Static/quasi-static/dynamic models of dislocations: wellposedness and exponential convergence to equilibrium

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, I will first talk about the mathematical validation of the static PN models for both straight and curved dislocation line by 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 minimizers. Then we study the relaxation process of dynamic Peierls-Nabarro dislocation model, which is a gradient flow with infinite nonlocal energy and double well potential describing how the materials relax to its equilibrium with the presence of a dislocation. We will show the dynamic solution will converge exponentially to a shifted steady profile which is uniquely determined.