On the Maxwell-Stefan Approach to Multicomponent Diffusion

Abstract

We consider the system of Maxwell-Stefan equations which describe multicomponent diffusive fluxes in non-dilute solutions or gas mixtures. We apply the Perron-Frobenius theorem to the irreducible and quasi-positive matrix which governs the flux-force relations and are able to show normal ellipticity of the associated multicomponent diffusion operator. This provides local-in-time wellposedness of the Maxwell-Stefan multicomponent diffusion system in the isobaric, isothermal case. (2000). Primary 35K59; Secondary 35Q35, 76R50, 76T30, 92E20 Mathematics Subject Classification

Stability of the Derivative of a Canonical Product

“…We suppose constant temperature and pressure. Our derivation follows [4]. For details on the modeling, we refer to the monographs [15,27].…”

“…Under some general assumptions on the nonlinearities, Giovangigli proved that there exists a unique global solution to the whole-space Maxwell-Stefan system if the initial datum is sufficiently close to the equilibrium state [15,Theorem 9.4.1]. Bothe [4] showed the existence of a unique local solution for general initial data. Boudin et al [7] considered a ternary system (N = 2) and assumed that two diffusivities are equal.…”

“…In this paper, we combine and extend the entropy-dissipation technique of [8,9,14,17,19] as well as ideas of Bothe [4] to overcome the above mathematical difficulties. We are able to prove the global-in-time existence of weak solutions to (1)-(3) for arbitrary diffusion matrices and general initial data.…”

“…Using the Perron-Frobenius theory for quasi-positive matrices, Bothe [4] characterized the spectrum of A(c) in case that c i > 0 for i = 1, . .…”

Compactness of Linearized Kinetic Operators

“…some specific physical phenomena (such as uphill diffusion in the purely diffusive case, see [26,29,16,3,5,24]). Consequently, it is not surprising that compactness properties in the mixture case cannot be deduced through a straightforward adaptation of the standard methods of proof from the mono-species case.…”