Simone Di

Alp Ár

Rich Árd

M Ész Áros

UNIQUENESS ISSUES FOR EVOLUTION EQUATIONS WITH DENSITY CONSTRAINTS

Keywords: crowd motion models, uniqueness results, density constraints, optimal transport, quasilinear elliptic-parabolic system. Mathematics Subject Classification: 35A02, 35K61, 35F25, 49J45

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 Fokker-Planck equation.

Introduction and preliminaries

Recently, modeling crowd behavior has received a lot of attention in applied mathematics. These models actually are in the heart of many other ones coming from biology (for instance cell migration, tumor growth, pattern formations in animal populations, etc.), particle physics and economics (see for example the recently introduced models of Mean Field Games, [START_REF] Lasry | Jeux à champ moyen I. Le cas stationnaire[END_REF][START_REF] Lasry | Jeux à champ moyen II. Horizon fini et contrôle optimal[END_REF][START_REF] Lasry | Mean field games[END_REF]). For more details on these models we direct the reader to the non-exhaustive list of works [START_REF] Chalons | Numerical Approximation of a macroscopic model of pedestrian flows[END_REF][START_REF] Colombo | Pedestrian Flows and non-classical shocks[END_REF][START_REF] Coscia | First-Order macroscopic modelling of human crowd dynamics[END_REF][START_REF] Cristiani | Multiscale Modeling of Pedestrian Dynamics[END_REF][START_REF] Helbing | A fluid dynamic model for the movement of pedestrians[END_REF][START_REF] Helbing | Social force model for pedestrian dynamics[END_REF][START_REF] Henderson | The statistics of crowd fluids[END_REF][START_REF] Hughes | A continuum theory for the flow of pedestrian[END_REF][START_REF] Hughes | The flow of human crowds[END_REF]. In all these models the question of congestion can play a crucial role. Indeed, from the modeling point of view one could have some 'singularities' if individuals want to occupy the same spot. In this paper, we will work with equations which model some type of congestion effects in crowd motion models (for a more detailed description of these models we direct the reader to the references [START_REF] Maury | A macroscopic crowd motion model of gradient flow type[END_REF][START_REF] Maury | Handling congestion in crowd motion modeling[END_REF][START_REF] Mészáros | Advection-diffusion equations with density constraints[END_REF]). These systems read as

     ∂ t ρ t -ν∆ρ t + ∇ • (ρ t v t ) = 0 ρ t ≤ 1, ρ| t=0 = ρ 0 v t = P adm(ρt) [u t ].
In the above system ρ t represents the density of a crowd (at time t) that moves in Ω ⊂ R d for a time T > 0 accordingly to the prescribed velocity u t , a given field that everyone would follow in the absence of the others. ν ∈ {0, 1} is just a parameter: if ν = 0, one deals with first order systems (the density of the population is just transported by some vector field) introduced in [START_REF] Maury | A macroscopic crowd motion model of gradient flow type[END_REF], while for ν = 1 one has a second order system (in addition to the transport, the population is also affected by some randomness, modeled by a non-degenerate diffusion) studied in [START_REF] Mészáros | Advection-diffusion equations with density constraints[END_REF]. In order to preserve a density constraint (we suppose that ρ t does not exceed a given threshold, let us say 1), at each moment one modifies u t to v t , a field that is the closest to u t (in the L 2 sense) and it is an admissible velocity field, i.e. the set adm(ρ t ) represents the fields that do not increase the density on already saturated zones {ρ t = 1}.

In generic congested models a pressure is also acting (Darcy's law): this pressure is preventing congestion and, according to various models, it is an increasing function of the density. In the usual soft congestion models we have some power law of the type p = ρ m (porous medium equation). However this choice cannot prevent ρ to be very high, while it is clear that a crowd of people cannot have a density higher than a certain threshold (studies say that the maximum density is 4.5 people/m 2 ), which we put for convenience equal to 1: we will refer to the constraint ρ ≤ 1 as an hard congestion effect. We shall present later how the pressure field appears in the models considered in this paper.

Introducing general multivalued monotone operators β, the above systems can be written in a compact form as (1.1)

     ∂ t ρ t + ∇ • (u t ρ t ) = ∆p t in (0, T ) × Ω (u t ρ t -∇p) • n = 0 on (0, T ) × ∂Ω p t ∈ β(ρ t ).
The two cases ν = 0 and ν = 1 correspond to two special operators β 1 and β 2 , namely

β 1 (ρ) =      0 if ρ < 1 [0, +∞] if ρ = 1 +∞ if ρ > 1, β 2 (ρ) =      ρ if ρ < 1 [1, +∞] if ρ = 1 +∞ if ρ > 1;
notice in particular that β 2 (ρ) = β 1 (ρ) + ρ. The hard congestion effect is due to the fact that β i (ρ) = +∞ whenever ρ > 1: in fact, this will force ρ to be always less than 1. It is worthwhile noticing that this problem has some features in common with the Hele-Shaw model and the Stefan problem (see for instance [START_REF] Igbida | Hele-Shaw type problems with dynamical boundary conditions[END_REF][START_REF] Igbida | Renormalized solution for Stefan type problems: existence and uniqueness[END_REF]), namely in both problems there is a degenerate monotone operator linking the density and the diffusive part. However there are also big differences: in our case we treat also the convection term, which can depend not smoothly on the position, while usually in the Hele-Shaw models there is a source term for the mass, which we do not treat here since we want to model a crowd moving inside a domain Ω. This modeling assumption is also the reason why we consider the no-flux Neumann boundary condition, since we want neither people exiting nor entering the domain.

