# A reaction-diffusion equation as a Hellinger-Kantorovich gradient flow

### (Weierstrass Institute for Applied Analysis and Stochastics and Humboldt University, Berlin)

**Event:** ERC Workshop on Optimal Transportation and Applications

**Date:** Oct 30, 2014,
**time:** 09:00

**Abstract.** Large classes of reaction-diffusion systems with reactions satisfying mass-

action kinetics and the detailed-balance condition can be written as a formal gradient

system with respect to the relative entropy. The the dual dissipation potential is the sum of a transport part for diffusion and a reaction part. We discuss the mathematical steps needed to turn the formal theory into a rigorous metric gradient system.

Motivated by scalar reaction-diffusion equations we construct the so-called Hellinger-

Kantorovich distance on the set of all non-negative measures. This distance can be

obtained (i) via transport and growth, (ii) by the inf-convolution of the Kantorovich-

Wasserstein distance and the Hellinger distance, and (iii) by minimizing a logarithmic-

entropy transport problem. We provide examples of entropies and such that induced

reaction-diffusion equation is a lambda-convex gradient ﬂow.

This is joint work with Matthias Liero (WIAS Berlin) and Giuseppe Savare (Pavia)