Calculus of Variations and Geometric Measure Theory

N. Ansini

Gradient Flows with Wiggly Potential: A Variational Approach to Dynamics

created by ansini on 26 Dec 2017



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


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.