Curvature driven flow of a family of interacting curves with applications

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.

On Diffusion and Transport Acting on Parameterized Moving Closed Curves in Space

“…The flowing finite-volume discretization was proposed by Mikula and Ševčovič [30] for the evolution of curves in the plane. It was further generalized and analyzed for evolving curves in 3D by Beneš, Kolář and Ševčovič in [7,8].…”

“…Here, we can see that the tangential motion of Γ t can also influence the distribution of the quantity ϱ along Γ t . Using ( 5) in (7) we finally obtain the advection-diffusion equation:…”

“…They constructed a numerical approximation scheme using a suitable tangential redistribution. Beneš, Kolář and Ševčovič investigated the role of tangential velocity in the context of material science [25] and the evolution of interacting curves [7], [8]. In [16] Garcke, Kohsaka and Ševčovič applied the uniform tangential redistribution in the theoretical proof of nonlinear stability of stationary solutions for curvature driven flow with triple junction in the plane.…”

“…[4, Lemma 2.5], DaPrato and Grisvard [9], Lunardi [28]). The method based on analytic semigroup and maximal regularity theory has been successfully applied to prove the existence, regularity and uniqueness of solutions representing evolving families of 2D and 3D curves in the series of papers co-authored by Ševčovič, Mikula, Yazaki, Beneš and Kolář [30], [31], [32], [41], [25], [7], [8]. Now, we can state the following result on the local existence and uniqueness of solutions.…”