Analysis of mathematical model of grain boundary motions in materials and evolution of mathematical theory

在本研究项目中，我们考虑材料中晶界运动的数学模型，这是一个Kobayashi-Warren-Carter型。然后，我们证明了解的渐近性，平稳解的存在性和稳定性，以及吸引子的存在性和表征。同时，我们证明了原始的小林-沃伦-卡特晶界运动模型的可解性。此外，我们还证明了Allen-Cahn方程系统和Kobayashi-Warren-Carter型晶界运动模型解的存在性。研究了以总变分函数为界面能的相场系统的最优控制问题。然后，我们证明了使非线性非光滑代价函数最小的最优控制的存在性。并给出了最优对的必要条件。

In this research project, we consider mathematical models of grain boundary motions in materials, which is a Kobayashi-Warren-Carter type. Then, we showed the asymptotic behavior of solutions, the existence and stability of stationary solutions and the existence and characterization of attractors. Also, we proved the solvability of the original Kobayashi-Warren-Carter model of grain boundary motions. Moreover, we showed the existence of solutions to a system of Allen-Cahn equation and grain boundary motion model of Kobayashi-Warren-Carter type. Also, we studied optimal control problems for phase field system with total variation functional as the interfacial energy. Then, we proved the existence of an optimal control that minimizes the nonlinear and nonsmooth cost functional. Moreover, we showed the necessary condition of the optimal pair.