Existence Analysis of Maxwell--Stefan Systems for Multicomponent Mixtures

Abstract

Maxwell--Stefan systems describing the dynamics of the molar concentrations of a gas mixture with an arbitrary number of components are analyzed in a bounded domain under isobaric, isothermal conditions. The systems consist of mass balance equations and equations for the chemical potentials, depending on the relative velocities, supplemented with initial and homogeneous Neumann boundary conditions. Global-in-time existence of bounded weak solutions to the quasi-linear parabolic system and their exponential decay to the homogeneous steady state are proved. The mathematical difficulties are due to the singular Maxwell--Stefan diffusion matrix, the cross-diffusion coupling, and the lack of standard maximum principles. Key ideas of the proofs are the Perron--Frobenius theory for quasi-positive matrices, entropy-dissipation methods, and a new entropy variable formulation allowing for the proof of nonnegative lower and upper bounds for the concentrations.