Existence Of Weak Solutions For A Phase-field Model Of Grain Boundary Motion

The regulation and control of the microstructure of materials is a key issue in guid-ing the optimal design and molding of materials.However,due to the complexity and diversity of the microstructure,it is difficult to achieve a profound understanding of the evo-lution mechanism and the formation law of the microstructure only by experimental means.Therefore,it is very important to establish a theoretical model to assist the analysis and simulation.Phase-field method is a non-equilibrium calculation method,which can be used to represent the transition state between two phases.As a method to simulate the evolution of microstructure,it has been widely used in the field of material functional design.The Alber-Zhu model in the case that the order parameter is conserved,proposed by Alber and Zhu Peicheng in 2008[7].This model is a phase-field model which describes the interface motion by interface diffusion in elastically deformed solids.For example,the basic phe-nomena occurring during this process,called sintering,are densification and grain growth.Since no atom exchange occurs at the interface,the volumes of the different regions sep-arated by the interface are conserved.Diffusion is driven only by a decrease in the bulk free.In this paper,our research will be based on the Alber-Zhu model which order pa-rameter is conserved.The evolution equation in this model is a fourth order,nonlinear degenerate partial differential equation of parabolic type.We ignore the elastic effect and reduce the original initial boundary value problem to a single equation of 1 dimensional situation,and prove the existence of the weak solution of the reduced initial boundary value problem of this model.Our proof of existence is based on the Galerkin method.In spite of the problem we consider is a single equation for order parameter without the elasticity sub-system,it,however,still keeps intrinsic difficulties.Since the equation considered is nonuniform,degenerate which the principle term with a non-smooth function of the gradient of unknown S.Hence,the estimate of the highest-order derivative obtained is dependent on|S_x|,this cause a lot of difficulties on the theoretical analysis and numerical simulation.