Global attractors for Cahn–Hilliard equations with nonconstant mobility

Abstract

We address, in a three-dimensional spatial setting, both the viscous and the standard Cahn–Hilliard equation with a nonconstant mobility coefficient. As shown in Barrett and Blowey (1999 Math. Comput. 68 487–517), one cannot expect the uniqueness of the solution to the related initial and boundary value problems. Nevertheless, referring to Ball's theory of generalized semiflows, we are able to prove the existence of compact quasi-invariant global attractors for the associated dynamical processes settled in the natural ‘finite energy’ space. A key point in the proof is a careful use of the energy equality, combined with the derivation of a ‘local compactness’ estimate for systems with supercritical nonlinearities, which may have an independent interest. Under growth restrictions on the configuration potential, we also show the existence of a compact global attractor for the semiflow generated by the (weaker) solutions to the nonviscous equation characterized by a ‘finite entropy’ condition.