Variational convergence of gradient flows and rate-independent evolutions in metric spaces

TL;DR

This paper investigates the limiting behavior of gradient flows in metric spaces as dissipation potentials degenerate, establishing a variational framework for BV solutions and demonstrating their emergence from p-gradient flows as p approaches 1.

Contribution

It introduces a general variational definition of BV solutions in metric spaces and proves their stability under perturbations, linking p-gradient flows to rate-independent evolutions.

Findings

BV solutions characterize the limit of p-gradient flows as p approaches 1.

The variational BV framework captures different solution regimes in metric evolutions.

BV solutions naturally arise as limits of gradient flows with linear dissipation.

Abstract

We study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metric-dissipation potentials degenerate in the limit to a dissipation with linear growth. We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, and along the jump transitions. By using tools of metric analysis, BV functions and blow-up by time rescaling, we show that this variational notion is stable with respect to a wide class of perturbations involving energies, distances, and dissipation potentials. As a particular application, we show that BV solutions to rate-independent problems arise naturally as a limit of $p$-gradient flows, $p>1$, when the exponents $p$ converge to 1.