Remarks on the Acoustic Limit for the Boltzmann Equation

TL;DR

This paper advances the understanding of the acoustic limit for the Boltzmann equation by broadening collision kernel classes, refining fluctuation scaling, and extending results to bounded domains with boundary conditions.

Contribution

It improves previous results by including all classical collision kernels, enhances the fluctuation scaling to match theoretical expectations, and extends the analysis to bounded domains with boundary conditions.

Findings

Treats all classical collision kernels compatible with DiPerna-Lions theory.

Refines the fluctuation scaling to O(ε^{1/2}).

Extends results to bounded domains with impermeable boundaries.

Abstract

We use some new nonlinear estimates found in \cite {LM} to improve the results of \cite{GL} that establish the acoustic limit for DiPerna-Lions solutions of Boltzmann equation in three ways. First, we enlarge the class of collision kernels treated to that found in \cite{LM}, thereby treating all classical collision kernels to which the DiPerna-Lions theory applies. Second, we improve the scaling of the kinetic density fluctuations with Knudsen number from $O(\epsilon^m)$ for some $m>\frac12$ to $O(\epsilon^\frac12)$. Third, we extend the results from periodic domains to bounded domains with impermeable boundaries, deriving the boundary condition for the acoustic system.