On Two Incompressible Hydrodynamic Limits of the Boltzmann–Enskog Equation. I. Formal Derivations

Abstract

Two macroscopic limits of the Boltzmann–Enskog kinetic equation with three small parameters: the Knudsen number, the Mach number and the diameter of particles, are considered in the whole physical space. Under two different assumptions about the relations between the small parameters, it is shown that the Boltzmann–Enskog equation results in the Navier–Stokes equation for incompressible fluids together with two different Boussinesq relations and temperature fluctuation equations. Differences between the resulting macroscopic systems of equations corresponding to the Boltzmann–Enskog equation and the Boltzmann equation are shown. In the present paper, the results are of a conditional (formal) nature: both existence of a solution and existence of appropriate limits are assumed. The proof of a corresponding rigorous result will be given in Jagodziński and Lachowicz (Jagodziński, S., Lachowicz, M. On two incompressible hydrodynamic limits of the Boltzmann–Enskog equation. II: A rigorous result, to appear).