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Curvature driven flow of a family of interacting curves with applications |
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| Authors: | Michal Beneš Miroslav Kolář Daniel Ševčovič |
| Institution: | 1. Department of Mathematics, Faculty of 2. Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Prague, 12000, Czech Republic;3. Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Prague, 12000, Czech Republic
Department of Mathematics, School of 4. Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa 214-8571, Japan;5. Department of Applied Mathematics and 6. Statistics, Faculty of 7. Mathematics Physics and Informatics, Comenius University in Bratislava, Bratislava, Slovakia |
| Abstract: | 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. |
| Keywords: | curvature driven flow flowing finite volume method Hölder smooth solutions interacting curves nonlocal flow |