Acoustic limit of the Boltzmann equation: classical solutions

TL;DR

This paper proves that solutions to the Boltzmann equation in a periodic domain converge to solutions of the acoustic system as the scaling parameter tends to zero, using uniform energy estimates for classical solutions.

Contribution

It establishes the global-in-time uniform energy estimates and strong convergence of solutions from the Boltzmann equation to the acoustic system in the classical solution framework.

Findings

Uniform energy estimates for solutions in the acoustic limit.

Strong convergence of Boltzmann solutions to acoustic system solutions.

Applicability to collision kernels including hard-sphere and inverse-power law with angular cutoff.

Abstract

We study the acoustic limit from the Boltzmann equation in the framework of classical solutions. For a solution $F_\varepsilon=\mu +\varepsilon \sqrt{\mu}f_\varepsilon$ to the rescaled Boltzmann equation in the acoustic time scaling \partial_t F_\varepsilon +\vgrad F_\varepsilon =\frac{1}{\varepsilon} \Q(F_\varepsilon,F_\varepsilon), inside a periodic box $\mathbb{T}^3$, we establish the global-in-time uniform energy estimates of $f_\varepsilon$ in $\varepsilon$ and prove that $f_\varepsilon$ converges strongly to $f$ whose dynamics is governed by the acoustic system. The collision kernel $\Q$ includes hard-sphere interaction and inverse-power law with an angular cutoff.