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

2010/07/05 by Régis Monneau, Gonzalez, Maria del Mar, Monneau, Regis · 1 citation
Computer Science · Materials Science · Mathematics · #35B40 #35D30 #35G25 #35J25 #35Q99 #70F99 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.1007.0740

openalex publication_date 2010/07/05 · openalex created_date 2019/07/30 · openalex updated_date 2026/07/28

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 ε. As ε 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.

Citations

• Homogenization of the Peierls-Nabarro model for dislocation dynamics
• Regularity of the obstacle problem for a fractional power of the laplace operator

Cited by

• The strongly nonlocal Allen-Cahn problem

Related

• Slow dynamics in reaction-diffusion systems
• Discrete Dislocations Dynamics with annihilation as the limit of the Peierls-Nabarro model in one dimension
• From the Peierls-Nabarro model to the equation of motion of the\n dislocation continuum
• Motion by mean curvature in interacting particle systems
• Slow Patterns in Multilayer Dislocation Evolution with Dynamic Boundary Conditions