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Phasefield theory for fractional diffusion-reaction equations and applications

Imbert, Cyril; Souganidis, Panagiotis E. (2009), Phasefield theory for fractional diffusion-reaction equations and applications. https://basepub.dauphine.fr/handle/123456789/7218

Type
Document de travail / Working paper
External document link
http://arxiv.org/abs/0907.5524v1
Date
2009
Publisher
Université Paris-Dauphine
Published in
Paris
Pages
41
Metadata
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Author(s)
Imbert, Cyril cc
Souganidis, Panagiotis E.
Abstract (EN)
This paper is concerned with diffusion-reaction equations where the classical diffusion term, such as the Laplacian operator, is replaced with a singular integral term, such as the fractional Laplacian operator. As far as the reaction term is concerned, we consider bistable non-linearities. After properly rescaling (in time and space) these integro-differential evolution equations, we show that the limits of their solutions as the scaling parameter goes to zero exhibit interfaces moving by anisotropic mean curvature. The singularity and the unbounded support of the potential at stake are both the novelty and the challenging difficulty of this work.
Subjects / Keywords
fractional diffusion-reaction equations; traveling wave; phasefield theory; anisotropic mean curvature motion; fractional Laplacian; Green-Kubo type formulae