Nonlocal First-Order Hamilton-Jacobi Equations Modelling Dislocations Dynamics

Abstract

We study nonlocal first-order equations arising in the theory of dislocations. We prove the existence and uniqueness of the solutions of these equations in the case of positive and negative velocities, under suitable regularity assumptions on the initial data and the velocity. These results are based on new $L^1$-type estimates on the viscosity solutions of first-order Hamilton-Jacobi Equations appearing in the so-called ``level-sets approach''. Our work is inspired by and simplifies a recent work of Alvarez, Cardaliaguet and Monneau.

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• Global Existence Results and Uniqueness for Dislocation Equations
• Uniqueness Results for Nonlocal Hamilton-Jacobi Equations
• Global existence for a system of non-linear and non-local transport equations describing the dynamics of dislocation densities
• Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations
• Convergence of nonlocal geometric flows to anisotropic mean curvature motion
• Weak Solutions for Dislocation Type Equations
• Asymptotic behaviour of rescaled nonlocal functionals and evolutions