An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems

¹ Mathematics Department, Université de Liége, Liége, Belgium.
² INRIA Paris MOKAPLAN, Paris, France.
³ McGill University, Montreal, Canada.
⁴ IECL Université de Lorraine, Nancy, France.
⁵ GFM Universidade de Lisboa, Lisboa, Portugal.

^* Corresponding author: leonard.monsaingeon@gmail.com

Received: 27 June 2017
Accepted: 21 December 2017

Abstract

In this paper, we show that unbalanced optimal transport provides a convenient framework to handle reaction and diffusion processes in a unified metric setting. We use a constructive method, alternating minimizing movements for the Wasserstein distance and for the Fisher-Rao distance, and prove existence of weak solutions for general scalar reaction-diffusion-advection equations. We extend the approach to systems of multiple interacting species, and also consider an application to a very degenerate diffusion problem involving a Gamma-limit. Moreover, some numerical simulations are included.