  We study nonlocal first-order equations arising in the theory of
dislocations. We prove the existence and uniqueness of the solutions of these
equations in the case of positive and negative velocities, under suitable
regularity assumptions on the initial data and the velocity. These results are
based on new $L^1$-type estimates on the viscosity solutions of first-order
Hamilton-Jacobi Equations appearing in the so-called ``level-sets approach''.
Our work is inspired by and simplifies a recent work of Alvarez, Cardaliaguet
and Monneau.
