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

TL;DR

This paper analyzes the long-term behavior of solutions to the Peierls-Nabarro dislocation model, showing exponential convergence to a unique steady state, advancing understanding of dislocation relaxation in materials.

Contribution

It proves the exponential convergence of dynamic solutions to a shifted steady profile in the Peierls-Nabarro dislocation model, a novel result in dislocation dynamics.

Findings

Solutions converge exponentially to a unique steady profile.

The steady profile is explicitly characterized.

The model captures the relaxation process of dislocations.

Abstract

In this paper 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. We prove the dynamic solution to Peierls-Nabarro model will converge exponentially to a shifted steady profile which is uniquely determined.