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 $F_{\epsilon }=\mu +\epsilon \sqrt{\mu }{f}_{\epsilon }$ 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 $f_{\epsilon }$ in $\epsilon$ and prove that $f_{\epsilon }$ 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.

14 pages, To appear on Discrete and Continuous Dynamical Systems - Series A