Discrete-to-continuum limit for nonlinear reaction-diffusion systems via EDP convergence for gradient systems
Authors
- Heinze, Georg
- Mielke, Alexander
ORCID: 0000-0002-4583-3888 - Stephan, Artur
ORCID: 0000-0001-9871-3946
2020 Mathematics Subject Classification
- 35A15 35K57 35A35 47J35 65M08
Keywords
- Variational methods applied to PDEs, evolutionary Gamma-convergence, reaction-diffusion equations, dissipative evolution equations, nonlinear evolution equations, theoretical approximation to solutions, finite volume methods
DOI
Abstract
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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