### Mathematics Colloquium

# Alexander Mielke

### (Weierstraß-Institut Berlin)

## "On a metric and geometric approach to reaction-diffusion systems as gradient systems"

** Date: ** |
Mon, December 1, 2014 |

** Time: ** |
17:15 |

** Place: ** |
Research II Lecture Hall |

**Abstract:** We discuss reaction-diffusion systems with reactions satisfying
mass-action kinetics and the detailed-balance condition. They
allow for a gradient structure where the dual dissipation potential
is the sum of a transport (or Kantorovich-Wasserstein) part for
diffusion and a reaction part. Similar structures are developed for
the temperature-dependent case when the internal energy is used as
variable.

The dissipation potential plays formally the role of an
infinite-dimensional Riemannian tensor, which induces a Riemannian
distance, called dissipation distance in this context.
In the pure diffusion case this gives the Kantorovich-Wasserstein
distance. We show that in a simple case including a reaction
the induced dissipation distance, called Hellinger-Kantorovich
distance, can be characterized explicitly.

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

*The colloquium is preceded by tea from 16:45 in the Resnikoff Mathematics Common Room, Research I, 127.*