On the separation property and the global attractor for the nonlocal Cahn-Hilliard equation in three dimensions

In this note, we consider the nonlocal Cahn-Hilliard equation with constant mobility and singular potential in three dimensional bounded and smooth domains. Given any global solution (whose existence and uniqueness are already known), we prove the so-called {\it instantaneous} and {\it uniform} separation property: any global solution with initial finite energy is globally confined (in the $L^{\mathrm{\infty }}$ metric) in the interval $[-1+\delta ,1-\delta ]$ on the time interval $[\tau ,\mathrm{\infty })$ for any $\tau >0$, where $\delta$ only depends on the norms of the initial datum, $\tau$ and the parameters of the system. We then exploit such result to improve the regularity of the global attractor for the dynamical system associated to the problem.