Curvature driven flow of a family of interacting curves with applications

In this paper, we investigate a system of geometric evolution equations describing a curvature‐driven motion of a family of planar curves with mutual interactions that can have local as well as nonlocal character, and the entire curve may influence evolution of other curves. We propose a direct Lagr...

Full description

Saved in:
Bibliographic Details
Published inMathematical methods in the applied sciences Vol. 43; no. 7; pp. 4177 - 4190
Main Authors Beneš, Michal, Kolář, Miroslav, Ševčovič, Daniel
Format Journal Article
LanguageEnglish
Published Freiburg Wiley Subscription Services, Inc 15.05.2020
Subjects
Curvature
curvature driven flow
Curves
Dislocation loops
Evolution
flowing finite volume method
Hölder smooth solutions
interacting curves
Nonlinear equations
nonlocal flow
Online AccessGet full text

Cover

More Information
Summary:In this paper, we investigate a system of geometric evolution equations describing a curvature‐driven motion of a family of planar curves with mutual interactions that can have local as well as nonlocal character, and the entire curve may influence evolution of other curves. We propose a direct Lagrangian approach for solving such a geometric flow of interacting curves. We prove local existence, uniqueness, and continuation of classical Hölder smooth solutions to the governing system of nonlinear parabolic equations. A numerical solution to the governing system has been constructed by means of the method of flowing finite volumes. We also discuss various applications of the motion of interacting curves arising in nonlocal geometric flows of curves as well as an interesting physical problem of motion of two interacting dislocation loops in the material science.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.6182