Generation of balanced viscosity solutions to rate-independent systems via variational convergence

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 $$(\Psi _n)_{n\in {\mathbb {N}}}$$ with superlinear growth at infinity and suitably converging to a 1-positively homogeneous potential $$\Psi _0$$, and a smooth energy functional $${\mathcal {E}}$$, we provide sufficient conditions on them ensuring that the solutions of the associated (generalized) gradient systems$$(\Psi _n,{\mathcal {E}})$$ converge as $$n\rightarrow \infty $$ to a Balanced Viscosity solution of the rate-independent system driven by $$\Psi _0$$ and $${\mathcal {E}}$$. In specific cases, we also obtain results on the reverse approximation of balanced viscosity solutions by means of solutions to gradient systems. Our approach is based on the key observation that solutions to gradient systems/that Balanced Viscosity solutions to rate-independent systems can be characterized as (null-)minimizers of suitable trajectory functionals, for which we indeed prove Mosco-convergence. As particular cases, our analysis encompasses both the vanishing-viscosity approximation of rate-independent systems from Mielke et al. (ESAIM Control Optim Calc Var 18(1):36–80, 2012, J Eur Math Soc 18(9):2107–2165, 2016), and their stochastic derivation developed in Bonaschi and Peletier (Contin Mech Thermodyn 28:1191–1219, 2016).