Exponential time decay of solutions to reaction-cross-diffusion systems of Maxwell-Stefan type

TL;DR

This paper studies the long-term behavior of Maxwell--Stefan reaction-cross-diffusion systems, proving exponential decay to equilibrium and establishing existence of solutions, with implications for chemically reacting fluids with complex interactions.

Contribution

It introduces a novel approach to analyze exponential decay in non-symmetric, non-positive definite diffusion matrices using an enlarged variable space and entropy methods.

Findings

Proved exponential decay to equilibrium with a constructive rate.

Established existence of global bounded weak solutions.

Extended results to complex-balanced reaction systems.

Abstract

The large-time asymptotics of weak solutions to Maxwell--Stefan diffusion systems for chemically reacting fluids with different molar masses and reversible reactions are investigated. The diffusion matrix of the system is generally neither symmetric nor positive definite, but the equations admit a formal gradient-flow structure which provides entropy (free energy) estimates. The main result is the exponential decay to the unique equilibrium with a rate that is constructive up to a finite-dimensional inequality. The key elements of the proof are the existence of a unique detailed-balanced equilibrium and the derivation of an inequality relating the entropy and the entropy production. The main difficulty comes from the fact that the reactions are represented by molar fractions while the conservation laws hold for the concentrations. The idea is to enlarge the space of $n$ partial concentrations by adding the total concentration, viewed as an independent variable, thus working with $n+1$ variables. Further results concern the existence of global bounded weak solutions to the parabolic system and an extension of the results to complex-balanced systems.