On Two Incompressible Hydrodynamic Limits of the Boltzmann–Enskog Equation II: A Rigorous Result

The present paper is a continuation of part one, in which two macroscopic limits of the Boltzmann–Enskog kinetic equation with three small parameters—the Knudsen number, Mach number, and diameter of particles—were considered at the formal level, and conditional theorems were stated. In this paper, the proof of a rigorous result corresponding to one of the two cases considered in part one is given. Under some suitable assumptions on the relations between the small parameters, it is proved that the Boltzmann–Enskog equation results in the Navier–Stokes equation for incompressible fluids together with the Boussinesq relation and the temperature fluctuations equation. The time and velocity averaging is exploited.Dedicated to Professor Janusz Mika on his 70th birthday. [ABSTRACT FROM AUTHOR]