Long time behavior for a curvature flow of networks related to grain bundary motion with the effect of lattice misorientations

TL;DR

This paper investigates the long-term behavior of a curvature flow model for grain boundary networks that incorporates lattice misorientations, ensuring energy dissipation and analyzing stability under the Herring condition.

Contribution

It introduces a curvature flow model with time-dependent mobilities accounting for lattice misorientations and studies its solvability and asymptotic behavior.

Findings

Established existence and uniqueness of solutions.

Analyzed the asymptotic stability of the flow.

Ensured energy dissipation law holds.

Abstract

The mathematical model of grain boundary motion, including lattice misorientations' effect, is considered. When time-dependent lattice misorientations are state variables of the surface tension of the grain boundary, to ensure the energy dissipation law, one can obtain a curvature flow of networks with time-dependent mobilities. This paper studies the solvability and long-time asymptotic behavior of the curvature flow subjected to the Herring condition which ensures that the constituent grain boundary surface tensions are balanced at the triple junction.