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.

arXiv comment36 pages

References in corpus (4)

• An extension problem related to the fractional Laplacian
• Second-Order Elliptic Integro-Differential Equations: Viscosity Solutions' Theory Revisited
• Dislocation dynamics: from microscopic models to macroscopic crystal plasticity
• Phasefield theory for fractional diffusion-reaction equations and applications

Cited by in corpus (11)

• Nonlocal diffusion and applications
• On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres
• Crystal dislocations with different orientations and collisions
• Chaotic orbits for systems of nonlocal equations
• Long-time asymptotics for evolutionary crystal dislocation models
• Mathematical validation of the Peierls--Nabarro model for edge dislocations
• Metastable Speeds in the Fractional Allen-Cahn Equation
• The Effect of Forest Dislocations on the Evolution of a Phase-Field Model for Plastic Slip
• A multiplicity result via Ljusternick-Schnirelmann category and morse theory for a fractional schrödinger equation in ${\mathbb{R}}^N$
• Uniform distribution of dislocations in Peierls-Nabarro models for semi-coherent interfaces
• A Transport-Constrained Gradient-Flow Description of Continuum Dislocation Dynamics