  In this paper, we perform mathematical validation of the Peierls--Nabarro
(PN) models, which are multiscale models of dislocations that incorporate the
detailed dislocation core structure. We focus on the static and dynamic PN
models of an edge dislocation. In a PN model, the total energy includes the
elastic energy in the two half-space continua and a nonlinear potential energy
across the slip plane, which is always infinite. We rigorously establish 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. The
shear displacement jump is determined only by the reduced problem on the slip
plane while the displacement fields in the two half spaces are determined by
linear elasticity. We establish the existence and sharp regularities of
classical solutions in Hilbert space. For both the reduced problem and the full
PN model, we prove that a static solution is a global minimizer in perturbed
sense. We also show that there is a unique classical, global in time solution
of the dynamic PN model.
