Gradient flows with wiggly potential: a variational approach to the dynamics

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_epsilon 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 ofE_epsilon in the limit equation as epsilon tends to zero. We perform a mutiscale analysis according to the mutual vanishing behaviour of the spatial parameter epsilon and the time step tau and we highlight three different regimes. We discuss for each case the existence of a pinning threshold and we derive the limit equation describing the motion.