The Incompressible Navier-Stokes Limit of the Boltzmann Equation for Hard Cutoff Potentials

TL;DR

This paper proves that solutions of the Boltzmann equation for hard cutoff potentials converge to Leray solutions of the Navier-Stokes equations as Mach and Knudsen numbers approach zero, extending previous results to a broader class of potentials.

Contribution

It establishes the incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials, completing earlier work on Maxwell molecules.

Findings

Convergence of Boltzmann solutions to Navier-Stokes solutions for hard cutoff potentials.

Extension of previous results to a broader class of intermolecular potentials.

Validation of the hydrodynamic limit in the kinetic theory for these potentials.

Abstract

The present paper proves that all limit points of sequences of renormalized solutions of the Boltzmann equation in the limit of small, asymptotically equivalent Mach and Knudsen numbers are governed by Leray solutions of the Navier-Stokes equations. This convergence result holds for hard cutoff potentials in the sense of H. Grad, and therefore completes earlier results by the same authors [Invent. Math. 155, 81-161(2004)] for Maxwell molecules.