一类序参数守恒的相场模型的弱解存在性

Existence of Weak Solutions for a Class of Phase-Field Models with Conservation of Order Parameter

摘要: 本文将基于序参数守恒情形下的 Alber-Zhu 模型展开研究, 该模型序参数的演化由退化的四阶抛物型方程控制, 可用于描述微观尺度下可弹性变形固体中晶界的运动, 这个过程的一个典型例子是烧结, 在烧结过程中会引发致密化及晶粒生长等基本微观组织结构变化. 由于在界面上没有发生原子交换, 因此被界面分开的不同区域的体积是守恒的, 扩散过程仅由体自由能的降低来驱动. 本文考虑简化后的模型, 即忽略弹性效应, 并将方程磨光为一维空间的单个非退化方程, 其中该序参数的边界条件为 Neumann 边界条件和无流条件相结合. 运用 Galerkin 方法证明了该模型在约化后的初边值问题弱解的存在性. 尽管所考虑的问题是序参数的单一方程, 但由于未知数 S 梯度项的存在，它仍存在固有的困难.

Abstract: In this paper, our research will be based on the Alber-Zhu model which order parameter is conserved. The evolution equation in this model is a fourth order, nonlinear degenerate partial differential equation of parabolic type. This model is a phasefield model which describes the interface motion by interface diffusion in elastically deformed solids. For example, the basic phenomena 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 separated by the interface are conserved. We ignore the elastic effect and reduce the original initial boundary value problem to a single non-degenerate equation of1dimensional situation, and prove the existence of the weak solution of the reduced initial boundary value problem of this model, where the boundary conditions of the order parameters are a combination of the Neumann boundary conditions and the no-flow conditions. The Galerkin method is used to prove the existence of weak solutions for the reduced initial boundary value problem of this model. Although the problem considered is a single equation of order parameters, it is inherently difficult due to the presence which is the gradient of unknown function S. Hence, we need to mollify the gradient term, which causes a lot of difficulties on the theoretical analysis and numerical simulation.