On diffusion and transport acting on parameterized moving closed curves in space

Abstract

We investigate the motion of closed smooth curves that evolve in space $\mathbb{R}^3$. The governing evolutionary equation for the evolution of the curve is accompanied by a parabolic equation for the scalar quantity evaluated over the evolving curve. We apply the direct Lagrangian approach to describe the flow of 3D curves, resulting in a system of degenerate parabolic equations. We prove the local existence and uniqueness of classical H\"older smooth solutions to the governing system of nonlinear parabolic equations. A numerical discretization scheme is constructed using the method of flowing finite volumes. We present several numerical examples of the evolution of curves in 3D with a scalar quantity. We consider the flow of curves with zero torsion evolving in rotating and parallel planes. Next, we present examples of the evolution of curves with initially knotted and unknotted curves.