The {\L}ojasiewicz-Simon inequality related to grain boundary motion and its applications

TL;DR

This paper investigates the {}ojasiewicz-Simon gradient inequality within a mathematical model of grain boundary motion, deriving a curve shortening equation with energy dissipation and applying the inequality to analyze grain boundary energy.

Contribution

It introduces a new curve shortening equation with time-dependent mobility and establishes the {}ojasiewicz-Simon inequality for grain boundary energy, advancing understanding of grain boundary dynamics.

Findings

Derived a curve shortening equation with energy dissipation law.

Established the {}ojasiewicz-Simon gradient inequality for grain boundary energy.

Applied the inequality to analyze energy behavior in grain boundary motion.

Abstract

In this paper, we study the {\L}ojasiewicz-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 {\L}ojasiewicz-Simon gradient inequality for the grain boundary energy. Finally, we give applications of the inequality to the energy.