Evolution of multiple closed knotted curves in space

TL;DR

This paper studies the evolution of multiple knotted curves in three-dimensional space driven by curvature and torsion, establishing mathematical existence results and providing computational insights into their flow dynamics.

Contribution

It introduces a novel analytical framework for the geometric flow of interacting curves, proving local existence and uniqueness of solutions for the system.

Findings

Proved local existence and uniqueness of smooth solutions.

Developed computational models for curve evolution.

Analyzed effects of nonlocal interactions on flow dynamics.

Abstract

We investigate a system of geometric evolution equations describing a curvature and torsion driven motion of a family of 3D curves in the normal and binormal directions. We explore the direct Lagrangian approach for treating the geometric flow of such interacting curves. Using the abstract theory of nonlinear analytic semi-flows, we are able to prove local existence, uniqueness, and continuation of classical H\"older smooth solutions to the governing system of non-linear parabolic equations modelling $n$ evolving curves with mutual nonlocal interactions. We present several computational studies of the flow that combine the normal or binormal velocity and considering nonlocal interaction.