On Uniform Decay of the Entropy for Reaction–Diffusion Systems

Abstract

This work provides entropy decay estimates for classes of nonlinear reactiondiffusion systems modeling reversible chemical reactions under the detailed-balance condition. We obtain explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the Log-Sobolev estimate and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: (i) vanishing diffusion constants in some chemical components and (ii) usage of different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions.

Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction–diffusion systems

“…2S 1 ⇌ S 2 ) the convexification argument seems not to allow for explicit estimates on the rate of convergence. The results of [44] were extended in [15] to complex balanced systems thanks to the derivation of the entropy dissipation (8).…”

“…• In a recent work [26], we proposed a constructive approach to show exponential convergence to equilibrium for general detailed balanced reaction-diffusion systems, which allows to obtain explicit bounds on the rates of convergence in contrast to the convexification argument of [44]. The applicability of the constructive approach was demonstrated for two typical example systems: i) a reversible reaction of arbitrary many chemical substances α 1 S 1 + · · · + α I S I ⇌ β 1 B 1 + · · · + β J B J ( * ) and ii) a reversible enzyme reaction S 1 + S 2 ⇌ S 3 ⇌ S 4 + S 5 .…”

Quenching for a parabolic system with general singular terms

“…For more research on quenching phenomena for parabolic system with Neumann boundary conditions, we refer readers to [3,5,15,17,19], and some advances in quenching phenomena those days, we refer readers to [1,2,[8][9][10][11] and references therein. In addition, for some research on decay, see [6,12,13] and corresponding references therein.…”

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

“…In comparison to previous results for reaction-diffusion systems, e.g. [19,35], the difference here is that the reactions are defined in terms of molar fractions, while the conservation laws are written in terms of concentrations. This difference causes the main difficulty in proving (13), except in very special cases, e.g., when all molar masses are equal (in this case, the molar fraction and concentration are proportional) or in case of equal homogeneities (see Section 3.4).…”