Uniqueness issues for evolution equations with density constraints

TL;DR

This paper establishes uniqueness results for evolution equations with density constraints, covering both first-order crowd motion models with monotonic velocities and diffusive models with regularization effects.

Contribution

It provides rigorous proofs of uniqueness under monotonicity assumptions and for diffusive models, extending known results to broader classes of velocity fields.

Findings

Uniqueness of solutions for crowd motion with hard congestion effects.

Uniqueness of solutions for diffusive models with $L^\infty$ velocity fields.

Use of Wasserstein distance and parabolic estimates to establish contraction properties.

Abstract

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.