基于Sobolev嵌入的模式形成模型的平衡验证
Equilibrium Validation in Models for Pattern Formation Based on Sobolev Embeddings
摘要：
在研究偏微分方程的平衡解时，有许多平衡点，人们不可能希望找到它们全部。因此，人们通常集中精力寻找平衡解的各个分支。一方面，对这些分支的严格的理论理解是理想的，但通常不容易处理。另一方面，数值分叉搜索是有用的，但不能保证给出准确的结构，因为它们可能会错过分支的一部分或找到不存在的虚假分支。在最近的一系列论文中，我们提出了第三种选择。也就是说，我们已经开发了一种计算机辅助证明的方法来证明平衡解的分支的存在性和隔离性。在目前的文件中，我们扩展这些技术的大田-川崎模型的动力学的二嵌段共聚物的尺寸一，二，三，通过给出一个详细的描述的分析基础的方法。虽然本文集中应用的方法Ohta-Kawasaki模型，功能分析的方法和技术可以推广到其他抛物型偏微分方程。
英文摘要：
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.
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