Weak solutions to an initial-boundary value problem for a continuum equation of motion of grain boundaries
Peicheng Zhu, Lei Yu, Yang Xiang · Discrete and Continuous Dynamical Systems - B · 2022
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.