The Acoustic Limit for the Boltzmann Equation

The acoustic equations are the linearization of the compressible Euler equations about a spatially homogeneous fluid state. We first derive them directly from the Boltzmann equation as the formal limit of moment equations for an appropriately scaled family of Boltzmann solutions. We then establish this limit for the Boltzmann equation considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L1) to a unique limit governed by a solution of the acoustic equations for all time, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L2 initial data of the acoustic equations. The associated local conservation laws are recovered in the limit.