scholarly journals Convergence Rates of the Allen--Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative Entropies

2020 ◽  
Vol 52 (6) ◽  
pp. 6222-6233
Author(s):  
Julian Fischer ◽  
Tim Laux ◽  
Theresa M. Simon
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Convergence Rates ◽  
Short Proof ◽  
Curvature Flow ◽  
Cahn Equation

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.

The limit of the anisotropic double-obstacle Allen–Cahn equation

1996 ◽  
Vol 126 (6) ◽  
pp. 1217-1234 ◽  
Author(s):  
Charles M. Elliott ◽  
Reiner Schätzle
Keyword(s):  
Convex Function ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Normal Velocity ◽  
Cahn Equation ◽  
The Second Fundamental Form ◽  
Image Position ◽  
Anisotropic Mean Curvature

In this paper, we prove that solutions of the anisotropic Allen–Cahn equation in doubleobstacle formwhere A is a strictly convex function, homogeneous of degree two, converge to an anisotropic mean-curvature flowwhen this equation admits a smooth solution in ℝn. Here VN and R respectively denote the normal velocity and the second fundamental form of the interface, and More precisely, we show that the Hausdorff-distance between the zero-level set of φ and the interface of the above anisotropic mean-curvature flow is of order O(ε2).

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.

INTERFACE ESTIMATES FOR THE FULLY ANISOTROPIC ALLEN-CAHN EQUATION AND ANISOTROPIC MEAN-CURVATURE FLOW

1996 ◽  
Vol 06 (08) ◽  
pp. 1103-1118 ◽  
Author(s):  
CHARLES M. ELLIOTT ◽  
MAURIZIO PAOLINI ◽  
REINER SCHÄTZLE
Keyword(s):  
Convex Function ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Kinetic Term ◽  
Normal Velocity ◽  
Cahn Equation ◽  
The Second Fundamental Form ◽  
Anisotropic Mean Curvature

In this paper, we prove that solutions of the anisotropic Allen-Cahn equation in double-obstacle form with kinetic term [Formula: see text] in {|φ|<1}, where A is a convex function, homogeneous of degree two, and β depends only on the direction of ∇φ, converge to an anisotropic mean-curvature flow [Formula: see text] Here VN and R denote respectively the normal velocity and the second fundamental form of the interface, and [Formula: see text]. We prove this in the case when the above flow admits a smooth solution, and we establish that the Hausdorff-distance between the zero-level set of φ and the interface of the flow is of order O(ε2).

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