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

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, combinatori...

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Bibliographic Details
Published inarXiv.org
Main Authors von Brecht, James H, Blair, Ryan
Format Paper Journal Article
LanguageEnglish
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
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1711.08104