  We study a mathematical model describing the dynamics of dislocation
densities in crystals. This model is expressed as a one-dimensional system of a
parabolic equation and a first order Hamilton-Jacobi equation that are coupled
together. We show the existence and uniqueness of a viscosity solution among
those assuming a lower-bound on their gradient for all time including the
initial data. Moreover, we show the existence of a viscosity solution when we
have no such restriction on the initial data. We also state a result of
existence and uniqueness of an entropy solution of the system obtained by
spatial derivation. The uniqueness of this entropy solution holds in the class
of ``bounded from below'' solutions. In order to prove these results, we use a
relation between scalar conservation laws and Hamilton-Jacobi equations, mainly
to get some gradient estimates. This study takes place on $\R$, and on a
bounded domain with suitable boundary conditions.
