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Author: Valdinoci, Enrico × End date: 2019 × Start date: 2010 × Subject: collisions ×

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    Long-time behavior for crystal dislocation dynamics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Patrizi, Stefania; Valdinoci, Enrico
    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).
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    Relaxation times for atom dislocations in crystals
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Patrizi, Stefania; Valdinoci, Enrico
    We study the relaxation times for a parabolic differential equation whose solution represents the atom dislocation in a crystal. The equation that we consider comprises the classical Peierls-Nabarro model as a particular case, and it allows also long range interactions. It is known that the dislocation function of such a model has the tendency to concentrate at single points, which evolve in time according to the external stress and a singular, long range potential. Depending on the orientation of the dislocation function at these points, the potential may be either attractive or repulsive, hence collisions may occur in the latter case and, at the collision time, the dislocation function does not disappear. The goal of this paper is to provide accurate estimates on the relaxation times of the system after collision. More precisely, we take into account the case of two and three colliding points, and we show that, after a small transition time subsequent to the collision, the dislocation function relaxes exponentially fast to a steady state. We stress that the exponential decay is somehow exceptional in nonlocal problems (for instance, the spatial decay in this case is polynomial). The exponential time decay is due to the coupling (in a suitable space/time scale) between the evolution term and the potential induced by the periodicity of the crystal.
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    Crystal dislocations with different orientations and collisions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Patrizi, Stefania; Valdinoci, Enrico
    We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superpositions of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well). We show that, at a long time scale, and at a macroscopic space scale, the dislocations have the tendency to concentrate as pure jumps at points which evolve in time, driven by the external stress and by a singular potential. Due to differences in the dislocation orientations, these points may collide in finite time.