Long-time behavior for crystal dislocation dynamics

Stefania Patrizi

doi:10.1142/S0218202517500427

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Long-time behavior for crystal dislocation dynamics

2017-09-20

Stefania Patrizi *

Enrico Valdinoci

DOI：10.1142/S0218202517500427

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Abstract

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We describe the asymptotic states for the solutions of a nonlocal equation of evolutionary type, which have the physical meaning of the atom dislocation function in a periodic crystal. More precisely, we can describe accurately the smoothing effect on the dislocation function occurring slightly after a particle collision (roughly speaking, two opposite transitions layers average out) and, in this way, we can trap the atom dislocation function between a superposition of transition layers which, as time flows, approaches either a constant function or a single heteroclinic (depending on the algebraic properties of the orientations of the initial transition layers). The results are endowed with explicit and quantitative estimates and, as a byproduct, we show that the ODE systems of particles that govern the evolution of the transition layers does not admit stationary solutions (i.e. roughly speaking, transition layers always move).

Keywords:

Peierls-Nabarro model

nonlocal integro-differential equations

dislocation dynamics

attractive/repulsive potentials

collisions

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