Gradient flows and Evolution Variational Inequalities in metric spaces. I: structural properties

TL;DR

This paper analyzes the structural properties of gradient flows in metric spaces characterized by Evolution Variational Inequalities, establishing key equivalences, convergence results, and stability properties without relying on geometric curvature assumptions.

Contribution

It introduces new structural results for EVI gradient flows, proves their equivalence with De Giorgi's characterization, and establishes convergence of the JKO scheme with explicit error estimates.

Findings

Proves the equivalence between EVI gradient flows and curves of maximal slope.

Establishes convergence of the JKO scheme with order 1/2 error estimate.

Introduces a relaxed Ekeland-based Minimizing Movement algorithm with uniform convergence.

Abstract

This is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space $(X,\mathsf{d})$ that can be characterized by Evolution Variational Inequalities. We present new results concerning the structural properties of solutions to the $\mathrm{EVI}$ formulation, such as contraction, regularity, asymptotic expansion, precise energy identity, stability, asymptotic behaviour and their link with the geodesic convexity of the driving functional. Under the crucial assumption of the existence of an $\mathrm{EVI}$ gradient flow, we will also prove two main results: the equivalence with the De Giorgi variational characterization of curves of maximal slope and the convergence of the Minimizing Movement-JKO scheme to the $\mathrm{EVI}$ gradient flow, with an explicit and uniform error estimate of order $1/2$ with respect to the step size, independent of any geometric hypothesis (such as upper or lower curvature bounds) on $\mathsf{d}$. In order to avoid any compactness assumption, we will also introduce a suitable relaxation of the Minimizing Movement algorithm obtained by the Ekeland variational principle, and we will prove its uniform convergence as well.