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

摘要

We propose the new notion of Visco-Energetic solutions to rate-independentsystems $(X,\mathcal E,\mathsf d)$ driven by a time dependent energy $\mathcalE$ and a dissipation quasi-distance $\mathsf d$ in a general metric-topologicalspace $X$. As for the classic Energetic approach, solutions can be obtained bysolving a modified time Incremental Minimization Scheme, where at each step thedissipation (quasi-)distance $\mathsf d$ is incremented by a viscous correction$\delta$ (e.g.~proportional to the square of the distance $\mathsf d$), whichpenalizes far distance jumps by inducing a localized version of the stabilitycondition. We prove a general convergence result and a typical characterization byStability and Energy Balance in a setting comparable to the standard energeticone, thus capable to cover a wide range of applications. The new refined EnergyBalance condition compensates the localized stability and provides a carefuldescription of the jump behavior: at every jump the solution follows an optimaltransition, which resembles in a suitable variational sense the discrete schemethat has been implemented for the whole construction.