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

Régis Monneau (1) , Maria del Mar Gonzalez (2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
2 UPC - Universitat Politècnica de Catalunya = Université polytechnique de Catalogne [Barcelona]

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 $\varepsilon$. As $\varepsilon$ 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.

Domaines: Equations aux dérivées partielles [math.AP]

Régis Monneau, Maria del Mar Gonzalez. Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. 2010. ⟨hal-00497492⟩