Perturbed minimizing movements of families of functionals

TL;DR

This paper investigates how perturbations in energy and dissipation terms affect the convergence of minimizing-movement schemes for gradient flows, providing conditions for limit independence and applications to homogenization and crystalline flows.

Contribution

It introduces a framework for analyzing perturbed minimizing-movement schemes with two small parameters and characterizes conditions for their limits to be well-defined and independent of perturbations.

Findings

Identifies regimes where limits depend on perturbations

Provides conditions ensuring limit independence from perturbations

Applies the theory to homogenization and crystalline flow models

Abstract

We consider the well-known minimizing-movement approach to the definition of a solution of gradient-flow type equations by means of an implicit Euler scheme depending on an energy and a dissipation term. We perturb the energy by considering a ($\Gamma$-converging) sequence and the dissipation by varying multiplicative terms. The scheme depends on two small parameters $\varepsilon$ and $\tau$, governing energy and time scales, respectively. We characterize the extreme cases when $\varepsilon/\tau$ and $\tau/\varepsilon$ converges to $0$ sufficiently fast, and exhibit a sufficient condition that guarantees that the limit is indeed independent of $\varepsilon$ and $\tau$. We give examples showing that this in general is not the case, and apply this approach to study some discrete approximations, the homogenization of wiggly energies and geometric crystalline flows obtained as limits of ferromagnetic energies.