Acoustic and Stokes limits for the Boltzmann equation

The Boltzmann equation is 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 or Stokes equations, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L² initial data of the acoustic or Stokes equations. The associated conservation laws are recovered in the limit.