Moreover our equations can be seen also as a quasilinear elliptic-parabolic system with very degenerate nonlinearity (see [START_REF] Otto | L 1 -contraction and uniqueness for quasilinear elliptic-parabolic equations[END_REF][START_REF] Igbida | Uniqueness for nonlinear degenerate problem[END_REF][START_REF] Carrillo | Entropy Solution for Nonlinear Degenerate Problems[END_REF]), for which the issue of uniqueness is still an open problem when we add a driving vector field without imposing an entropy condition. In this context we may write our equations using the variable u = p + ρ as

∂ t g(u) + ∇ • (Φ(t, x, u)) = ∆b i (u), where b 1 (u) = (u -1) + , g(u) = u -b 1 (u), b 2 (u) = u and Φ(t, x, u) = u t (x)g(u).
In the case i = 1 we have a double degeneracy since both b 1 and g have a flat part, instead for the case i = 2 we have only a degenerate part in g(u). However again the (possibly) rough coefficient in Φ rules out results already present in the literature. For problems of this form we expect an L 1 -contraction result, see for example [START_REF] Carrillo | Entropy Solution for Nonlinear Degenerate Problems[END_REF]; notice also that we do not need a concept of entropic solution since the equation reduces to a linear one in the joint variable (ρ, p) (see the systems (1.2) below). Thus the usual concept of weak solutions can be considered for our purposes.

In fact, we will be dealing always with the following reformulations of the systems, using the fact that p ∈ β(ρ) if and only if ρ ≤ 1 and

p ∈ H 1 ρ (Ω) (see its definition in (1.3)): (1.2)      ∂ t ρ t + ∇ • (u t ρ t ) = ∆p t in (0, T ) × Ω (u t ρ t -∇p) • n = 0 on (0, T ) × ∂Ω ρ t ≤ 1, p t ∈ H 1 ρt (Ω).      ∂ t ρ t + ∇ • (u t ρ t ) = ∆(p t + ρ t ) in (0, T ) × Ω (u t ρ t -∇p -∇ρ t ) • n = 0 on (0, T ) × ∂Ω ρ t ≤ 1, p t ∈ H 1 ρt (Ω).
In Subsection 1.1 we will derive in another way these systems: that derivation justifies also the regularity assumption of p t .

A very powerful tool to attack these type of macroscopic hard-congestion problems -where we impose a density constraint on the density of the population -is the theory of optimal transport (see [START_REF] Villani | Topics in Optimal Transportation[END_REF][START_REF] Santambrogio | Optimal Transport for Applied Mathematicians[END_REF]), as we can see in the recent works [START_REF] Maury | A macroscopic crowd motion model of gradient flow type[END_REF][START_REF] Maury | Congestion-driven dendritic growth, Discrete Contin[END_REF][START_REF] Maury | Handling congestion in crowd motion modeling[END_REF][START_REF] Mészáros | Advection-diffusion equations with density constraints[END_REF][START_REF] Alexander | Quasi-static evolution and congested crowd transport[END_REF]. In this framework, the density of the agents satisfies a continuity, or a Fokker-Planck equation (with a velocity field taking into account the congestion effects) and can be seen as a curve in the Wasserstein space.

Our aim in this paper is to prove some basic results of uniqueness of the solutions in this setting. As far as we are aware of, this question is a missing puzzle in full generality in the models studied in [START_REF] Maury | A macroscopic crowd motion model of gradient flow type[END_REF][START_REF] Maury | Congestion-driven dendritic growth, Discrete Contin[END_REF][START_REF] Maury | Handling congestion in crowd motion modeling[END_REF][START_REF] Mészáros | Advection-diffusion equations with density constraints[END_REF]. Let us remark that the uniqueness question is crucial if one wants to include this type of models into a larger system and one aims to show existence results by fixed point methods, as it is done for Mean Field Games in general for instance (for example as in [START_REF] Porretta | Weak Solutions to Fokker-Planck Equations and Mean Field Games[END_REF]).

We will treat two different cases, with very different approaches: in the first one we simply consider a crowd driven by a given velocity field and subject to a density constraint. In this case the assumption that the velocity field is monotone will be crucial in order to prove a λ-contraction result for the solutions, that will imply uniqueness. In the second case we add a diffusive term, which models some randomness in the crowd movement (see [START_REF] Mészáros | Advection-diffusion equations with density constraints[END_REF] for recent developments and existence results in this setting); in this case we prove a standard L 1 -contraction property passing to the dual problem and proving there existence for sufficiently generic data. In this case a major role is played by the regularizing effect of the Laplacian, which allows us to prove the result even if the velocity field is merely bounded.

