Ibrahim, Hassan (2008) Analysis of parabolic/Hamilton-Jacobi systems modelizing the dynamics of dislocation densities inabounded domain. PhD thesis, ENPC p.263.

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Abstract

This thesis is concerned with the theoretical study of a mathematical model arising from the study of the dynamics of dislocation densities in crystals of small size. This dynamics is modelized by parabolic/Hamilton-Jacobi nonlinear coupled system. Dislocations are linear defects which move in crystals when those are subjected to exterior stresses. Independently, at the end of the thesis, we present, in a short chapter, a numerical method for the transport of fronts.

In this thesis, three types of equations are considered : non-linear first order Hamilton-Jacobi equations, scalar conservations laws, and singular parabolic equations.

We treat a singular parabolic/Hamilton-Jacobi system where the singularity appears from the presence of the inverse of the gradient. Our system takes into consideration the short range dislocation-dislocation interactions, as well as the formation of boundary layers. We study the existence, uniqueness and the regularity of the solutions of this system. This study relies essentially on the theory of viscosity solutions; the theory of entropy and classical solutions. Two main cases are considered : the case of zero exterior stresses, and the case of constant exterior stresses (not necessarily zero).

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