  In this paper we present some basic uniqueness results for evolutive
equations under density constraints. First, we develop a rigorous proof of a
well-known result (among specialists) in the case where the spontaneous
velocity field satisfies a monotonicity assumption: we prove the uniqueness of
a solution for first order systems modeling crowd motion with hard congestion
effects, introduced recently by \emph{Maury et al.} The monotonicity of the
velocity field implies that the $2-$Wasserstein distance along two solutions is
$\lambda$-contractive, which in particular implies uniqueness. In the case of
diffusive models, we prove the uniqueness of a solution passing through the
dual equation, where we use some well-known parabolic estimates to conclude an
$L^1-$contraction property. In this case, by the regularization effect of the
non-degenerate diffusion, the result follows even if the given velocity field
is only $L^\infty$ as in the standard Fokker-Planck equation.
