The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part I: Mathematical analysis
The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part I: Mathematical analysis
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DOI：
10.1017/s095679250000053x
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发表时间：
1991-09
影响因子：
1.9
通讯作者：
J. F. Blowey;C. M. Elliott
中科院分区：
文献类型：
--
作者：
J. F. Blowey;C. M. Elliott
本文对Cahn-Hilliard方程进行了数学分析，其中的自由能为无限壁的双势垒势函数。证明了具有Lyapunov泛函的问题的弱提法的存在唯一性。给出了弱形式的正则性结果，并考虑了时间无穷大时的渐近行为。对相关的平稳问题进行了研究，证明了非平凡平稳解的存在性，并进一步证明了任意平稳解的正则性。构造了一维和二维的定常解，导出了一维定常解个数的计算公式。因此，很自然地将其作为唯象参数λ→0来研究其渐近行为，其主要结果是两相之间的界面具有最小面积。
A mathematical analysis is carried out for the Cahn–Hilliard equation where the free energy takes the form of a double well potential function with infinite walls. Existence and uniqueness are proved for a weak formulation of the problem which possesses a Lyapunov functional. Regularity results are presented for the weak formulation, and consideration is given to the asymptotic behaviour as the time becomes infinite. An investigation of the associated stationary problem is undertaken proving the existence of a nontrivial stationary solution and further regularity results for any stationary solution. Stationary solutions are constructed in one and two dimensions; a formula for the number of stationary solutions in one dimension is derived. It is then natural to study the asymptotic behaviour as the phenomenological parameter λ→0, the main result being that the interface between the two phases has minimal area.