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 monot
onicity 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.
This paper intents to present the state of art and recent developments of the optimal transportation theory with many marginals for a class of repulsive cost functions. We introduce some aspects of the Density Functional Theory (DFT) from a mathemati
cal point of view, and revisit the theory of optimal transport from its perspective. Moreover, in the last three sections, we describe some recent and new theoretical and numerical results obtained for the Coulomb cost, the repulsive harmonic cost and the determinant cost.
