Methods and Applications of Analysis

Volume 27 (2020)

Number 2

Long time behavior of dynamic solution to Peierls–Nabarro dislocation model

Pages: 161 – 198

DOI: https://dx.doi.org/10.4310/MAA.2020.v27.n2.a4

Authors

Yuan Gao (Department of Mathematics and Department of Physics, Duke University, Durham North Carolina, U.S.A.)

Jian-Guo Liu (Department of Mathematics and Department of Physics, Duke University, Durham North Carolina, U.S.A.)

Abstract

In this paper we study the relaxation process of the Peierls–Nabarro dislocation model, which is a gradient flow with a singular nonlocal energy and a double well potential describing how the materials relax to its equilibrium with the presence of a dislocation. We prove the dynamic solution to the Peierls–Nabarro model will converge exponentially to a shifted steady profile which is uniquely determined.

Keywords

global attractor, spectral gap, nonlocal Ginzburg Laudau, Omega limit set with vanishing dissipation

2010 Mathematics Subject Classification

35B40, 35Q74, 35R11, 74S25

Received 23 March 2020

Accepted 27 July 2020

Published 20 August 2020