Uniqueness issues for evolution equations with density constraints

In this paper, we present some basic uniqueness results for evolution 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 Maury et al. The monotonicity of the velocity field implies that the [Formula: see text]-Wasserstein distance along two solutions is [Formula: see text]-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 [Formula: see text]-contraction property. In this case, by the regularization effect of the nondegenerate diffusion, the result follows even if the given velocity field is only [Formula: see text] as in the standard Fokker–Planck equation.