Gradient flows: from theory to application

Abstract

This volume contains selected contributions related to the international workshop on Gradient flows: from theory to application, held at the International Centre for Mathematical Sciences in Edinburgh during 20-24 April 2015. Partial differential equations (PDE) have been used successfully to describe a variety of important phenomena in physics, engineering, life and social sciences. Many of these processes are driven by minimising energies with respect to certain costs, following the common rule in nature to be as efficient as possible. Hence a large class of nonlinear PDEs can be interpreted as gradient flows in certain metrics, in which the energy along solutions decreases as fast as possible. The choice of the energy as well as the dissipation mechanism allows for a variety of formulations. The heat equation for example can be interpreted as an L 2-gradient flow of the H 1-seminorm E = 1 2 ||u| 2 , but also as a Wasserstein gradient flow of the entropy E = u log u. Both interpretations have their merits, although the latter might be considered a more 'natural' measure to describe the state of a system. Wasserstein gradient flows have become a popular tool in PDE analysis, especially since the seminal work of Jordan, Kinderlehrer and Otto, cf. [21]. They demonstrated that solutions of the Fokker-Planck equation can be interpreted as a steepest descent of the entropy functional with respect to the Wasserstein metric. The connection between Wasserstein metrics and dynamic systems involving dissipation or diffusion initiated a lot of research on the analysis of gradient flows, see for example the monograph by Ambrosio, Gigli and Savaré [2]. It allowed for further developments in the field of optimal transportation problems, see [36, 34] and set the basis for the development of numerical schemes. Also research on the connection of gradient flow structures to the underlying microscopic particle systems by studying the large deviation behaviour [1] or the GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) framework to describe, in one I Article published online by EDP Sciences and available at