WIAS Preprint No. 2302, (2016)

Long-time behavior for crystal dislocation dynamics


  • Patrizi, Stefania
  • Valdinoci, Enrico
    ORCID: 0000-0001-6222-2272

2010 Mathematics Subject Classification

  • 82D25 35R09 74E15 35R11 47G20


  • Peierls-Nabarro model, nonlocal integro-differential equations, dislocation dynamics, attractive/repulsive potentials, collisions


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).

Appeared in

  • Math. Models Methods Appl. Sci., 27:12 (2017) pp. 2185--2228.

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