# Reaction-diffusion via optimal transport on the cone space.

### (Weierstrass Institute Berlin)

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

**Date:** Oct 28, 2014,
**time:** 16:00

**Abstract.** It was recently shown by Mielke that a wide class of reaction-diﬀusion systems

can be formulated in a natural way via gradient structures for the relative entropy or free energy. The metric gradient of the driving functional is determined via a state-dependent Onsager operator containing a diﬀusion part of Wasserstein type and an additional reaction term. With the Onsager operator we can associate a dissipation distance in the sense of Benamou-Brenier by inﬁmizing the total dissipation over all connecting curves. The question of attainment of this inﬁmum, which is the same as the existence of geodesic curves, is an open question in most cases. In this talk we present a full characterization of a dissipation distance induced by a reaction-diﬀusion Onsager operator that depends linearly on the state. In particular, we show that the distance is given by the Kantorovich Wasserstein distance on an extended space, which is given by the cone construction over the underlying domain. This is joint work with Alexander Mielke and Giuseppe Savar.