Existence of solutions for a model describing the dynamics of junctions between dislocations

Abstract. We study a dynamical version of a multi-phase field model of Koslowski and Ortiz for planar dislocation networks. We consider a two-dimensional vector field which describes phase transitions between constant phases. Each phase transition corresponds to a dislocation line, and the vectorial field description allows the formation of junctions between dislocations. This vector field is assumed to satisfy a non-local vectorial Hamilton-Jacobi equation with non-zero viscosity. For this model, we prove the existence for all time of a weak solution. Key words. Dislocation dynamics, non-local equations, junctions, parabolic system of equa-tions. AMS subject classifications. 35K15, 74K30. 1. Introduction. 1.1. Physical motivation. Dislocations are line defects in crystal, and their motion is at the origin of plastic properties of metals. In these crystals, we can observe self-organised structures, like Frank networks, i.e. networks of dislocations related by junctions. See for instance page 190 in Hull, Bacon [13] for such networks i