Dynamics of Embedded Curves by Doubly-Nonlocal Reaction-Diffusion Systems

• Published in: arXiv.org
• Main Authors: von Brecht, James H, Blair, Ryan
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 22.11.2017
• Subjects: Combinatorial analysis; Complexity; Economic models; Embedded systems; Mathematical models; Mathematics - Analysis of PDEs; Mathematics - Geometric Topology; Partial differential equations; Well posed problems
• DOI: 10.48550/arxiv.1711.08104

We study a class of nonlocal, energy-driven dynamical models that govern the motion of closed, embedded curves from both an energetic and dynamical perspective. Our energetic results provide a variety of ways to understand physically motivated energetic models in terms of more classical, combinatorial measures of complexity for embedded curves. This line of investigation culminates in a family of complexity bounds that relate a rather broad class of models to a generalized, or weighted, variant of the crossing number. Our dynamic results include global well-posedness of the associated partial differential equations, regularity of equilibria for these flows as well as a more detailed investigation of dynamics near such equilibria. Finally, we explore a few global dynamical properties of these models numerically.