论文标题
基于Sobolev嵌入的模型形成模型中的平衡验证
Equilibrium Validation in Models for Pattern Formation Based on Sobolev Embeddings
论文作者
论文摘要
在针对部分微分方程的平衡解决方案的研究中，有太多的平衡，以至于人们不能希望找到所有的平衡。因此，通常集中于寻找平衡溶液的单个分支。一方面，对这些分支的严格理论理解是理想的，但通常不可行。另一方面，数值分叉搜索是有用的，但不能保证具有准确的结构，因为它们可能会错过一部分分支或找到不存在的虚假分支。在最近的一系列论文中，我们的目标是第三种选择。也就是说，我们开发了一种计算机辅助证明的方法，以证明平衡解决方案的分支的存在和隔离。在当前的论文中，我们通过详细描述该方法的分析基础，将这些技术扩展到二嵌段共聚物动力学的OHTA-KAWASAKI模型。尽管本文集中于将方法应用于Ohta-Kawasaki模型，但功能分析方法和技术可以推广到其他抛物线偏微分方程。
In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual branches of equilibrium solutions. On the one hand, a rigorous theoretical understanding of these branches is ideal but not generally tractable. On the other hand, numerical bifurcation searches are useful but not guaranteed to give an accurate structure, in that they could miss a portion of a branch or find a spurious branch where none exists. In a series of recent papers, we have aimed for a third option. Namely, we have developed a method of computer-assisted proofs to prove both existence and isolation of branches of equilibrium solutions. In the current paper, we extend these techniques to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three, by giving a detailed description of the analytical underpinnings of the method. Although the paper concentrates on applying the method to the Ohta-Kawasaki model, the functional analytic approach and techniques can be generalized to other parabolic partial differential equations.
