Rigorous Numerics for the Cahn-Hilliard Equation on the Unit Square
Rigorous Numerics for the Cahn-Hilliard Equation on the Unit Square
复制标题
单位平方上 Cahn-Hilliard 方程的严格数值
DOI：
10.5209/rev_rema.2008.v21.n2.16380
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发表时间：
2008
影响因子：
0.8
通讯作者：
Thomas Wanner
中科院分区：
文献类型：
--
作者：
S. Maier;Ullrich Miller;K. Mischaikow;Thomas Wanner
在一维域上Cahn-Hilliard方程的平稳解集的结构是完全了解的，而在二维基域上只有部分结果可用。在本文中，我们证明了如何严格的计算技术可以被用来建立计算机辅助的存在证明的平衡cann - hilliard方程在单位平方。我们的方法基于Mischaikow和Zgliczy ' nski[22]的结果，并结合了严格的计算和Conley指数技术。我们能够建立平衡的分支，在更严格的条件下，甚至可以建立特定平衡解的局部唯一性。给出了几个分支的示例计算，说明了结果模式。
While the structure of the set of stationary solutions of the Cahn-Hilliard equation on one-dimensional domains is completely understood, only partial results are available for two-dimensional base domains. In this paper, we demonstrate how rigorous computational techniques can be employed to establish computerassisted existence proofs for equilibria of the Cahn-Hilliard equation on the unit square. Our method is based on results by Mischaikow and Zgliczy´nski [22], and combines rigorous computations with Conley index techniques. We are able to establish branches of equilibria and, under more restrictive conditions, even the local uniqueness of specific equilibrium solutions. Sample computations for several branches are presented, which illustrate the resulting patterns.