On the separation property and the global attractor for the nonlocal Cahn-Hilliard equation in three dimensions (Q6570505)

The author considers a nonlocal Cahn-Hilliard equation with constant mobility and singular potential in a domain bounded in three dimensions. What is proven here is the following version of the strict separation property (see [\textit{C. G. Gal} et al., J. Differ. Equations 263, No. 9, 5253--5297 (2017; Zbl 1400.35178)] for the two-dimensional case): For any \(\tau>0\) there exists \(\delta\in (0,1)\) such that any global solution with finite initial energy is globally bounded, in \(L^\infty\), to the interval \([-1+\delta, 1-\delta]\), for all \(t\ge\tau\). Here, \(\delta>0\) depends on the initial data, \(\tau\), and other (physical) parameters. The separation property guarantees that values of the order-parameter do not reach singular values of the potential. Finally, the separation property, which is related to the gradient flow structure of the Cahn-Hilliard model, is shown to provide further regularity for the global attractor associated with the dynamical system.