Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks

The quantitative convergence to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks with mass action kinetics is studied by using the so-called entropy method. In the first part of the paper, by deriving explicitly the entropy dissipation, we show that for complex balanced systems without boundary equilibria, each trajectory converges exponentially fast to the unique complex balance equilibrium. Moreover, a constructive proof is proposed to explicitly estimate the rate of convergence in the special case of a cyclic reaction. In the second part of the paper, complex balanced systems with boundary equilibria are considered. We investigate two specific cases featuring two and three chemical substances respectively. In these cases, the boundary equilibria are shown to be unstable in some sense, so that exponential convergence to the unique strictly positive equilibrium can also be proven.Comment: 33 page

The entropy method for reaction-diffusion systems without detailed balance: first order chemical reaction networks

In this paper, the applicability of the entropy method for the trend towards equilibrium for reaction-diffusion systems arising from first order chemical reaction networks is studied. In particular, we present a suitable entropy structure for weakly reversible reaction networks without detail balance condition. We show by deriving an entropy-entropy dissipation estimate that for any weakly reversible network each solution trajectory converges exponentially fast to the unique positive equilibrium with computable rates. This convergence is shown to be true even in cases when the diffusion coefficients all but one species are zero. For non-weakly reversible networks consisting of source, transmission and target components, it is shown that species belonging to a source or transmission component decay to zero exponentially fast while species belonging to a target component converge to the corresponding positive equilibria, which are determined by the dynamics of the target component and the mass injected from other components. The results of this work, in some sense, complete the picture of trend to equilibrium for first order chemical reaction networks.Comment: 26 page

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

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, 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.Comment: 25 page

Trend to equilibrium of renormalized solutions to reaction-cross-diffusion systems

The convergence to equilibrium of renormalized solutions to reaction-cross-diffusion systems in a bounded domain under no-flux boundary conditions is studied. The reactions model complex balanced chemical reaction networks coming from mass-action kinetics and thus do not obey any growth condition, while the diffusion matrix is of cross-diffusion type and hence nondiagonal and neither symmetric nor positive semi-definite, but the system admits a formal gradient-flow or entropy structure. The diffusion term generalizes the population model of Shigesada, Kawasaki and Teramoto to an arbitrary number of species. By showing that any renormalized solution satisfies the conservation of masses and a weak entropy-entropy production inequality, it can be proved under the assumption of no boundary equilibria that {\it all} renormalized solutions converge exponentially to the complex balanced equilibrium with a rate which is explicit up to a finite dimensional inequality.Comment: The convergence of $I_6(M)$ is correcte

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

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.Comment: New version; Proof of mass conservation for renormalised solutions is include

Close-to-equilibrium regularity for reaction-diffusion systems

The close-to-equilibrium regularity of solutions to a class of reaction-diffusion systems is investigated. The considered systems typically arise from chemical reaction networks and satisfy a complex balanced condition. Under some restrictions on spatial dimensions ($d\leq 4$) and order of nonlinearities ($\mu = 1 + 4/d$), we show that if the initial data is close to a complex balanced equilibrium in $L^2$-norm, then classical solutions are shown global and converging exponentially to equilibrium in $L^{\infty}$-norm. Possible extensions to higher dimensions and order of nonlinearities are also discussed. The results of this paper improve the recent work [M.J. C\'aceres and J.A. Ca\~nizo, Nonlinear Analysis: TMA 159 (2017): 62-84]

Uniform boundedness for reaction-diffusion systems with mass dissipation

We study the global existence and uniform-in-time bounds of classical solutions in all dimensions to reaction-diffusion systems dissipating mass. By utilizing the duality method and the regularization of the heat operator, we show that if the diffusion coefficients are close to each other, or if the diffusion coefficients are large enough compared to initial data, then the local classical solution exists globally and is bounded uniformly in time. Applications of the results include the validity of the Global Attractor Conjecture for complex balanced reaction systems with large diffusion

Chemical reaction-diffusion networks; convergence of the method of lines

We show that solutions of the chemical reaction-diffusion system associated to $A+B\rightleftharpoons C$ in one spatial dimension can be approximated in $L^2$ on any finite time interval by solutions of a space discretized ODE system which models the corresponding chemical reaction system replicated in the discretization subdomains where the concentrations are assumed spatially constant. Same-species reactions through the virtual boundaries of adjacent subdomains lead to diffusion in the vanishing limit. We show convergence of our numerical scheme by way of a consistency estimate, with features generalizable to reaction networks other than the one considered here, and to multiple space dimensions. In particular, the connection with the class of complex-balanced systems is briefly discussed here, and will be considered in future work

Generalized potential games

In this paper, we introduce a notion of generalized potential games that is inspired by a newly developed theory on generalized gradient flows. More precisely, a game is called generalized potential if the simultaneous gradient of the loss functions is a nonlinear function of the gradient of a potential function. Applications include a class of games arising from chemical reaction networks with detailed balance condition. For this class of games, we prove an explicit exponential convergence to equilibrium for evolution of a single reversible reaction. Moreover, numerical investigations are performed to calculate the equilibrium state of some reversible chemical reactions which give rise to generalized potential games.Comment: 23 pages, 6 figures. Comments are welcom

Mathematical Analysis of Chemical Reaction Systems

The use of mathematical methods for the analysis of chemical reaction systems has a very long history, and involves many types of models: deterministic versus stochastic, continuous versus discrete, and homogeneous versus spatially distributed. Here we focus on mathematical models based on deterministic mass-action kinetics. These models are systems of coupled nonlinear differential equations on the positive orthant. We explain how mathematical properties of the solutions of mass-action systems are strongly related to key properties of the networks of chemical reactions that generate them, such as specific versions of reversibility and feedback interactions.Comment: 17 pages, 7 figures, revie