Convergence of the Allen-Cahn Equation to Multiphase Mean Curvature Flow
نویسندگان
چکیده
منابع مشابه
Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature
We investigate the singular limit, as ε→ 0, of the Allen-Cahn equation uεt = ∆u +εf(u), with f a balanced bistable nonlinearity. We consider rather general initial data u0 that is independent of ε. It is known that this equation converges to the generalized motion by mean curvature — in the sense of viscosity solutions— defined by Evans, Spruck and Chen, Giga, Goto. However the convergence rate...
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We study perturbations of the Allen–Cahn equation and prove the convergence to forced mean curvature flow in the sharp interface limit. We allow for perturbations that are square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and apply previous results for the Allen–Cahn action functional. Finally we discuss...
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We consider a mass conserved Allen-Cahn equation ut = ∆u + ε−2(f(u)− ελ(t)) in a bounded domain with no flux boundary condition, where ελ(t) is the average of f(u(·, t)) and −f is the derivative of a double equal well potential. Given a smooth hypersurface γ0 contained in the domain, we show that the solution uε with appropriate initial data approaches, as ε ց 0, to a limit which takes only two...
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We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke’s mean curvature flow with a generalized right angle condition on the boundary.
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This paper develops and analyzes two fully discrete interior penalty discontinuous Galerkin (IP-DG) methods for the Allen-Cahn equation, which is a nonlinear singular perturbation of the heat equation and originally arises from phase transition of binary alloys in materials science, and its sharp interface limit (the mean curvature flow) as the perturbation parameter tends to zero. Both fully i...
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ژورنال
عنوان ژورنال: Communications on Pure and Applied Mathematics
سال: 2018
ISSN: 0010-3640
DOI: 10.1002/cpa.21747