Received: 7 February 2017 Revised: 8 December 2017 Accepted: 21 January 2018 DOI: 10.1002/zamm.201700038 ORIGINAL PAPER Formal derivation and existence result of an approximate model on dislocation densities Hassan Ibrahim 1 Vivian Rizik 1,2 Zaynab Salloum 1 1 Lebanese University, Faculty of Science - I, Mathematics Department, Hadath, Lebanon 2 Université de technologie de Compiègne, LMAC Laboratory of Applied Mathematics of Compiègne, Centre de recherche Royallieu - CS, 60 319 - 60 203, Compiègne cedex, France Correspondence Lebanese University, Faculty of Science - I, Mathematics Department, Hadath, Lebanon. Email: hsnibrahim81@gmail.com Funding information Lebanese University 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 [8], 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. KEYWORDS boundary value problems, comparison principle, parabolic systems AMS CLASSIFICATION 35K50, 35K40, 35K55 1 INTRODUCTION Dislocations are line defects, or microscopic irregularity inside materials were the atoms are out of position in the crystalline structure. They represent a non-stationary phenomena when stress fields are applied. The typical thickness of a dislocation line is of order of 10 -9 m and its typical length is of order of 10 -6 m. Mathematically, this theory was originally developed by Volterra [21] in 1907. In 1934, Orowan, [18] Polanyi [19] and Taylor [20] introduced the principal explanation of macroscopic plastic deformation in crystals as a result of the motion of dislocations. The first direct observation of dislocations using electron microscope goes back to Hirsch, Horne, and Whelan [11] and Bollmann [1] in 1956. For a recent study of the physical theory of dislocations, we refer the reader to Nabaro [17] and Hirth and Lothe. [10] However, for a mathematical study of various models on dislocations, we refer to [2–5,16]. In our work, we are interested in the mathematical study of a model on dislocation densities developed by Groma, Czikor, and Zaiser (GCZ for short) in [8] that describes the short-range interactions. This model is written as a coupled system of nonlinear parabolic equations on a bounded domain in one-space dimension. Several variants of the GCZ models have been treated in the work of [6,7,12–14] where particular assumptions on the exterior stress field have been involved. In fact, in their work, they only considered either constant or bounded space-time dependent stresses. This case was far from the real stress in [8] which scales as a nonlinear term (that we precise later) depending on the dislocation density. To our knowledge, our result is the first theoretical one of its type, treating the real physical stress field where we overcome the nonlinear difficulty. Our main result is to prove the short time existence and uniqueness using a fixed point argument and a comparison principle on the gradient of the solution. This paper is organized as follows. Section 2 is devoted to the derivation and the general setting of our problem. The principal notations and results are detailed in Section 3. A comparison principle, on the gradient of the solution, is given in Section 4. In Section 5, we use a fixed point argument on convenient spaces to establish the short time existence and uniqueness. We end up in Section 6 with some conclusions. Z Angew Math Mech. 2018;1–18. © 2018 WILEY-VCH Verlag GmbH &Co. KGaA, Weinheim 1 www.zamm-journal.org