Calculus of Variations and Geometric Measure Theory

L. Minotti - G. Savaré

Viscous corrections of the Time Incremental Minimization Scheme and Visco-Energetic Solutions to Rate-Independent Evolution Problems

created by savare on 11 Jun 2016


Submitted Paper

Inserted: 11 jun 2016
Last Updated: 11 jun 2016

Pages: 1-60
Year: 2016


We propose the new notion of Visco-Energetic solutions to rate-independent systems (X,E,d) driven by a time dependent energy E and a dissipation quasi-distance d in a general metric-topological space X.

As for the classic Energetic approach, solutions can be obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation (quasi-) distance d is incremented by a viscous correction (e.g. proportional to the square of the distance d which penalizes far distance jumps by inducing a localized version of the stability condition. We prove a general convergence result and a typical characterization by Stability and Energy Balance in a setting comparable to the standard energetic one, thus capable to cover a wide range of applications.

The new refined Energy Balance condition compensates the localized stability and provides a careful description of the jump behavior: at every jump the solution follows an optimal transition, which resembles in a suitable variational sense the discrete scheme that has been implemented for the whole construction.