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

Alexander Mielke

(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)