Weak solutions to an initial-boundary value problem for a continuum equation of motion of grain boundaries

TL;DR

This paper establishes the existence, uniqueness, and asymptotic analysis of weak solutions for a continuum model of grain boundary motion, addressing degeneracy and non-local singularities.

Contribution

It proves global weak solutions for the model with positive parameters and analyzes their behavior as the parameter approaches zero, handling complex degeneracies.

Findings

Proved global existence and uniqueness of weak solutions.

Analyzed asymptotic behavior as the disconnection density parameter approaches zero.

Addressed mathematical challenges due to degeneracy and non-local singularities.

Abstract

We investigate an initial-(periodic-)boundary value problem for a continuum equation, which is a model for motion of grain boundaries based on the underlying microscopic mechanisms of line defects (disconnections) and integrated the effects of a diverse range of thermodynamic driving forces. We first prove the global-in-time existence and uniqueness of weak solution to this initial-boundary value problem in the case with positive equilibrium disconnection density parameter B, and then investigate the asymptotic behavior of the solutions as B goes to zero. The main difficulties in the proof of main theorems are due to the degeneracy of B=0, a non-local term with singularity, and a non-smooth coefficient of the highest derivative associated with the gradient of the unknown. The key ingredients in the proof are the energy method, an estimate for a singular integral of the Hilbert type, and a compactness lemma.