Calculus of Variations and Geometric Measure Theory

V. Agostiniani - R. Rossi

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

created by rossi on 25 Nov 2016
modified on 23 Nov 2017


Published Paper

Inserted: 25 nov 2016
Last Updated: 23 nov 2017

Journal: J. Differential Equations
Pages: 7815–7855
Year: 2017


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 nonconvex, 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.