Remarks on the Acoustic Limit for the Boltzmann Equation

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.