具有动力边界条件的Cahn-Hilliard方程及其与流体耦合系统的数学分析

复杂多相体系的界面现象广泛存在于自然界、日常生活和现代技术应用之中。组分间自由界面的行为决定了多相体系的特性，因此研究界面现象及其演化机制具有重要的理论意义和应用价值。本项目拟对源于两相混合体系中界面问题的两类偏微分方程开展研究：（A）具有新动力边界条件的Cahn-Hilliard方程，它描述了带边界效应的相分离过程且具有同时满足质量守恒、能量耗散以及力的平衡三类物理约束的重要特点；（B）关于两相不可压流体移动接触线问题的Cahn-Hilliard-Navier-Stokes方程组，它满足关于速度场的广义Navier边界条件和关于相场函数的动力边界条件，是描述动态润湿现象的重要数学模型。本项目研究相应初边值问题的适定性、解的正则性以及长时间行为等偏微分方程领域的基本问题。描述固体边界上界面现象的偏微分方程及其边界条件具有复杂的结构，对其开展研究需要数学方法上的创新，具有重要的意义和价值。

The interfacial phenomena in multiphase systems widely exist in the nature, our daily life and modern engineering applications. The behavior of interfaces between different phases determines important features of the system. As a consequence, the study on the interfacial phenomena and the mechanism of its evolution is not only important from the theoretic point of view, but also valuable for real applications. The aim of this project is to investigate two classes of partial differential equations that arise from interfacial problems in certain two-phase systems, including: (A) the Cahn-Hilliard equation subject to a class of new dynamic boundary condition describing the phase separation process with nontrivial boundary interaction, which fulfills three important physical properties such as mass conservation, energy dissipation and force balance; (B) the Cahn-Hilliard-Navier-Stokes system for incompressible two-phase flows with moving contact lines, subject to the generalized Navier boundary condition for the velocity as well as the dynamic boundary condition for the phase-field function, which serves as an important model for the dynamic wetting phenomena. We study well-posedness of the corresponding initial boundary value problems as well as the regularity and long-time behavior of solutions. These are all fundamental problems in the field of partial differential equations. For those partial differential equations that describe interfacial phenomena on the solid boundary, their boundary conditions present specific and complicated structures, which require new methods and techniques to handle the associated mathematical difficulties. Our research will make an important contribution to the development of the theory of nonlinear evolution equations. Besides, it will be helpful for the related numerical analysis and yields a better understanding of the complex interfacial phenomena.