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-balance 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-balance systems.
Highlights
The analysis of the large-time behavior of dynamical networks is important to the understanding of their stability properties
While there is a vast literature on the large-time asymptotics of reaction–diffusion systems, much less is available for reaction systems with cross-diffusion terms
We prove the exponential decay of solutions to reaction-cross-diffusion systems of Maxwell–Stefan form by combining recent techniques for cross-diffusion systems [37] and reaction–diffusion equations [25]
Summary
The analysis of the large-time behavior of dynamical networks is important to the understanding of their stability properties. Of particular interest are reversible chemical reactions interacting with diffusion. While there is a vast literature on the large-time asymptotics of reaction–diffusion systems, much less is available for reaction systems with cross-diffusion terms. Such systems arise naturally in multicomponent fluid modeling and population dynamics [38]. We prove the exponential decay of solutions to reaction-cross-diffusion systems of Maxwell–Stefan form by combining recent techniques for cross-diffusion systems [37] and reaction–diffusion equations [25]. The main feature of our result is that the decay rate is constructive up to a finite-dimensional inequality and that the result holds for detailed-balance or complex-balance systems
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