Multiscale analysis of singularly perturbed finite dimensional gradient flows: the minimizing movement approach

TL;DR

This paper analyzes a discrete minimization scheme for singularly perturbed gradient flows, establishing convergence to different types of solutions depending on the scaling between viscosity and time step.

Contribution

It provides a rigorous convergence analysis of the scheme under various scalings, including the derivation of Balanced Viscosity solutions and characterization of limit evolutions.

Findings

Convergence to Balanced Viscosity solutions when viscosity dominates

Characterization of limit evolution at finite scale ratios

Optimization of interfacial energy at jump times

Abstract

We perform a convergence analysis of a discrete-in-time minimization scheme approximating a finite dimensional singularly perturbed gradient flow. We allow for different scalings between the viscosity parameter $\varepsilon$ and the time scale $\tau$. When the ratio $\frac{\varepsilon}{\tau}$ diverges, we rigorously prove the convergence of this scheme to a (discontinuous) Balanced Viscosity solution of the quasistatic evolution problem obtained as formal limit, when $\varepsilon\to 0$, of the gradient flow. We also characterize the limit evolution corresponding to an asymptotically finite ratio between the scales, which is of a different kind. In this case, a discrete interfacial energy is optimized at jump times.