Abstract
The convergence to equilibrium for renormalised solutions to nonlinear reaction–diffusion systems is studied. The considered reaction–diffusion systems arise from chemical reaction networks with mass action kinetics and satisfy the complex balanced condition. By applying the so-called entropy method, we show that if the system does not have boundary equilibria, i.e. equilibrium states lying on the boundary of {mathbb {R}}_+^N, then any renormalised solution converges exponentially to the complex balanced equilibrium with a rate, which can be computed explicitly up to a finite-dimensional inequality. This inequality is proven via a contradiction argument and thus not explicitly. An explicit method of proof, however, is provided for a specific application modelling a reversible enzyme reaction by exploiting the specific structure of the conservation laws. Our approach is also useful to study the trend to equilibrium for systems possessing boundary equilibria. More precisely, to show the convergence to equilibrium for systems with boundary equilibria, we establish a sufficient condition in terms of a modified finite-dimensional inequality along trajectories of the system. By assuming this condition, which roughly means that the system produces too much entropy to stay close to a boundary equilibrium for infinite time, the entropy method shows exponential convergence to equilibrium for renormalised solutions to complex balanced systems with boundary equilibria.
Highlights
The convergence to equilibrium for renormalised solutions to nonlinear reaction–diffusion systems is studied
In the first main results of this paper, we prove for general, complex balanced reaction–diffusion systems (1)–(3) without boundary equilibria, that any so-called renormalised solution converges exponentially to the complex balanced equilibrium with a rate which can be explicitly estimated in terms of the systems’ parameters and a constant obtained from a finite-dimensional inequality with mass conservation constraints
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 [45]
Summary
ZAMP proved to be very useful in studying convergence to equilibrium for many PDE systems, in particular reaction–diffusion systems which feature a suitable dissipative structure. |C| is the set of chemical complexes, which are either reactants and/or products of a chemical reaction and y = (yj)Nj=1 denotes a vector of stoichiometric coefficients for the substances S1, . Many chemical reaction networks exhibit mass conservation laws. To state the main results of this paper, we need the following definitions concerning equilibria of chemical reaction networks. A chemical reaction network is called complex balanced if for each strictly positive mass vector M ∈ Rm >0 it possesses a strictly positive (i.e. not a boundary) complex balanced equilibrium. The complex balanced condition was considered by Boltzmann [5] under the name semi-detailed balanced condition or cyclic balanced condition and was systematically used by Horn, Jackson and Feinberg in the seventies for chemical reaction network theory, see, e.g. The complex balanced condition was considered by Boltzmann [5] under the name semi-detailed balanced condition or cyclic balanced condition and was systematically used by Horn, Jackson and Feinberg in the seventies for chemical reaction network theory, see, e.g. [21,40,42]
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