Quantitative Pointwise Estimate of the Solution of the Linearized Boltzmann Equation

TL;DR

This paper provides detailed pointwise estimates and decay rates for solutions of the linearized Boltzmann equation across different potential types, extending classical results with new weighted energy techniques.

Contribution

It introduces refined estimates and extends classical decay results to hard and soft potentials using exponential velocity weights.

Findings

Fluid structure for hard potentials and Maxwellian molecules

Optimal decay in fluid part for soft potentials

Sub-exponential decay outside finite Mach number region

Abstract

We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad's angular cutoff assumption. More precisely, for solutions inside the finite Mach number region, we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region, we obtain sub-exponential decay in the space variable. The singular wave estimate, regularization estimate and refined weighted energy estimate play important roles in this paper. Our results largely extend the classical results of Liu-Yu \cite{[LiuYu], [LiuYu2], [LiuYu1]} and Lee-Liu-Yu \cite% {[LeeLiuYu]} to hard and soft potentials by imposing suitable exponential velocity weight on the initial condition.