Long-time behavior for crystal dislocation dynamics

DOI10.1142/S0218202517500427zbMath1377.82040arXiv1609.04441MaRDI QIDQ4588313

Stefania Patrizi, Enrico Valdinoci

Publication date: 2 November 2017

Published in: Mathematical Models and Methods in Applied Sciences (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1609.04441

zbMATH Keywords

collisionsdislocation dynamicsPeierls-Nabarro modelattractive/repulsive potentialsnonlocal integro-differential equations

Mathematics Subject Classification ID

Interacting particle systems in time-dependent statistical mechanics (82C22) Crystalline structure (74E15) Statistical mechanics of crystals (82D25) Integro-differential operators (47G20) Fractional partial differential equations (35R11) Integro-partial differential equations (35R09)

Related Items (15)

A fractional glance to the theory of edge dislocations ⋮ Relaxation times for atom dislocations in crystals ⋮ Variational evolution of dislocations in single crystals ⋮ Asymptotic Stability for Diffusion with Dynamic Boundary Reaction from Ginzburg–Landau Energy ⋮ Derivation of the 1-D Groma-Balogh equations from the Peierls-Nabarro model ⋮ Decay estimates for a perturbed two-terms space-time fractional diffusive problem ⋮ On a class of non-local elliptic equations with asymptotically linear term ⋮ Existence and uniqueness of solutions to the Peierls–Nabarro model in anisotropic media ⋮ Discrete Dislocation 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 dislocation continuum ⋮ Decay estimates for evolution equations with classical and fractional time-derivatives ⋮ The effect of forest dislocations on the evolution of a phase-field model for plastic slip ⋮ Metastable speeds in the fractional Allen-Cahn equation ⋮ From atomistic model to the Peierls-Nabarro model with \({\gamma}\)-surface for dislocations ⋮ Long-time asymptotics for evolutionary crystal dislocation models

Cites Work

• Local and global minimizers for a variational energy involving a fractional norm
• Hitchhiker's guide to the fractional Sobolev spaces
• Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one
• A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations
• Crystal dislocations with different orientations and collisions
• Homogenization and Orowan's law for anisotropic fractional operators of any order
• Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting
• Relaxation times for atom dislocations in crystals
• Strongly Nonlocal Dislocation Dynamics in Crystals
• Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions
• One-dimensional solutions of non-local Allen-Cahn-type equations with rough kernels

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