Calculus of Variations and Geometric Measure Theory

G. Scilla - F. Solombrino

Multiscale analysis of singularly perturbed finite dimensional gradient flows: the minimizing movement approach

created by scilla on 17 Dec 2017
modified on 05 Oct 2018


Published Paper

Inserted: 17 dec 2017
Last Updated: 5 oct 2018

Journal: Nonlinearity
Volume: 31
Number: 11
Pages: 5036-5074
Year: 2018
Doi: 10.1088/1361-6544/aad6ac

ArXiv: 1712.06182 PDF


We perform a convergence analysis of a discrete-in-time minimization scheme approximating a finite dimensional singularly perturbed gradient flow. We allow for different scalings between the viscosity parameter $\varepsilon$ and the time scale $\tau$. When the ratio $\frac{\varepsilon}{\tau}$ diverges, we rigorously prove the convergence of this scheme to a (discontinuous) Balanced Viscosity solution of the quasistatic evolution problem obtained as formal limit, when $\varepsilon\to 0$, of the gradient flow. We also characterize the limit evolution corresponding to an asymptotically finite ratio between the scales, which is of a different kind. In this case, a discrete interfacial energy is optimized at jump times.

Keywords: singular perturbations, Gradient Flow, Variational methods, rate-independent systems, minimizing movement, Balanced Viscosity solutions, crease energy