Existence and Uniqueness Results to a System of Hamilton–Jacobi Equations with Application to Dislocation Dynamics

We study the existence and uniqueness of a nonlinear system of eikonal equations in one space dimension for any BV initial data. We present two results. In the first one, we prove the existence of a discontinuous viscosity solution without any monotony conditions neither on the velocities nor on the initial data. In the second, we show the continuity of the constructed solution under continuous initial data, and continuous velocities verifying a certain monotony condition. We present an application to a system modeling the dynamics of dislocations densities