论文标题
正则化和分离，用于不断发展的表面Cahn-Hilliard方程
Regularisation and separation for evolving surface Cahn-Hilliard equations
论文作者
论文摘要
我们考虑在二维不断发展的闭合表面上嵌入$ \ Mathbb r^3 $以及相关的加权模型上的Cahn-Hilliard方程，具有恒定的迁移率和对数电势。前两位作者已经建立了给定时间间隔$ [0，t] $在给定时间间隔$ [0，t] $上相应初始值问题的弱解决方案的适当性。在这里，我们首先在有限的时间内证明了弱解决方案的某些正则化属性。然后，我们显示了两个问题的严格分离属性的有效性。这意味着从任何积极的时间开始，解决方案都远离纯阶段$ \ pm1 $。该属性在实现解决方案的高阶规则性方面起着至关重要的作用。同样，这是对标准双孔近似的严格验证。目前的结果是平面域中经典方程的众所周知的结果。
We consider the Cahn-Hilliard equation with constant mobility and logarithmic potential on a two-dimensional evolving closed surface embedded in $\mathbb R^3$, as well as a related weighted model. The well-posedness of weak solutions for the corresponding initial value problems on a given time interval $[0,T]$ have already been established by the first two authors. Here we first prove some regularisation properties of weak solutions in finite time. Then, we show the validity of the strict separation property for both the problems. This means that the solutions stay uniformly away from the pure phases $\pm1$ from any positive time on. This property plays an essential role to achieve higher-order regularity for the solutions. Also, it is a rigorous validation of the standard double-well approximation. The present results are a twofold extension of the well-known ones for the classical equation in planar domains.
