Global attractors for gradient flows in metric spaces

TL;DR

This paper develops a comprehensive long-time analysis for gradient flows in metric spaces, introducing new solution concepts and proving the existence of global attractors across various mathematical contexts.

Contribution

It introduces generalized solutions for metric gradient flows, extends attractor theory to non-unique evolutions, and applies results to diverse spaces including Banach and Wasserstein spaces.

Findings

Existence of global attractors for energy and generalized solutions

Applicability to Banach and Wasserstein spaces

Analysis of phase-change evolutions driven by mean curvature

Abstract

We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope, we introduce the latter to include limits of time-incremental approximations constructed via the Minimizing Movements approach. For both notions of solutions we prove the existence of the global attractor. Since the evolutionary problems we consider may lack uniqueness, we rely on the theory of generalized semiflows introduced by J.M. Ball. The notions of generalized and energy solutions are quite flexible and can be used to address gradient flows in a variety of contexts, ranging from Banach spaces to Wasserstein spaces of probability measures. We present applications of our abstract results by proving the existence of the global attractor for the energy solutions both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature.