From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system

We establish a Navier-Stokes-Fourier limit for solutions of the Boltzmann equation considered over any periodic spatial domain of dimension two or more. We do this for a broad class of collision kernels that relaxes the Grad small deflection cutoff condition for hard potentials and includes for the first time the case of soft potentials. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that are compact. Every limit point is governed by a weak solution of a Navier-Stokes-Fourier system for all time.