Singular vanishing-viscosity limits of gradient flows: the finite-dimensional case

TL;DR

This paper investigates the limit behavior of gradient flows with vanishing viscosity in finite-dimensional spaces, showing convergence to stationary points and characterizing jumps via Dissipative Viscosity solutions.

Contribution

It introduces a variational framework for analyzing singular vanishing-viscosity limits and characterizes the solutions' behavior at jump points, extending the understanding of gradient flow limits.

Findings

Solutions converge to stationary points as viscosity vanishes

Dissipative Viscosity solutions describe jump behavior

Conditions for Balanced Viscosity solutions are provided

Abstract

In this note we study the singular vanishing-viscosity limit of a gradient flow set in a finite-dimensional Hilbert space and driven by a smooth, but possibly non convex, time-dependent energy functional. We resort to ideas and techniques from the variational approach to gradient flows and rate-independent evolution to show that, under suitable assumptions, the solutions to the singularly perturbed problem converge to a curve of stationary points of the energy, whose behavior at jump points is characterized in terms of the notion of Dissipative Viscosity solution. We also provide sufficient conditions under which Dissipative Viscosity solutions enjoy better properties, which turn them into Balanced Viscosity solutions. Finally, we discuss the generic character of our assumptions.