E. Daus, A. Jüngel, B. Quoc Tang:

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

in: "ASC Report 8/2018", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2018, 1 - 43.

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.

Strongly coupled parabolic systems, Maxwell-Stefan systems, global existence

http://www.asc.tuwien.ac.at/preprint/2018/asc08x2018.pdf

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