The acoustic limit for the Boltzmann equation

The acoustic equations are the linearization of the compressible Euler equa tions about a spatially homogeneous fluid state. We first derive them direc tly from the Boltzmann equation as the formal limit of moment equations for an appropriately scaled family of Boltzmann solutions. We then establish t his limit for the Boltzmann equation considered over a periodic spatial dom ain for bounded collision kernels. Appropriately scaled families of DiPerna -Lions renormalized solutions are shown to have fluctuations that converge entropically land hence strongly in L-1) to a unique limit governed by a so lution of the acoustic equations for all time, provided that its initial fl uctuations converge entropically to an appropriate limit associated to any given L-2 initial data of the acoustic equations. The associated local cons ervation laws are recovered in the limit.