The dynamics of drops and attached interfaces for the constrained Allen–Cahn equation

2001 ◽  
Vol 12 (1) ◽  
pp. 1-24 ◽  
Author(s):  
D. STAFFORD ◽  
M. J. WARD ◽  
B. WETTON
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Asymptotic Result ◽  
Interface Motion ◽  
Front Tracking Method ◽  
Cahn Equation ◽  
Area Preserving ◽  
Dimensional Domain

The motion of interfaces for a mass-conserving Allen–Cahn equation that are attached to the boundary of a two-dimensional domain is studied. In the limit of thin interfaces, the interface motion for this problem is known to be governed by an area-preserving mean curvature flow. A numerical front-tracking method, that allows for a numerical solution of this type of curvature flow, is used to compute the motion of interfaces that are attached orthogonally to the boundary. Results obtained from these computations are favourably compared with a previously-derived asymptotic result for the motion of attached interfaces that enclose a small area. The area-preserving mean curvature flow predicts that a semi-circular interface is stationary when it is attached to a flat segment of the boundary. For this case, the interface motion is shown to be metastable and an explicit characterization of the metastability is given.

scholarly journals Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, II: Brakke's flows

2015 ◽  
Vol 17 (05) ◽  
pp. 1450041
Author(s):  
Adriano Pisante ◽  
Fabio Punzo
Keyword(s):  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Initial Conditions ◽  
Curvature Flow ◽  
Monotonicity Formula ◽  
Uniform Estimates ◽  
Cahn Equation ◽  
Motion By Mean Curvature ◽  
Density Of Solutions

We prove convergence of solutions to the parabolic Allen–Cahn equation to Brakke's motion by mean curvature in Riemannian manifolds with Ricci curvature bounded from below. Our results hold for a general class of initial conditions and extend previous results from [T. Ilmanen, Convergence of the Allen–Cahn equation to the Brakke's motion by mean curvature, J. Differential Geom. 31 (1993) 417–461] even in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen–Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by nonpositivity of the limiting energy discrepancy and a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, I, uniform estimates, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.; arXiv:1308.0569], to get various density bounds for the limiting measures.

Mean curvature flow by the Allen–Cahn equation

2015 ◽  
Vol 26 (4) ◽  
pp. 535-559 ◽  
Author(s):  
D. S. LEE ◽  
J. S. KIM
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Curve Shortening Flow ◽  
Cahn Equation ◽  
Angle Condition ◽  
Motion By Mean Curvature ◽  
Curve Shortening ◽  
Three Space ◽  
Stable Hybrid

In this paper, we investigate motion by mean curvature using the Allen–Cahn (AC) equation in two and three space dimensions. We use an unconditionally stable hybrid numerical scheme to solve the equation. Numerical experiments demonstrate that we can use the AC equation for applications to motion by mean curvature. We also study the curve-shortening flow with a prescribed contact angle condition.

Translating Solitons of the Mean Curvature Flow Asymptotic to Hyperplanes in ℝn+1

2018 ◽  
Vol 2020 (24) ◽  
pp. 10114-10153 ◽  
Author(s):  
Eddygledson S Gama ◽  
Francisco Martín
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Normal Component ◽  
Vertical Cylinder ◽  
Curvature Flow ◽  
Grim Reaper ◽  
The Family ◽  
The Mean ◽  
Translating Solitons

Abstract A translating soliton is a hypersurface $M$ in ${\mathbb{R}}^{n+1}$ such that the family $M_t= M- t \,\mathbf e_{n+1}$ is a mean curvature flow, that is, such that normal component of the velocity at each point is equal to the mean curvature at that point $\mathbf{H}=\mathbf e_{n+1}^{\perp }.$ In this paper we obtain a characterization of hyperplanes that are parallel to the velocity and the family of tilted grim reaper cylinders as the only translating solitons in $\mathbb{R}^{n+1}$ that are $C^1$-asymptotic to two half-hyperplanes outside a non-vertical cylinder. This result was proven for translators in $\mathbb{R}^3$ by the 2nd author, Perez-Garcia, Savas-Halilaj, and Smoczyk under the additional hypotheses that the genus of the surface was locally bounded and the cylinder was perpendicular to the translating velocity.

The Surface Area Preserving Mean Curvature Flow in Quasi-Fuchsian Manifolds

2012 ◽  
Vol 32 (6) ◽  
pp. 2191-2202 ◽  
Author(s):  
Tian Daping ◽  
Li Guanghan ◽  
Wu Chuanxi
Keyword(s):  
Surface Area ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Area Preserving

scholarly journals Characterization of facet breaking for nonsmooth mean curvature flow in the convex case

2001 ◽  
pp. 415-446 ◽  
Author(s):  
Giovanni Bellettini ◽  
Matteo Novaga ◽  
Maurizio Paolini
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Convex Case
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