From the Boltzmann equation for gas mixture to the two-fluid incompressible hydrodynamic system

TL;DR

This paper rigorously derives the two-fluid incompressible Navier-Stokes-Fourier system from the Boltzmann equation for gas mixtures, using dimensionless analysis, moments' method, and energy estimates.

Contribution

It introduces a rigorous derivation of the two-fluid hydrodynamic model from kinetic theory for gas mixtures, including novel models and detailed limit analysis.

Findings

Validated the two-fluid incompressible Navier-Stokes-Fourier system as a hydrodynamic limit.

Developed refined energy estimates based on Macro-Micro decomposition.

Derived new hydrodynamic models through moments' method.

Abstract

In this paper, we study the hydrodynamic limit transition from the Boltzmann equation for gas mixtures to the two-fluid macroscopic system. Employing a meticulous dimensionless analysis, we derive several novel hydrodynamic models via the moments' method. For a certain class of scaled Boltzmann equations governing gas mixtures of two species, we rigorously establish the two-fluid incompressible Navier-Stokes-Fourier system as the hydrodynamic limit. This validation is achieved through the Hilbert expansion around the global Maxwellian and refined energy estimates based on the Macro-Micro decomposition.