemma to avoid constructing explicit solution of the damped transport equations