Formal derivation and existence result of an approximate model on dislocation densities
Hassan Ibrahim, Vivian Rizik, Zaynab Salloum · ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik · 2018
Abstract In this work, we are interested in a mathematical problem arising from the dynamics of dislocation densities in crystals. The model, originally developed by Groma, Czikor, and Zaiser in [ ], is a coupled singular parabolic system that describes the motion of dislocations in a bounded crystal, taking into account the short‐range interactions and the effect of exterior stresses. We show the derivation and the short time existence and uniqueness of a regular solution in a Hölder space using a fixed point argument and a particular comparison principle.