The Łojasiewicz-Simon inequality related to grain boundary motion and its applications

In this paper, we study the Łojasiewicz-Simon gradient inequality for the mathematical model of grain boundary motion. We first derive a curve shortening equation with time-dependent mobility, which guarantees the energy dissipation law for the grain boundary energy, including the difference between orientations of the constituent grains as a state variable. Next, we discuss the Łojasiewicz-Simon gradient inequality for the grain boundary energy. Finally, we give applications of the inequality to the energy.