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

Masashi Mizuno, Ayumi Sakiyama, Keisuke Takasao

TL;DR

This work analyzes the Łojasiewicz-Simon gradient inequality for a one-dimensional grain boundary model with misorientation. By embedding the grain boundary energy into Yagi's abstract gradient framework and constructing suitable function spaces, the authors prove a local gradient inequality near equilibria and characterize the critical manifold. The LS inequality is then used to establish long-time convergence to equilibrium for the evolution system without requiring convexity of the energy density σ, and it yields a quantitative grain-length bound in terms of curvature and misorientation near equilibrium. These results extend previous convexity-restricted analyses to periodic, nonconvex energy densities relevant to grain boundary dynamics and provide a framework for analyzing algebraic decay in complex gradient-flow systems.

Abstract

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.

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

TL;DR

This work analyzes the Łojasiewicz-Simon gradient inequality for a one-dimensional grain boundary model with misorientation. By embedding the grain boundary energy into Yagi's abstract gradient framework and constructing suitable function spaces, the authors prove a local gradient inequality near equilibria and characterize the critical manifold. The LS inequality is then used to establish long-time convergence to equilibrium for the evolution system without requiring convexity of the energy density σ, and it yields a quantitative grain-length bound in terms of curvature and misorientation near equilibrium. These results extend previous convexity-restricted analyses to periodic, nonconvex energy densities relevant to grain boundary dynamics and provide a framework for analyzing algebraic decay in complex gradient-flow systems.

Abstract

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.
Paper Structure (11 sections, 22 theorems, 195 equations, 1 figure)

This paper contains 11 sections, 22 theorems, 195 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Ł ojasiewicz-Simon inequality
  3. Abstract theory of the Ł ojasiewicz-Simon inequality
  4. Function spaces and the Ł ojasiewicz-Simon inequality
  5. Auxiliary lemma
  6. Proof of the Ł ojasiewicz-Simon inequality
  7. Properties of the Gâteaux derivative of $\dot{E}$
  8. Properties of Banach space $Y$
  9. Applications
  10. Long-time asymptotic behavior of the evolution problem
  11. A grain length estimate by the grain boundary energy

Key Result

Theorem 2.1

Let $X$ be a real Hilbert space. Let $\Phi:X\rightarrow\mathbb{R}$ be a continuously differentiable function with the derivative $\dot{\Phi}:X\rightarrow X$, and let $\bar{u}$ be its critical point. Assume that $u\mapsto\dot{\Phi}(u)$ is Gâteaux differentiable at $\bar{u}$ with derivative $L=D\dot{\ Define the critical manifold $S$ as and assume that $\Phi$ is analytic on $S$. Then, in a neighbor

Figures (1)

  • Figure 1: The left figure is a schematic of two grains (Grain 1, Grain 2) and a single grain boundary. The right figure is the one with $\frac{\pi}{2}$ rotation for the crystal lattice of Grain 1. As can be seen, the structure of the grain boundary does not change by $\frac{\pi}{2}$ rotation, so the grain boundary energy density $\sigma$ should be periodic in misorientation.

Theorems & Definitions (44)

  • Theorem 2.1: Theorem 3.1 in MR4274456
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Lemma 2.5: Proposition 1.3 in MR4274456
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • ...and 34 more