In this talk, we discuss a singular limit of the Kobayashi-Warren-Carter system which models the evolutions of structures of grains. We characterize a sharp interface limit of the Kobayashi-Warren-Carter (KWC) energy. We also consider a sharp interface limit of its gradient flow. Under a suitable scaling, we obtain a gradient flow involving fractional time derivative. We further study a minimizer of a fidelity term plus total variation type energy obtained by minimizing the singular limit of the KWC energy with respect to the order parameter. It turns out in one-dimensional setting that all minimizer must be piecewise
constant. This talk is based on several joint works with A. Kubo (Hokkaido U.), H. Kuroda (Hokkaido U.), J. Okamoto (Kyoto U.), K. Sakakibara (Kanazawa U./RIKEN) and M. Uesaka (DetaLabs Inc.).