*Proceedings*

**Inserted:** 26 dec 2017

**Journal:** Mathematical Analysis of Continuum Mechanics and Industrial Applications II. Proceedings of the International Conference CoMFoS16.

**Volume:** 30

**Pages:** 139--151

**Year:** 2018

**Doi:** 10.1007/978-981-10-6283-4

**Abstract:**

Free energies with many small wiggles, arising from small scale micro-structural changes, appear often in phase transformations, protein folding and friction problems. In this paper we investigate gradient flows with energies $E_\varepsilon$ given by the superposition of a convex functional and fast small oscillations. We apply the time-discrete minimising movement scheme to capture the effect of the local minimizers of $E_\varepsilon$ in the limit equation as $\varepsilon$ tends to zero. We perform a mutiscale analysis according to the mutual vanishing behaviour of the spatial parameter $\varepsilon$ and the time step $\tau$ and we highlight three different regimes $\tau\ll\varepsilon$, $\varepsilon\ll\tau$ (Braides 2014) and $\tau\sim\varepsilon$ (Ansini-Braides-Zimmer 2016) . We discuss for each case the existence of a pinning threshold and we derive the limit equation describing the motion.