Robust exponential attractors for Cahn‐Hilliard type equations with singular potentials

作者：Alain Miranville; Sergey Zelik

来源：Mathematical Methods in the Applied Sciences, 2004, 27(5): 545-582.

DOI：10.1002/mma.464

• Cahn‐Hilliard equation
• singular potential
• logarithmic potential
• exponential attractor
• global attractor

Our aim in this article is to study the long time behaviour of a family of singularly perturbed Cahn‐Hilliard equations with singular (and, in particular, logarithmic) potentials. In particular, we are able to construct a continuous family of exponential attractors (as the perturbation parameter goes to 0). Furthermore, using these exponential attractors, we are able to prove the existence of the finite dimensional global attractor which attracts the bounded sets of initial data for all the possible values of the spatial average of the order parameter, hence improving previous results which required strong restrictions on the size of the spatial domain and to work on spaces on which the average of the order parameter is prescribed. Finally, we are able, in one and two space dimensions, to separate the solutions from the singular values of the potential, which allows us to reduce the problem to one with a regular potential. Unfortunately, for the unperturbed problem in three space dimensions, we need additional assumptions on the potential, which prevents us from proving such a result for logarithmic potentials.