Uniqueness issues for evolution equations with density constraints

Abstract

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 Maury et al. The monotonicity of the velocity field implies that the 2−Wasserstein distance along two solutions is λ-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 ∞ as in the standard FokkerPlanck equation.

On Cross-Diffusion Systems for Two Populations Subject to a Common Congestion Effect

“…First if m < +∞, since ρ 1,m +ρ 2,m solves (5.1), then it is unique and according to Proposition 5.2, p m := F m (ρ 1,m + ρ 2,m ) is in L 2 ((0, T ), H 1 (Ω)). Moreover, we have already shown in Theorem 2.3 that the pressure p ∞ associated to the constraint ρ 1,∞ + ρ 2,∞ 1 is in L 2 ((0, T ), H 1 (Ω)) and, according to [21], (ρ 1,∞ + ρ 2,∞ , p ∞ ) is unique. Then, for m ∈ [1, +∞], ρ i 1,m solves…”

{Euclidean, metric, and Wasserstein} gradient flows: an overview

“…Variants where the motion of the crowd also undergoes diffusion are also studied, as in [38,76] (in particular, [76] provides interesting uniqueness results) as well as variants with several species (see [25]). …”

Crowd motion and evolution PDEs under density constraints

“…However, uniqueness for (1.3) with non-smooth vector fields u is so far only a conjecture, and the only result in this direction is the one obtained in the second part of [22], about the diffusive case (σ > 0 in (2.5)):…”

“…If u satisfies (u(x) − u(y)) · (x − y) ≤ −λ|x − y| 2 (which is the case if u = −∇V and D 2 V ≥ λI, but also if u is (−λ)-Lispchitz), the first term gives −λW 2 2 ( 1 t , 2 t ); for the second, we havep 1 t > 0 ⇒ 1 t = 1, 2 t ≤ 1 ⇒ det(I −D 2 ϕ) = det(DT ) ≥ 1 ⇒ ∆ϕ ≤ 0,and this rest is negative (the term with p 2 t can be dealt with in a similar manner). This computation (made precise in[22]) allows to prove the following result.…”

“…The idea to write such a survey comes from the need to clarify and summarize the evolution of the theory since [34]. For instance, we now understand better uniqueness issues, thanks in particular to [22], but we are also able to formulate clear conjectures on this matter. In what concerns existence and approximation, [34] was only concerned with the gradient-flow case; this is an important setting but not the only reasonable one: we will see how the gradient structure simplifies the setting but the PDE can also be studied without it.…”