✦ LIBER ✦

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

✍ Scribed by Alain Miranville; Sergey Zelik

Publisher: John Wiley and Sons
Year: 2004
Tongue: English
Weight: 270 KB
Volume: 27
Category: Article
ISSN: 0170-4214
DOI: 10.1002/mma.464

No coin nor oath required. For personal study only.

✦ Synopsis

Abstract

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. Copyright © 2004 John Wiley & Sons, Ltd.

📜 SIMILAR VOLUMES

Exponential attractors for the Cahn–Hill

Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions

✍ A. Miranville; S. Zelik 📂 Article 📅 2005 🏛 John Wiley and Sons 🌐 English ⚖ 216 KB

We consider in this article the Cahn-Hilliard equation endowed with dynamic boundary conditions. By interpreting these boundary conditions as a parabolic equation on the boundary and by considering a regularized problem, we obtain, by the Leray-Schauder principle, the existence and uniqueness of sol

A Cahn–Hilliard type equation with perio

A Cahn–Hilliard type equation with periodic gradient-dependent potentials and sources

✍ Yinghua Li; Jingxue Yin; Chunhua Jin 📂 Article 📅 2009 🏛 John Wiley and Sons 🌐 English ⚖ 156 KB

This paper is concerned with the existence, uniqueness and attractability of time periodic solutions of a Cahn-Hilliard type equation with periodic gradient-dependent potentials and sources.

A “Sup + C Inf” inequality for Liouville

A “Sup + C Inf” inequality for Liouville-type equations with singular potentials

✍ Daniele Bartolucci 📂 Article 📅 2011 🏛 John Wiley and Sons 🌐 English ⚖ 142 KB

We generalize a Harnack-type inequality (I. Shafrir, C. R. Acad. Sci. Paris, 315 (1992), 159-164), for solutions of Liouville equations to the case where the weight function may admit zeroes or singularities of power-type |x| 2 α , with α ∈ (-1, 1).