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

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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.