Stability of background perturbation for Boltzmann equation

TL;DR

This paper studies how small changes in the background Maxwellian affect solutions to the Boltzmann equation, providing stability results and decay estimates for the solution under background perturbations.

Contribution

It establishes the continuous dependence of solutions on background variations and derives sharp decay estimates, advancing understanding of solution stability in kinetic theory.

Findings

Solutions depend continuously on background variations.

Sharp decay estimates for errors due to background perturbations.

Refined linearized estimates and decomposition techniques used in proofs.

Abstract

Consider the Boltzmann equation in the perturbation regime. Since the macroscopic quantities in the background global Maxwellian are obtained through measurements, there are typically some errors involved. This paper investigates the effect of background variations on the solution for a given initial perturbation. Our findings demonstrate that the solution changes continuously with variations in the background and provide a sharp time decay estimate of the associated errors. The proof relies on refined estimates for the linearized solution operator and a proper decomposition of the nonlinear solution.