Work

Analysis validation of a new continuum model for the evolution of grain boundaries in polycrystalline materials

Year: 2026

Type: preprint

Abstract: To describe the evolution of grain boundaries based on the underlying microscopic mechanisms of line defects (disconnections) and the integrated effects of a diverse range of thermodynamic driving forces, Zhang, et al. in 2017 formulated a continuum equation. We prove the global-in-time existence, uniqueness, and regularity of the weak solution to an initial-boundary value problem for this model. The existence, uniqueness of the stationary solution are also established. Finally, we investigate the large-time behavior of the weak solution of the evolution problem, and show that the solution converges to the stationary solution in a suitable sense. Numerical simulations are carried out to validate the analysis results. The main difficulties in the proof of main theorems are due to a non-local term with singularity, a non-smooth coefficient of the highest derivative associated with the gradient of the unknown, and the special form of free energy which is not uniformly bounded from below. The key ingredients in the proof are the energy method, an estimate for a singular integral of the Hilbert type, Fourier transform of Hilbert transformations, and an estimate with a weight, for time-derivative of the unknown.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2608.28954

Authors: Zhu, Peicheng, Qin, Xiaoxue, Xiang, Yang

Institutions: Shanghai University, Hong Kong University of Science and Technology

Language: English

Topic: Solidification and crystal growth phenomena

Subfield: Materials Chemistry

Field: Materials Science

Domain: Physical Sciences

Open Access status: green

Funders: National Natural Science Foundation of China, Science and Technology Commission of Shanghai Municipality, Tongji University

Awards: G5199732171, G6849697376

OpenAlex ID: W7204877555

Cites:

Cited by:

Related to: 10