Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction

TL;DR

This paper demonstrates a novel Wasserstein gradient-flow approach to analyze the singular limit of a reaction-diffusion system, showing convergence to a simpler ODE system without relying on linearity or second-order structure.

Contribution

The paper re-proves the singular limit result using Wasserstein gradient-flow techniques, avoiding linearity and second-order assumptions of previous methods.

Findings

Proves convergence of rescaled solutions to a limit system.

Establishes Gamma-convergence of the action functional.

Identifies the limiting differential equations for the system.

Abstract

We study a singular-limit problem arising in the modelling of chemical reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 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 Peletier, Savar\'e, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear structure of the equation. In this paper 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 {\epsilon}-problem converge to a solution of the limiting problem.