Acoustic limit of the Boltzmann equation: classical solutions

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.