Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity

Abstract

In this paper, we investigate a time‐dependent family of plane closed Jordan curves evolving in the normal direction with a velocity that is assumed to be a function of the curvature, tangential angle, and position vector of a curve. We follow the direct approach and analyze the system of governing PDEs for relevant geometric quantities. We focus on a class of the so‐called curvature adjusted tangential velocities for computation of the curvature driven flow of plane closed curves. Such a curvature adjusted tangential velocity depends on the modulus of the curvature and its curve average. Using the theory of abstract parabolic equations, we prove local existence, uniqueness, and continuation of classical solutions to the system of governing equations. We furthermore analyze geometric flows for which normal velocity may depend on global curve quantities such as the length, enclosed area, or total elastic energy of a curve. We also propose a stable numerical approximation scheme on the basis of the flowing finite volume method. Several computational examples of various nonlocal geometric flows are also presented in this paper. Copyright © 2012 John Wiley & Sons, Ltd.

Evolution of plane curves with a curvature adjusted tangential velocity

“…On the other hand, as far as the time complexity of computation and simplicity of a numerical approximation scheme are concerned, discretization of (2.2) together with Recall that in the case where α is given by the expression (3.3) with ϕ ≡ 1 and ω = 0, the short time existence and uniqueness of smooth solutions to the system of PDEs (2.2), (2.3), (2.4) and (2.5) has been shown in [15]. In the forthcoming paper [25] the authors will prove local existence and uniqueness of a classical solution to the full system of governing equations (2.2)-(2.5) and (3.3), (3.4).…”

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

“…The following lemma deals with properties of the tangential velocity functional and it is due toŠevčovič and Yazaki [28]. To formulate its statement we need to introduce the scale of Banach spaces…”

“…In [28, Theorem 1]Ševčovič and Yazaki proved a rather general result on local existence and uniqueness and continuation of classical Hölder smooth solutions to the non-local flow driven by the normal velocity which is the sum of local and nonlocal parts provided that the nonlocal part is however independent of X and n Γ . This is why we have to slightly modify the proof of [28,Theorem 1] in order to handle the case when the nonlocal part has the form: c(X, n Γ )F.…”

“…Following the proof of [28,Theorem 1], the linearization H 0 (Ȳ ) of the principal part atȲ = (κ Γ ,ν,r, X) can be written in the form H 0 (Ȳ ) =D(Ȳ )∂ 2 u +C(Ȳ ), whereD (Ȳ ) = diag(āḡ −2 ,āḡ −2 , 0,āḡ −2 ) ∈ c ε (S 1 ) andC(Ȳ ) is a 5 × 5 matrix with coefficients smoothly depending onκ Γ ,ν,r, X and thus belonging to c ε (S 1 ). The rest of the proof is similar to that of [28,Theorem 1]. The principal partD∂ 2 u is a generator of an analytic semigroup on E 0 with the domain E 1 and it belongs to the maximal regularity class M(E 1 , E 0 ) on the pair of spaces (E 1 , E 0 ).…”

Solution to the inverse Wulff problem by means of the enhanced semidefinite relaxation method

“…Following [22,31] we introduce a notation for parameterization of planar curves. Let Γ ⊂ R 2 be a C 1 smooth curve of a finite length, i. e. Γ can be parameterized by a C 1 mapping x :…”