Regularisation and separation for evolving surface Cahn-Hilliard equations

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.