The 3D strict separation property for the nonlocal Cahn-Hilliard equation with singular potential

We consider the nonlocal Cahn-Hilliard equation with singular (logarithmic) potential and constant mobility in three-dimensional bounded domains and we establish the validity of the instantaneous strict separation property. This means that any weak solution, which is not a pure phase initially, stays uniformly away from the pure phases +/- 1 from any positive time on. This work extends the result in dimension two for the same equation and gives a positive answer to the long-standing open problem of the validity of the strict separation property in dimensions higher than 2. In conclusion, we show how this property plays an essential role to achieve higher-order regularity for the solutions and to prove that any weak solution converges to a single equilibrium.