# An unbalanced Optimal Transport splitting scheme for general advection-reaction-diffusion problems

## TL;DR

This paper introduces a novel unbalanced optimal transport splitting scheme that unifies reaction, diffusion, and advection processes in scalar and multi-species systems, with proven existence of solutions and numerical demonstrations.

## Contribution

It develops a new constructive splitting scheme using Wasserstein and Fisher-Rao distances for reaction-diffusion-advection equations, extending to systems and degenerate diffusion cases.

## Key findings

- Existence of weak solutions for general scalar reaction-diffusion-advection equations.
- Extension of the scheme to systems of multiple interacting species.
- Numerical simulations demonstrating the method's effectiveness.

## Abstract

In this paper, we show that unbalanced optimal transport provides a convenient framework to handle reaction and diffusion processes in a unified metric framework. 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.