Generation of interface for solutions of the mass conserved Allen-Cahn equation

Danielle Hilhorst; Hiroshi Matano; Thanh Nam Nguyen; Hendrik Weber

doi:10.1137/18M1204747

Generation of interface for solutions of the mass conserved Allen-Cahn equation

Danielle Hilhorst, Hiroshi Matano, Thanh Nam Nguyen, Hendrik Weber

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Abstract

In this paper, we study the generation of interface for the solution of the mass conserved Allen-Cahn equation involving a nonlocal integral term. We show that, for a rather general class of initial functions that are independent of ϵ, the solution generally develops a steep transition layer of thickness O(ϵγ) (0 < γ ≤ 1) at a certain time of order ϵ2 |ln ϵ|. In some cases, we prove that the thickness of the interface is exactly of order ϵ, which is the optimal thickness estimate. We note that the comparison principle does not hold for our equation because of the nonlocal term so that the methods that were employed in the earlier studies of the standard Allen-Cahn equation do not work. We will therefore take a different approach, which is based on the fine analysis of the long-time behavior of the corresponding nonlocal ODEs and some energy estimates.

Original language	English
Pages (from-to)	2624-2654
Number of pages	31
Journal	Siam Journal on Mathematical Analysis
Volume	52
Issue number	3
DOIs	https://doi.org/10.1137/18M1204747
Publication status	Published - 31 Dec 2020

Bibliographical note

Funding Information:
The research of the second author was partially supported by KAKENHI (16H02151). The research of the first, second, and third authors was supported by the GDRI ReaDiNet. The research of the third author was supported by ERC Adora and by Plan Cancer HTE program EcoAML. The research of the fourth author was supported by EPSRC First Grant EP/L018969/1 and by the Royal Society through the University Research Fellowship UF140187.

Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.

Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

Funding

The research of the second author was partially supported by KAKENHI (16H02151). The research of the first, second, and third authors was supported by the GDRI ReaDiNet. The research of the third author was supported by ERC Adora and by Plan Cancer HTE program EcoAML. The research of the fourth author was supported by EPSRC First Grant EP/L018969/1 and by the Royal Society through the University Research Fellowship UF140187.

Keywords

Absence of interface
Allen-Cahn
Generation of interface
Nonlinear PDE
Reaction-diffusion equation
Singular perturbation

ASJC Scopus subject areas

Analysis
Computational Mathematics
Applied Mathematics

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10.1137/18M1204747

Cite this

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title = "Generation of interface for solutions of the mass conserved Allen-Cahn equation",

abstract = "In this paper, we study the generation of interface for the solution of the mass conserved Allen-Cahn equation involving a nonlocal integral term. We show that, for a rather general class of initial functions that are independent of ϵ, the solution generally develops a steep transition layer of thickness O(ϵγ) (0 < γ ≤ 1) at a certain time of order ϵ2 |ln ϵ|. In some cases, we prove that the thickness of the interface is exactly of order ϵ, which is the optimal thickness estimate. We note that the comparison principle does not hold for our equation because of the nonlocal term so that the methods that were employed in the earlier studies of the standard Allen-Cahn equation do not work. We will therefore take a different approach, which is based on the fine analysis of the long-time behavior of the corresponding nonlocal ODEs and some energy estimates.",

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author = "Danielle Hilhorst and Hiroshi Matano and Nguyen, {Thanh Nam} and Hendrik Weber",

note = "Funding Information: The research of the second author was partially supported by KAKENHI (16H02151). The research of the first, second, and third authors was supported by the GDRI ReaDiNet. The research of the third author was supported by ERC Adora and by Plan Cancer HTE program EcoAML. The research of the fourth author was supported by EPSRC First Grant EP/L018969/1 and by the Royal Society through the University Research Fellowship UF140187. Publisher Copyright: {\textcopyright} 2020 Society for Industrial and Applied Mathematics. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.",

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AU - Matano, Hiroshi

AU - Nguyen, Thanh Nam

AU - Weber, Hendrik

N1 - Funding Information: The research of the second author was partially supported by KAKENHI (16H02151). The research of the first, second, and third authors was supported by the GDRI ReaDiNet. The research of the third author was supported by ERC Adora and by Plan Cancer HTE program EcoAML. The research of the fourth author was supported by EPSRC First Grant EP/L018969/1 and by the Royal Society through the University Research Fellowship UF140187. Publisher Copyright: © 2020 Society for Industrial and Applied Mathematics. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/12/31

Y1 - 2020/12/31

N2 - In this paper, we study the generation of interface for the solution of the mass conserved Allen-Cahn equation involving a nonlocal integral term. We show that, for a rather general class of initial functions that are independent of ϵ, the solution generally develops a steep transition layer of thickness O(ϵγ) (0 < γ ≤ 1) at a certain time of order ϵ2 |ln ϵ|. In some cases, we prove that the thickness of the interface is exactly of order ϵ, which is the optimal thickness estimate. We note that the comparison principle does not hold for our equation because of the nonlocal term so that the methods that were employed in the earlier studies of the standard Allen-Cahn equation do not work. We will therefore take a different approach, which is based on the fine analysis of the long-time behavior of the corresponding nonlocal ODEs and some energy estimates.

AB - In this paper, we study the generation of interface for the solution of the mass conserved Allen-Cahn equation involving a nonlocal integral term. We show that, for a rather general class of initial functions that are independent of ϵ, the solution generally develops a steep transition layer of thickness O(ϵγ) (0 < γ ≤ 1) at a certain time of order ϵ2 |ln ϵ|. In some cases, we prove that the thickness of the interface is exactly of order ϵ, which is the optimal thickness estimate. We note that the comparison principle does not hold for our equation because of the nonlocal term so that the methods that were employed in the earlier studies of the standard Allen-Cahn equation do not work. We will therefore take a different approach, which is based on the fine analysis of the long-time behavior of the corresponding nonlocal ODEs and some energy estimates.

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Generation of interface for solutions of the mass conserved Allen-Cahn equation

Abstract

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Keywords

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