  In the well-known Kohn-Sham theory in Density Functional Theory, a fictitious
non-interacting system is introduced that has the same particle density as a
system of $N$ electrons subjected to mutual Coulomb repulsion and an external
electric field. For a long time, the treatment of the kinetic energy was not
correct and the theory was not well-defined for $N$-representable particle
densities. In the work of [Hadjisavvas and Theophilou, Phys. Rev. A, 1984, 30,
2183], a rigorous Kohn-Sham theory for $N$-representable particle densities was
developed using the Levy-Lieb functional. Since a Levy-Lieb-type functional can
be defined for Current Density Functional Theory formulated with the
paramagnetic current density, we here develop a rigorous $N$-representable
Kohn-Sham approach for interacting electrons in magnetic field. Furthermore, in
the one-electron case, criteria for $N$-representable particle densities to be
$v$-representable are given.
