Explicit exponential convergence to equilibrium for nonlinear reaction–diffusion systems with detailed balance condition

Abstract

The convergence to equilibrium of mass action reaction-diffusion systems arising from networks of chemical reactions is studied. The considered reaction networks are assumed to satisfy the detailed balance condition and have no boundary equilibria. We propose a general approach based on the so-called entropy method, which is able to quantify with explicitly computable rates the decay of an entropy functional in terms of an entropy entropy-dissipation inequality based on the totality of the conservation laws of the system.As a consequence follows convergence to the unique detailed balance equilibrium with explicitly computable convergence rates. The general approach is further detailed for two important example systems: a single reversible reaction involving an arbitrary number of chemical substances and a chain of two reversible reactions arising from enzyme reactions.

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

“…• 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 .…”

Global existence for reaction–diffusion systems with dissipation of mass and quadratic growth

“…(e) A related, active topic is the study, by means of entropy methods, of the stabilization as t → ∞ of (classical or weak) global solutions of systems with dissipation of mass. For this, we refer to, e.g., [5,1,13,7,8,9,17,16].…”

Global classical solutions to quadratic systems with mass control in arbitrary dimensions

“…This is true, for instance for complex balanced chemical reaction networks, see e.g. [FT17,FT17a,DFT17]. Indeed, for such systems, interpolation of slowly (i.e.…”

“…It was proved in several works, e.g. [DF08,FT17,FT17a,PSZ16], that any solution (renormalised, weak or strong) to (50) with initial mass M converges exponentially in L 1 -norm to the equilibrium u 8 defined above, i.e.…”