Abstract
We investigate the motion of closed, smooth non-self-intersecting curves
that evolve in space R 3 . The geometric 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 geometric flow of 3D curves
resulting in a system of degenerate parabolic equations. We prove the
local existence and uniqueness of classical Hölder smooth solutions to
the governing system of nonlinear parabolic equations. A numerical
discretization scheme has been constructed using the method of flowing
finite volumes. We present several numerical examples of the evolution
of curves in 3D with a scalar quantity. In this paper, we analyze the
flow of curves with no torsion evolving in rotating and parallel planes.
Next, we present examples of the evolution of curves with initially
knotted and unknotted curves.