TY - JOUR
T1 - Passing to the limit in a Wasserstein gradient flow : from diffusion to reaction
AU - Arnrich, S.
AU - Mielke, A.
AU - Peletier, M.A.
AU - Savaré, G.
AU - Veneroni, M.
PY - 2012
Y1 - 2012
N2 - 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 e --> 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 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 e-problem converge to a solution of the limiting problem.
AB - 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 e --> 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 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 e-problem converge to a solution of the limiting problem.
U2 - 10.1007/s00526-011-0440-9
DO - 10.1007/s00526-011-0440-9
M3 - Article
VL - 44
SP - 419
EP - 454
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
SN - 0944-2669
IS - 3-4
ER -