Phasefield theory for fractional diffusion-reaction equations and applications

Imbert, Cyril; Souganidis, Panagiotis E.

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

Metadata

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Author(s)

Imbert, Cyril

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