Longtime behavior of nonlocal Cahn-Hilliard equations

Abstract

Here we consider the nonlocal Cahn-Hilliard equation with constant mobilityin a bounded domain. We prove that the associated dynamical system has anexponential attractor, provided that the potential is regular. In order todo that a crucial step is showing the eventual boundedness of the orderparameter uniformly with respect to the initial datum. This is obtainedthrough an Alikakos-Moser type argument. We establish a similar result forthe viscous nonlocal Cahn-Hilliard equation with singular (e.g.,logarithmic) potential. In this case the validity of the so-calledseparation property is crucial. We also discuss the convergence of asolution to a single stationary state. The separation property in thenonviscous case is known to hold when the mobility degenerates at the purephases in a proper way and the potential is of logarithmic type. Thus, theexistence of an exponential attractor can be proven in this case as well.