On the relaxation of the Maxwell–Stefan system to linear diffusion

Abstract

In this note, we rigorously prove the relaxation limit of the Maxwell-Stefan system to a system of heat equations when all binary diffusion coefficients tend to the same positive value.

From the Boltzmann Description for Mixtures to the Maxwell–Stefan Diffusion Equations

“…A relationship between the Fick diffusion and Maxwell-Stefan diffusion has been investigated by Salvarani and Soares in [27]. They rigorously prove the relaxation of the Maxwell-Stefan system, together with the equimolar closure relationship (2), towards a system of uncoupled linear diffusion equations of Fickian type.…”

“…The binary diffusion coefficients depend on the reduced mass of the species, on the temperature and on the cross sections of the kinetic model. Therefore, by tuning the coefficients a n in the kinetic cross section (27), it is possible to obtain several forms of temperature dependency for the binary diffusion coefficients and compare them with the experiments. When a i = 0 for all i ≥ 1, the formula written above corresponds to the binary diffusion coefficients already deduced in Equation (26).…”

Maximum entropy principle approach to a non-isothermal Maxwell-Stefan diffusion model

“…Along with applications, mathematical analysis of Maxwell-Stefan equations emerged as equally important but relatively new subject [5,6]. It covers the aspects of existence and uniqueness of solutions [7,8], large-time asymptotics [9] and relaxation limits [10]. Our concern, however, will be the formal derivation of Maxwell-Stefan diffusion equations and analysis of their dissipative character.…”

On the Maxwell-Stefan diffusion limit for a reactive mixture of polyatomic gases in non-isothermal setting

“…In particular, the first studies have been devoted to the matrix formulation of the gradient-flux relationships (see [23] and the references therein). Subsequently, the wellposedness and the long-time behavior of the solutions of the Maxwell-Stefan system (or some variants) have been studied in [11,14,16,25,28,29], the numerical simulation of the Maxwell-Stefan system has been the subject of [9,14,22,32], and the relationships between Fickian diffusion and the Maxwell-Stefan model have been analyzed in [12,37]. By following the research line initiated by Bardos, Golse and Levermore in the Nineties -whose goal was the derivation of the equations of fluid mechanics starting from the Boltzmann equation [2,3] -several articles have carried out the formal derivation of isothermal multicomponent Maxwell-Stefan type diffusion equations starting from the Boltzmann system for monatomic non-reactive gaseous mixtures [12,13,15,26,27].…”