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Title of Journal: Arch Rational Mech Anal

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Abbravation: Archive for Rational Mechanics and Analysis

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Springer-Verlag

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DOI

10.1002/sres.3850030303

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1432-0673

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Dislocation Dynamics Shorttime Existence and Uni

Authors: Olivier Alvarez Philippe Hoch Yann Le Bouar Régis Monneau
Publish Date: 2006/03/22
Volume: 181, Issue: 3, Pages: 449-504
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Abstract

We study a mathematical model describing dislocation dynamics in crystals We consider a single dislocation line moving in its slip plane The normal velocity is given by the PeachKoehler force created by the dislocation line itself The mathematical model is an eikonal equation with a velocity which is a nonlocal quantity depending on the whole shape of the dislocation line We study the special case where the dislocation line is assumed to be a graph or a closed loop In the framework of discontinuous viscosity solutions for Hamilton–Jacobi equations we prove the existence and uniqueness of a solution for small time We also give physical explanations and a formal derivation of the mathematical model Finally we present numerical results based on a levelsets formulation of the problem These results illustrate in particular the fact that there is no general inclusion principle for this model

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  1. Optimal Design in Conductivity Under Locally Constrained Heat Flux
  2. Regularity Issues in the Problem of Fluid Structure Interaction
  3. Negative Eigenvalues of Two-Dimensional Schrödinger Operators
  4. Dobrushin States in the $${\phi^4_1}$$ Model
  5. The Liouville Theorem and the L 2 Decay for the FENE Dumbbell Model of Polymeric Flows
  6. The Liouville Theorem and the L 2 Decay for the FENE Dumbbell Model of Polymeric Flows
  7. Counterexamples to the Sonic Criterion
  8. On Nonlinear Stochastic Balance Laws
  9. Weak Solutions of the Navier–Stokes Equations for Compressible Flows in a Half-Space with No-Slip Boundary Conditions
  10. Generalized Transition Fronts for One-Dimensional Almost Periodic Fisher-KPP Equations
  11. The Gel’fand Problem for the Biharmonic Operator
  12. The Existence Problem for Steiner Networks in Strictly Convex Domains
  13. Stability of Detonation Profiles in the ZND Limit
  14. Uniqueness of Absolute Minimizers for $${L^\infty}$$-Functionals Involving Hamiltonians $${H(x,p)}$$H(x,p)
  15. A Condition for the Hölder Regularity of Local Minimizers of a Nonlinear Elastic Energy in Two Dimensions
  16. Kinetic Stabilization¶of a Nonlinear Sonic Phase Boundary
  17. The Toda System and Clustering Interfaces in the Allen–Cahn equation
  18. A Stability Criterion for Two-Fluid Interfaces and Applications
  19. Long-Time Convergence of an Adaptive Biasing Force Method: The Bi-Channel Case
  20. Displacement Convexity and Minimal Fronts at Phase Boundaries
  21. Gradient Bounds for Elliptic Problems Singular at the Boundary
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