Existence analysis of Maxwell-Stefan systems for multicomponent mixtures

TL;DR

This paper proves the global existence and exponential decay of solutions for Maxwell-Stefan systems modeling multicomponent gas mixtures, overcoming mathematical challenges posed by singular diffusion matrices and cross-diffusion effects.

Contribution

It introduces a novel entropy variable formulation and employs entropy-dissipation methods to establish well-posedness and decay properties of the system.

Findings

Proved global-in-time existence of bounded weak solutions.

Established exponential decay to steady state.

Developed a new entropy variable approach.

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 quasilinear 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.