An Energetic Variational Approach for the Cahn-Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis
An Energetic Variational Approach for the Cahn-Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis
复制标题
具有动态边界条件的 Cahn-Hilliard 方程的能量变分方法：模型推导和数学分析
DOI：
10.1007/s00205-019-01356-x
复制
发表时间：
2019
影响因子：
2.5
通讯作者：
Wu Hao
中科院分区：
文献类型：
--
作者：
Liu Chun;Wu Hao
Cahn-Hilliard方程是描述二元混合物相分离过程的基本模型。近年来，为了考虑材料与固体壁面之间可能存在的短程相互作用，人们提出了几种动态边界条件。本文的第一个目的是在相当一般的情况下，对Cahn-Hilliard方程提出一类新的动态边界条件。这一推导是基于能量变分方法，该方法结合了最小作用量原理和Onsager最大能量耗散原理。我们模型的一个特点是它自然地满足三个重要的物理约束：质量守恒、能量耗散和力平衡关系。接下来，我们对所得到的偏微分方程组进行全面的分析。在适当的假设下，我们证明了有或无表面扩散的初边值问题整体弱/强解的存在唯一性。进一步，我们建立了渐近极限的唯一性，并刻画了系统局部能量极小值的稳定性。
The Cahn–Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in order to account for possible short-range interactions of the material with the solid wall. Our first aim in this paper is to propose a new class of dynamic boundary conditions for the Cahn–Hilliard equation in a rather general setting. The derivation is based on an energetic variational approach that combines the least action principle and Onsager’s principle of maximum energy dissipation. One feature of our model is that it naturally fulfills three important physical constraints: conservation of mass, dissipation of energy and force balance relations. Next, we provide a comprehensive analysis of the resulting system of partial differential equations. Under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary value problem with or without surface diffusion. Furthermore, we establish the uniqueness of the asymptotic limit asand characterize the stability of local energy minimizers for the system.