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

TL;DR

This paper investigates the energetic and dynamical properties of nonlocal reaction-diffusion systems modeling the evolution of embedded curves, establishing complexity bounds, well-posedness, equilibrium regularity, and numerical dynamics insights.

Contribution

It introduces a comprehensive analysis of nonlocal models for embedded curves, linking energetic measures to crossing numbers and providing new dynamical and regularity results.

Findings

Established global well-posedness of the PDEs.

Derived complexity bounds related to crossing numbers.

Numerical exploration of global dynamics.

Abstract

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.