Exponential Time Decay of Solutions to Reaction-Cross-Diffusion Systems of Maxwell–Stefan Type

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.

Convergence of an implicit Euler Galerkin scheme for Poisson–Maxwell–Stefan systems

“…shows that the pressure vanishes, which is consistent with our choice of the driving force (see [2, (211)]). The driving force in [30, (7)] contains a non-vanishing pressure that is related to our expression for the total body force. The resulting driving force (19), however, is the same.…”

“…To obtain a positive definite diffusion matrix, we need to transform the system. With the so-called entropy variables (7) w…”

“…In [15,Theorem 9.7.4] and [18,Theorem 4.3], the large-time asymptotics for initial data close to equilibrium was analyzed. The convergence to equilibrium for any initial data was investigated in [6,21] without production terms and in [7] with production terms for reversible reactions. Salvarani and Soares proved a relaxation limit of the Maxwell-Stefan system to a system of linear heat equations [31].…”

“…Assumption (A4) is needed to derive the entropy production inequality (10). It is satisfied for reversible reactions; see [7,Lemma 6].…”

“…At time k = 0, we assume that ρ 0 i ≥ η > 0 in Ω. This allows us to define w 0 via definition (7). The condition can be removed by performing the limit η → 0 in the proof; see [6] for details.…”