  Making use of $\phi$-mapping topological current method, we discuss the
self-dual vortices in the Abelian Chern-Simons model with two complex scalar
fields. For each scalar field, an exact nontrivial equation with a topological
term which is missing in many references is derived analytically. The general
angular momentum is obtained. The magnetic flux which relates the two scalar
fields is calculated. Furthermore, we investigate the vortex evolution
processes, and find that because of the present of the vortex molecule, these
evolution processes is more complicated than the vortex evolution processes in
the corresponding single scalar field model.
