WIAS Preprint No. 1593, (2011)

Passing to the limit in a Wasserstein gradient flow: From diffusion to reaction



Authors

  • Arnrich, Steffen
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Peletier, Mark A.
  • Savaré, Giuseppe
  • Veneroni, Marco

2010 Mathematics Subject Classification

  • 35K67 35B25 35B27 49S99 35K20 35K57 60F10 70F40 70G75 37L05

Keywords

  • Fokker-Planck equation, transport equation, metric evolution, Gamma convergence

Abstract

We study a singular-limit problem arising in the modelling of chemical reactions. At finite $e>0$, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by $1/e$, and in the limit $eto0$, 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, Savaré, and Veneroni, em 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 propety of being a emphcurve 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 $e$-problem converge to a solution of the limiting problem.

Appeared in

  • Calc. Var. Partial Differ. Equ., 44 (2012) pp. 419--454.

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