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Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction. (English) Zbl 1270.35055
Summary: 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 \({\varepsilon \to 0}\), 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 [the third author et al., SIAM J. Math. Anal. 42, No. 4, 1805–1825 (2010; Zbl 1221.35045)], 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.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K67 Singular parabolic equations
35B25 Singular perturbations in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K57 Reaction-diffusion equations
60F10 Large deviations
70F40 Problems involving a system of particles with friction
70G75 Variational methods for problems in mechanics
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
35Q84 Fokker-Planck equations
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