Abstract: Materials defects such as dislocations are important line defects in crystalline materials and they play essential roles in understanding materials properties like plastic deformation. In this talk, we study the relaxation process of Peierls-Nabarro dislocation model, which is a gradient flow with singular nonlocal energy and double well potential describing how the materials relax to its equilibrium with the presence of a dislocation. The difficulties of this problem rising from bistable profile in R which naturally leads to a singular nonlocal energy. We first perform mathematical validation of the PN models by rigorously establishing the relationship between the PN model in the full space and the reduced problem on the slip plane in terms of both governing equations and energy variations. Then we characterize the omega-limit set, present spectral analysis for nonlocal Schordinger operator and show the dynamic solution to Peierls-Nabarro model will converge exponentially to a shifted steady profile which is uniquely determined.
