Robust exponential attractors for Cahn-Hilliard type equations with singular potentials. (English) Zbl 1050.35113

Miranville, Alain; Zelik, Sergey

Math. Methods Appl. Sci. 27, No. 5, 545-582 (2004).

The authors are interested in the study of the long time behaviour of the Cahn-Hilliard equation with singular potentials (and, in particular, with logarithmic potentials). They present an abstract result on the construction of a continuous (with respect to perturbations) family of exponential attractors. Note that here, they do not have the continuity up to time shifts as it was in previous studies. The authors are able to construct the existence of the finite-dimensional global attractor without any assumption on the size of the spatial domain. Having uniform (with respect to the perturbation parameter) \(L^\infty\)-bounds on the solutions, they manage to separate the solutions from the singular points of the potential and thus to reduce the problem to one with a regular potential. Unfortunately, in 3D cases the authors need additional assumptions on the potential and are thus not able to consider a logarithmic potential. However, they are able to prove such a result for logarithmic potentials in one and two space dimensions.

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

35Q72	Other PDE from mechanics (MSC2000)
35B40	Asymptotic behavior of solutions to PDEs
35B41	Attractors
76D27	Other free boundary flows; Hele-Shaw flows

Keywords:

Cahn-Hilliard equation; singular potential; logarithmic potential; exponential attractor; global attractor