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Curvature driven flow of a family of interacting curves with applications
Author(s) -
Beneš Michal,
Kolář Miroslav,
Ševčovič Daniel
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6182
Subject(s) - mathematics , uniqueness , curvature , nonlinear system , flow (mathematics) , motion (physics) , geometric flow , mathematical analysis , continuation , planar , mean curvature flow , geometry , mean curvature , classical mechanics , physics , computer graphics (images) , quantum mechanics , computer science , programming language
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.