Abstract
We study a singular-limit problem arising in the modelling of chemical reactions. At finite \({\varepsilon\, > \,0,}\) the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by \({1 / \varepsilon,}\) and in the limit \({\varepsilon\to0,}\) the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier et al. (SIAM J Math Anal, 42(4):1805–1825, 2010), using the linear structure of the equation. In this study we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the \({\varepsilon}\) -problem converge to a solution of the limiting problem.
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Acknowledgements
A. Mielke was partially supported by the European Research Council via “ERC-2010-AdG 267802” (AnaMultiScale). The research of M. A. Peletier has received funding from the Initial Training Network “FIRST” of the Seventh Framework Programme of the European Community (grant agreement number 238702). G. Savaré has been partly supported by a grant from MIUR for the PRIN08-project Optimal transport theory, geometric and functional inequalities and applications.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Arnrich, S., Mielke, A., Peletier, M.A. et al. Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction. Calc. Var. 44, 419–454 (2012). https://doi.org/10.1007/s00526-011-0440-9
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DOI: https://doi.org/10.1007/s00526-011-0440-9