Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems

Our focus are energy estimates for discretized reaction-diffusion systems for a finite number of species. We introduce a discretization scheme (Voronoi finite volume in space and fully implicit in time) which has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For a class of Voronoi finite volume meshes we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the discrete free energy to its equilibrium value with a unified rate of decay for this class of discretizations. The fundamental idea is an estimate of the free energy by the dissipation rate which is proved indirectly by taking into account sequences of Voronoi finite volume meshes. Essential ingredient in that proof is a discrete SobolevPoincaré inequality. 1 Model equations, notation, and assumptions Let Ω ⊂ RN be a bounded domain, Γ := ∂Ω. We consider m species Xi with initial densities Ui. These species undergo chemical reactions and underly diffusion processes. We assume Boltzmann statistics giving the relation between the densities ui of the species Xi and the corresponding chemical potentials vi, ui = uiei , i = 1, . . . ,m, (1.1) where the reference densities ui may depend on the spatial position and express the possible heterogeneity of the system under consideration. For the fluxes ji of the species Xi we make the ansatz ji = −μiui∇vi, i = 1, . . . ,m, (1.2) with mobility coefficients μi. To describe chemical reactions we assume that R ⊂ Z+×Z+ is a finite subset. A pair (α, β) ∈ R represents the vectors of stoichiometric coefficients of reversible reactions, usually written in the form α1X1 + · · ·+ αmXm β1X1 + · · ·+ βmXm. According to the mass action law, the net rate of this pair of reactions is of the form kαβ(a − aβ), where kαβ is a reaction coefficient, ai := exp(vi) is the chemical activity of Xi, and aα := ∏m i=1 a αi i . The net production rate of species Xi corresponding to the reaction rates for all reactions taking place is Ri := ∑ (α,β)∈R kαβ(a − a)(βi − αi). (1.3) The m continuity equation can be written as follows: ∂ui ∂t +∇ · ji = Ri in R+ × Ω, ν · ji = 0 on R+ × Γ, ui(0) = Ui in Ω, i = 1, . . . ,m. (1.4)