Homogenization of Wasserstein gradient flows

Abstract

We prove the convergence of a Wasserstein gradient flow of a free energy in inhomogeneous media. Both the energy and media can depend on the spatial variable in a fast oscillatory manner. In particular, we show that the gradient-flow structure is preserved in the limit which is expressed in terms of an effective energy and Wasserstein metric. The gradient flow and its limiting behavior are analyzed through an energy dissipation inequality (EDI). The result is consistent with asymptotic analysis in the realm of homogenization. However, we note that the effective metric is in general different from that obtained from the Gromov-Hausdorff convergence of metric spaces. We apply our framework to a linear Fokker-Planck equation but we believe the approach is robust enough to be applicable in a broader context.

31 pages

References in corpus (10)

• Multilayer Networks
• Multiscale mobility networks and the large scale spreading of infectious diseases
• Large deviation properties of weakly interacting processes via weak convergence methods
• Large Deviations for Multiscale Diffusions via Weak Convergence Methods
• Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction
• On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions
• Modeling of chemical reaction systems with detailed balance using gradient structures
• Exploring families of energy-dissipation landscapes via tilting -- three types of EDP convergence
• The Scharfetter--Gummel scheme for aggregation-diffusion equations
• Variational convergence of the Scharfetter-Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit