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
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.
Keywords
three-dimensional nonlocal Cahn–Hilliard equation, singular
potential, strict separation property, regularization of
weak solutions, convergence to equilibrium
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