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Discrete-to-continuum limit for nonlinear reaction-diffusion systems via EDP convergence for gradient systems
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We investigate the convergence of spatial discretizations for reaction-diffusion systems with mass-action law satisfying a detailed balance condition. Considering systems on the d-dimensional torus, we construct appropriate space-discrete processes and show convergence not only on the level of solutions, but also on the level of the gradient systems governing the evolutions. As an important step, we prove chain rule inequalities for the reaction-diffusion systems as well as their discretizations, featuring a non-convex dissipation functional. The convergence is then obtained with variational methods by building on the recently introduced notion of gradient systems in continuity equation format.
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Cited by 2 Pith papers
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Derivation of the fourth-order DLSS equation with nonlinear mobility via chemical reactions
A lattice chemical reaction network converges, by energy-dissipation-principle convergence, to the generalized fourth-order DLSS equation ∂tρ = −∂xx(ρ^α ∂xx log ρ) for every α>0.
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From diffusion to transmission via EDP-convergence: a paradigmatic multiscale limit
EDP-convergence of Otto-type gradient structures for nonlinear diffusion yields a unique effective membrane kinetic relation that can be exponential even when the microscopic dissipation is quadratic.
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