Curvature evolution of plane curves with prescribed opening angle

Ishimura, Naoyuki

doi:10.1017/s0004972700014714

Cited by 27 publications

(20 citation statements)

References 12 publications

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“…They proved that the graphical interface u(x, t) converges to an extracting self-similar solution, if an unbounded initial graph u 0 (x) satisfies linear growth and further assumptions. Ishimura [4] also studied the same problem in detail.…”

Section: Introductionmentioning

confidence: 98%

The condition on the stability of stationary lines in a curvature flow in the whole plane

Nara

Taniguchi

2007

Journal of Differential Equations

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The long time behavior of a curve in the whole plane moving by a curvature flow is studied. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. Here the initial curves are given by bounded functions on the x-axis. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order O(t −1/2 ) as time goes to infinity. The proof is based on the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary condition on the stability of constant solutions that represent stationary lines of the curvature flow in the whole plane.

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Section: Introductionmentioning

confidence: 98%

The condition on the stability of stationary lines in a curvature flow in the whole plane

Nara

Taniguchi

2007

Journal of Differential Equations

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“…At larger times, the solution is close to a single sinusoidal wave. In the regime of the sinusoidal profile, from (20) decay times are shorter, of the order of the standard time (ℓ/nπ) 2 /B that is familiar from exponential decay of Fourier modes in linear diffusion models (e.g. [10]) when the half-wavelength ℓ/n is the typical distance between neighboring regions of high and low mass concentration.…”

Section: Decaying Periodic Solution Initially Close To Square Wavementioning

confidence: 99%

“…Two decades ago [5], the exact solution was constructed in parametric integral form for nonlinear surface evolution near a symmetric grain boundary, with constant slope at the grain boundary groove, initial flatness and zero displacement at infinity. A similar procedure produces the more general "open angle" solutions for deposition in a wedge [20]. Subsequently, the fourth-order Mullins equation for curvature-driven surface diffusion on an almost-isotropic material, was solved with boundary conditions representing a symmetric grain boundary [6,28,7].…”

Section: Surface Evolution By Evaporation and Condensationmentioning

confidence: 99%

The Role of Symmetry and Separation in Surface Evolution and Curve Shortening

Broadbridge

Vassiliou

2011

SIGMA

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Abstract. With few exceptions, known explicit solutions of the curve shortening flow (CSE) of a plane curve, can be constructed by classical Lie point symmetry reductions or by functional separation of variables. One of the functionally separated solutions is the exact curve shortening flow of a closed, convex "oval"-shaped curve and another is the smoothing of an initial periodic curve that is close to a square wave. The types of anisotropic evaporation coefficient are found for which the evaporation-condensation evolution does or does not have solutions that are analogous to the basic solutions of the CSE, namely the grim reaper travelling wave, the homothetic shrinking closed curve and the homothetically expanding grain boundary groove. Using equivalence classes of anisotropic diffusion equations, it is shown that physical models of evaporation-condensation must have a diffusivity function that decreases as the inverse square of large slope. Some exact separated solutions are constructed for physically consistent anisotropic diffusion equations.

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“…Instead of the last condition of (6.5), we could specify a M such that lim x→∞ φ (x) = M > 0. There is a bijective map between φ 0 and M [8]. We next explain how (6.5) arises.…”

mentioning

confidence: 99%

“…given in [8], where the ODE was previously studied. The improved bound implies we do not need to take x N so large.…”

mentioning

confidence: 99%

On the Voronoi Implicit Interface Method

Zaitzeff¹,

Esedoḡlu²,

Garikipati³

2019

SIAM J. Sci. Comput.

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We present careful numerical convergence studies, using parameterized curves to reach very high resolutions in two dimensions, of a level set method for multiphase curvature motion known as the Voronoi implicit interface method. Our tests demonstrate that in the unequal, additive surface tension case, the Voronoi implicit interface method does not converge to the desired limit. We then present a variant that maintains the spirit of the original algorithm, and appears to fix the non-convergence. As a bonus, the new variant extends the Voronoi implicit interface method to unequal mobilities.

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Curvature evolution of plane curves with prescribed opening angle

Abstract: We discuss the evolution of plane curves which are described by entire graphs with prescribed opening angle. We show that a solution converges to the unique self-similar solution with the same asymptotics.

Cited by 27 publications

References 12 publications

The condition on the stability of stationary lines in a curvature flow in the whole plane

The condition on the stability of stationary lines in a curvature flow in the whole plane

The Role of Symmetry and Separation in Surface Evolution and Curve Shortening

On the Voronoi Implicit Interface Method

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