Relating a rate-independent system and a gradient system for the case of one-homogeneous potentials

TL;DR

This paper establishes a precise mathematical relationship between gradient flows and rate-independent systems for one-homogeneous energies, demonstrating their equivalence through a solution-dependent time reparametrization and analyzing jump behaviors.

Contribution

It introduces an exact relation between gradient flows and energetic solutions for one-homogeneous potentials, including rigorous analysis of jumps and a new existence and uniqueness result.

Findings

Established a solution-dependent reparametrization linking the two systems.

Proved equivalence of gradient and rate-independent flows for specific examples.

Provided a rigorous mathematical framework for analyzing jumps in energetic solutions.

Abstract

We consider a non-negative and one-homogeneous energy functional $\mathcal J$ on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-inpendent system given in terms of the time-dependent functional $\mathcal E(t,u)=t \mathcal J(u)$ and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutins of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.