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 present spectral analysis for nonlocal Schrodinger operator and show the dynamic solution to Peierls-Nabarro model will converge exponentially to a shifted steady profile which is uniquely determined.
Biography:
Dr. Yuan Gao obtained her Ph.D. from Fudan University in 2017 with dissertation "Some Nonlinear Evolution Equations in Material Science with Dissipative Structures" under supervising by Prof. Ti-Jun Xiao at Fudan University and Prof. Jian-Guo Liu at Duke University. Currently she is a William W. Elliott Assistant Research Professor at Duke University. Using calculus of variation, gradient flows, operator theory and control theory, she works on diverse PDEs including degenerate parabolic equation, nonlocal Allen- Cahn equation arising from models in material science and surface science. She has published in some top journals including SIAM J. Math. Anal., J. Nonlinear Sci. and SIAM J. Control Optim., and several other journals. She is an active member of applied mathematics and materials science community.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai