Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

Abstract

Abstract. We prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. The equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, which arises in the theory of dislocation dynamics. We show that if an anisotropic mean curvature motion is approximated by equations of this type then it is always of variational type, whereas the converse is true only in dimension two.

A posteriori error estimates for the effective Hamiltonian of dislocation dynamics

“…The main idea is that there is a comparison principle for the non-linear non-local right hand side of (3.1), essentially because the instability created by the discontinuity of the integer part E is somehow compensated by the vanishing of the gradient at the same points (see the proofs of the comparison principle in [15], [7], [11]). …”

Well-posedness and numerical analysis of a one-dimensional non-local transport equation modelling dislocations dynamics

“…Then, we have Claim 1. Φ is a (viscosity) supersolution of problem (14). As a matter of fact, since Φ is smooth, Φ is a classical supersolution of problem (14).…”

“…In the special case where the kernel c 0 is assumed nonnegative, some existence and uniqueness results for all time in any dimension are available in a "Slepčev formulation" (see [8,14]). …”

An Error Estimate for a New Scheme for Mean Curvature Motion

“…In order to propose a numerical scheme for anisotropic mean curvature motion, we will use the work of Da Lio et al [13] that we briefly recall here. Given a function g defined on the unit sphere…”

“…The idea is to use a recent work of Da Lio, Monneau and the author [13] concerning the convergence of dislocations dynamics to mean curvature motion. Dislocations are linear defects which move in crystals.…”