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ON DIFFUSION AND TRANSPORT ACTING ON PARAMETERIZED MOVING CLOSED CURVES IN SPACE
  • Michal Benes,
  • Miroslav Kolar,
  • Daniel Sevcovic
Michal Benes
Ceske vysoke uceni technicke v Praze Fakulta jaderna a fyzikalne inzenyrska
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Miroslav Kolar
Ceske vysoke uceni technicke v Praze Fakulta jaderna a fyzikalne inzenyrska
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Daniel Sevcovic
Univerzita Komenskeho v Bratislave Oddelenie aplikovanej matematiky

Corresponding Author:sevcovic@fmph.uniba.sk

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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.
02 Apr 2024Submitted to Mathematical Methods in the Applied Sciences
03 Apr 2024Submission Checks Completed
03 Apr 2024Assigned to Editor
11 Apr 2024Reviewer(s) Assigned