From DiPerna-Lions to Leray

David Levermore
University of Maryland
Department of Mathematics

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 made by Golse and Saint-Raymond, and includes for the first time kernel arising from soft potentials. All appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that are compact. Every limit point of such a family is governed by a Leray weak solution of a Navier-Stokes-Fourier system for all time. Key tools include the relative entropy cutoff method of Saint Raymond, the L^1 velocity averaging result of Golse and Saint Raymond, and some new estimates.

Presentation (PDF File)

Back to Workshop II: The Boltzmann Equation: DiPerna-Lions Plus 20 Years