Gradient flows in asymmetric metric spaces

We present a theory of gradient flows in asymmetric metric spaces presupposing no linear or differential structure. This extends the classical theory in inner product spaces and the theory in metric spaces recently proposed by Ambrosio, Gigli and Savare. We begin by extending to the asymmetric situation notions that arise in analysis on metric spaces (such as absolute continuity, metric derivaties, local slopes and curves of maximal slope). In particular, curves of maximal slope are a natural extension of gradient flows to asymmetric spaces. Then, we study variational approximations (time discretizations, error estimates and convergence of solutions of the discrete problem) of curves of maximal slope. Finally, we specialize our results to the context of asymmetric extensions of Wasserstein spaces of probability measures. We end by outlining possible applications to solid mechanics. The motivating observation is that the time evolution of probability measures that arise in the study of materials with microstructure, phase transformations, plasticity or damage is better described on asymmetrically metrizable (but not metrizable) topological spaces than on metrizable topological spaces.

This is joint work with Marc Oliver Rieger, University of Zurich and Johannes Zimmer, University of Bath.