Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one

Abstract

We consider a reaction-diffusion equation with a half-Laplacian. In the case where the solution is independent on time, the model reduces to the Peierls-Nabarro model describing dislocations as transition layers in a phase field setting. We introduce a suitable rescaling of the evolution equation, using a small parameter $\epsilon$. As $\epsilon$ goes to zero, we show that the limit dynamics is characterized by a system of ODEs describing the motion of particles with two-body interactions. The interaction forces are in $1/x$ and correspond to the well-known interaction between dislocations.

36 pages

References in corpus (2)

• Second-Order Elliptic Integro-Differential Equations: Viscosity Solutions' Theory Revisited
• Phasefield theory for fractional diffusion-reaction equations and applications

Cited by in corpus (2)

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