Area preserving geodesic curvature driven flow of closed curves on a surface

Abstract

We investigate a non-local geometric flow preserving surface area enclosed by a curve on a given surface evolved in the normal direction by the geodesic curvature and the external force. We show how such a flow of surface curves can be projected into a flow of planar curves with the non-local normal velocity. We prove that the surface area preserving flow decreases the length of the evolved surface curves. Local existence and continuation of classical smooth solutions to the governing system of partial differential equations is analysed as well. Furthermore, we propose a numerical method of flowing finite volume for spatial discretization in combination with the Runge-Kutta method for solving the resulting system. Several computational examples demonstrate variety of evolution of surface curves and the order of convergence.2010 Mathematics Subject Classification. Primary: 35K57, 35K65, 65N40, 65M08; Secondary: 53C80.

Modeling of Double Cross-Slip by Means of Geodesic Curvature Driven Flow

“…Motion law (1.1) is treated by means of the direct method (also known as the parametric or the Lagrangian method -see [4]). The dislocation curve G t evolving on a surface given by the graph of a smooth function ϕ is parametrized by means of the vectorial maping X : I u × I t → R 2 as the following:…”

“…For the technical details on this approach we refer the reader to, e.g. [4]. However, the physical nature of double cross-slip suggests that the given surface is piece-(667) wise planar, i.e., non-smooth.…”

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

“…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.…”