Generation via variational convergence of Balanced Viscosity solutions to rate-independent systems

TL;DR

This paper explores how Balanced Viscosity solutions for rate-independent systems can be derived as limits of gradient systems with vanishing viscosity, providing a unified framework for understanding their origin.

Contribution

It establishes conditions under which gradient systems with dissipation potentials Gamma-converge to Balanced Viscosity solutions, linking vanishing viscosity and stochastic derivations.

Findings

Derived sufficient conditions for Gamma-convergence to Balanced Viscosity solutions.

Unified framework for vanishing-viscosity and stochastic derivations.

Applicable to finite-dimensional rate-independent systems.

Abstract

In this paper we investigate the origin of the Balanced Viscosity solution concept for rate-independent evolution in the setting of a finite-dimensional space. Namely, given a family of dissipation potentials $(\Psi_n)_n$ with superlinear growth at infinity and a smooth energy functional $\mathcal{E}$, we enucleate sufficient conditions on them ensuring that the associated gradient systems $(\Psi_n,\mathcal{E})$ Evolutionary Gamma-converge to a limiting rate-independent system, understood in the sense of Balanced Viscosity solutions. In particular, our analysis encompasses both the vanishing-viscosity approximation of rate-independent systems and their stochastic derivation.