Robust exponential attractors for Cahn‐Hilliard type equations with singular potentials
Robust exponential attractors for Cahn‐Hilliard type equations with singular potentials
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DOI：
10.1002/mma.464
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发表时间：
2004-03
影响因子：
2.9
通讯作者：
A. Miranville;S. Zelik
中科院分区：
文献类型：
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作者：
A. Miranville;S. Zelik
本文的目的是研究一类具有奇异(特别是对数)势的奇摄动Cahn-Hilliard方程的长时间行为。特别地，我们能够构造一个指数吸引子的连续族(当扰动参数变为0时)。此外，利用这些指数吸引子，我们能够证明有限维全局吸引子的存在，该吸引子对于序参数的空间平均的所有可能值吸引有界的初始数据集，从而改进了以前的结果，这些结果要求对空间域的大小有很强的限制，并且工作在给定了序参数的平均的空间上。最后，我们能够在一维和二维空间中将解与势的奇异值分开，这允许我们将问题归结为具有规则势的问题。不幸的是，对于三维空间中的未扰动问题，我们需要关于势的额外假设，这使得我们无法证明对数势的这样一个结果。版权所有©2004 John Wiley&Sons，Ltd.
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.