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

  • *Corresponding author: Yang Xiang

    *Corresponding author: Yang Xiang

The work of P. C. Zhu was supported in part by Science and Technology Commission of Shanghai Municipality (Grant No. 20JC1413600). The work of Y. Xiang was supported by the Hong Kong Research Grants Council General Research Fund 16302818 and Collaborative Research Fund C1005-19G, and the Project of Hetao Shenzhen-HKUST Innovation Cooperation Zone HZQB-KCZYB-2020083

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  • 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.

    Mathematics Subject Classification: Primary: 35K65, 35R11; Secondary: 35Q74.


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