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

TL;DR

This paper proves that solutions to the 3D nonlocal Cahn-Hilliard equation with singular potential stay away from pure phases over time, extending previous 2D results and aiding in understanding solution regularity and convergence.

Contribution

It establishes the strict separation property in three dimensions for the first time, addressing a long-standing open problem and extending 2D results to higher dimensions.

Findings

Solutions remain uniformly away from pure phases after positive time

The strict separation property aids in proving higher-order regularity

Weak solutions converge to a single equilibrium

Abstract

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 $\pm1$ 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 two. 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.