Fluid Dynamic Limits for Gas Mixture I: Formal Derivations

Abstract

A macroscopic limit for a binary gas mixture in terms of the Boltzmann system with three small parameters: the Knudsen number, the Mach number and the diameter of particles, is considered in the whole physical space. When the small positive parameter ∊ goes to zero, it is shown that the Boltzmann system results in the compressible Euler equations decoupled with Navier–Stokes equations. In this first part of our paper, the results are of a conditional (formal) nature: both existence of a solution and existence of appropriate limits are assumed.

On the Boltzmann gas mixture equation: Linking the kinetic and fluid regimes

“…In what follows we summarize the main properties of the collision operator Q that are fundamental in the study of the mixture equation (for details see, for instance the articles by [22], [21], [31], [2], [17]):…”

“…Specifically the paper deals with the derivation of the macroscopic equations for a binary gas mixture of hard-sphere gases from asymptotic limits of the Boltzmann equation. By extending the methodology of the single-component gases case proposed in [17] and by employing different time and space scalings, we show that it is possible to recover, under suitable technical assumptions, various fluid dynamics equations like the incompressible linearized and nonlinear Navier-Stokes…”

Existence of Solutions of a Partial Integrodifferential Equation with Thermostat and Time Delay

“…Research perspectives include the possibility of performing an asymptotic analysis by parabolic (low-field) and hyperbolic (high-field) scalings (see [24][25][26][27][28][29][30][31][32][33][34]) with the aim of obtaining the dynamics of the system at the macroscopic scale. This is a work in progress and results will be presented in due course.…”

Thermostatted Kinetic Models: Open Problems

“…The second open problem refers to the modeling of biological systems, where genetic mutations can modify the interactions among the cancer cells, therefore the underlying description offered by the thermostatted kinetic theory models needs to be related to the evolution at the molecular (microscopic) and tissue (macroscopic) scale. In this context the asymptotic limits methods can be performed, see papers [10][11][12] and references cited therein.…”