Long-time behavior for crystal dislocation dynamics

TL;DR

This paper analyzes the long-term behavior of solutions to a nonlocal evolutionary equation modeling atom dislocation in crystals, revealing smoothing effects, asymptotic states, and the non-existence of stationary particle systems.

Contribution

It provides explicit estimates for the asymptotic states and demonstrates that transition layers always move, extending understanding of dislocation dynamics in periodic crystals.

Findings

Dislocation functions approach constant or heteroclinic states over time.

Transition layers tend to smooth out after collisions.

Particle systems governing transition layers have no stationary solutions.

Abstract

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 of explicit and quantitative estimates and, as a byproduct, we show that the ODE systems of particles that governs the evolution of the transition layers does not admit stationary solutions (i.e., roughly speaking, transition layers always move).