A variational approach to the quasistatic limit of viscous dynamic evolutions in finite dimension

TL;DR

This paper investigates the limit behavior of second-order systems with inertia and viscosity in finite dimensions, showing convergence to stationary points and describing the evolution at jump times using a variational approach.

Contribution

It introduces a variational method to analyze the quasistatic limit of viscous dynamic evolutions with nonconvex potentials, including characterization of jump discontinuities.

Findings

Solutions converge to stationary points of the energy

Behavior at jump times involves heteroclinic solutions

Provides a framework for nonconvex potential analysis

Abstract

In this paper we study the vanishing inertia and viscosity limit of a second order system set in an Euclidean space, driven by a possibly nonconvex time-dependent potential satisfying very general assumptions. By means of a variational approach, we show that the solutions of the singularly perturbed problem converge to a curve of stationary points of the energy and characterize the behavior of the limit evolution at jump times. At those times, the left and right limits of the evolution are connected by a finite number of heteroclinic solutions to the unscaled equation.