Energy diffusion and evolution equations

25.01.2017, 15:00  –  Haus 9, Raum 2.22
Institutskolloquium

Ralph Chill (Universität Dresden), Alexander Mielke (WIAS Berlin)

• 15:00 Ralph Chill: Dissipative evolution equations: recent progress on their qualitative behaviour
• 16:00 Coffee break
• 16:30 Alexander Mielke: A geometric approach to reaction-diffusion systems  

Abstract:

Dissipative evolution equations: recent progress on their qualitative behaviour  (Ralph Chill - Universität Dresden)

Dissipation of an energy is  a typical feature of many physical systems and gradient systems are the prototypes of dissipative evolution equations. These are evolution equations driven by an energy function which dissipates (decreases) along solutions. A possible mathematical question is how to translate properties of the underlying energy function into qualitative properties of solutions of the respective gradient system. Examples of such properties we have in mind are  positivity of solutions or comparison principles, regularity, or the asymptotic behaviour of solutions. In this talk we give a survey on some of the possible translations and their practical consequences.

A geometric approach to reaction-diffusion systems  (Alexander Mielke - WIAS Berlin)  

We introduce the concept of gradient systems as a special class of ordinary or partial differential equations for the time evolution of a system. We also highlight  the thermodynamical foundations starting from Lars Onsager's work in 1931, The evolution is given the gradient of the free energy with respect to a Riemannian metric describing the dissipative effects.  We show that certain chemical reactions systems can be written as such a gradient flow   with respect to the relative entropy as the physically relevant free energy. For spatially extended system one can add diffusion as an additional dissipative effect leading to a generalized type of Riemannian structure following the seminal work of Felix Otto in 2000.  For a specific simple example we show that the Riemannian tensor generates as new distance between arbitrary measures, that can be viewed as a reaction-diffusion distance and which we call Hellinger-Kantorovich distance.  This geometric point of view provides new existence and uniqueness results for the associated partial differential equation.  This is joint work with Matthias Liero (WIAS) and Giuseppe Savare (Pavia).