Let us underline that the core of both methods is classical in the literature.

(i) On one hand we use the differentiation of the squared Wasserstein distance along two solutions of continuity-type equations, and then use a Grönwall-type argument to show a contractive property and hence uniqueness of solutions for these evolution equations. This type of proof is very common in the optimal transport theory; nevertheless our analysis requires some finer nontrivial new ideas, mainly because of the appearance of the new pressure variable which we can formally identify as a term associated to the subdifferential of the indicatrix function of the (geodesically convex) set {ρ ∈ P(Ω) : ρ ≤ 1}. Moreover, we believe that by the nature of our problem the machinery of the L 1 -contraction through the doubling and re-doubling of the variables (successfully used for instance in [START_REF] Otto | L 1 -contraction and uniqueness for quasilinear elliptic-parabolic equations[END_REF] and [START_REF] Carrillo | Entropy Solution for Nonlinear Degenerate Problems[END_REF]) seems to be very difficult, complicated and heavy to adapt to our situation, while the method proposed by us is simple and elegant. In addition, in this context we give new proofs for some of the used results from the theory of optimal transport. (ii) On the other hand, in the second order case (when we add a non-degenerate diffusion term into the model) the idea to pass to the dual equation to show the uniqueness is very similar to the techniques developed already in [START_REF] Crowley | On the weak solution of moving boundary problems[END_REF] for instance; the method to obtain the L 1 -contraction follows the same lines of [START_REF] Bouillet | Nonuniqueness in L ∞ : an example[END_REF]. However the PDE studied here seems to be new in this context and the result of uniqueness is per se interesting in the theory of crowd motion.

We underline moreover that the two methods used in the two models are mutually exclusive: the W 2 distance along two solutions of the second order model would be contractive if the vector field would be monotone, but we do not require this assumption; the parabolic estimates used for the second model highly rely on the fact that one has a non-degenerate diffusion in the system, which is clearly not the case for the first order model.

We remark also that we expect a L 1 -contraction result also in the first case, since it can be seen as a doubly degenerate quasilinear elliptic-parabolic equation [START_REF] Otto | L 1 -contraction and uniqueness for quasilinear elliptic-parabolic equations[END_REF]. In Section 4 we provide a sketch of this result in the case in which the velocity field is monotone: we underline that we use the uniqueness proved in Section 2.

1.1. The density constraint: admissible velocities and pressures. In order to model crowd movement in the macroscopic setting with hard congestion, we work in a convex bounded domain Ω ⊂ R d such that |Ω| > 1. The evolution of the crowd will be analyzed by the evolution of its density, which is assumed to be a probability measure on Ω. The condition we want to impose is a bound on the density of the crowd, which is considered to be always less than or equal to 1. In particular the set of admissible measures will be denoted by

K 1 := {ρ ∈ P(Ω) : ρ ≤ 1 a.e.}
(here and after we identify ρ with its density with respect to the Lebesgue measure L d ).

As for the velocities, we need to impose that the density is not increasing when it is saturated: informally we would say that v is an admissible velocity for the measure ρ ∈ K 1 if ∇ • v ≥ 0 in the set {ρ = 1} and v • n ≤ 0 on ∂Ω where n is the outward normal. In order to make a rigorous definition we have to introduce the set of pressures:

(1.3) H 1 ρ (Ω) := {p ∈ H 1 (Ω) : p ≥ 0, p(1 -ρ) = 0 a.e.}.
Then, using the integration by parts formula, formally for p ∈ H 1 ρ (Ω) and v admissible one should have 0

≤ ˆΩ p∇ • v dx - ˆ∂Ω pv • n dH d-1 = -ˆΩ v • ∇p dx
and so we can define

adm(ρ) := v ∈ L 2 (Ω; R d ) : ˆΩ v • ∇p dx ≤ 0 for every p ∈ H 1 ρ (Ω) .
In the sequel we will denote by P adm(ρ) the L 2 (Ω, L d ; R d )-projection onto the cone adm(ρ). Now, in order to preserve the constraint ρ ≤ 1, we impose that the velocity always belongs to adm(ρ) and so a generic evolution equation with density constraint will be

∂ t ρ t + ∇ • (ρ t v t ) = 0 ρ t ≤ 1, v t ∈ adm(ρ t ).
One of the simplest such model is when we have a prescribed time-dependent velocity field u t and we want to impose that the velocity v t is the nearest possible to u t , time by time. This describes a situation where the crowd wants to have the velocity u t but it cannot, because of the density constraint, and so it adapts its velocity trying to deviate as little as possible: this will result in an highly nonlocal and discontinuous effect. The first order problem hence reads as

(1.4)      ∂ t ρ t + ∇ • (ρ t v t ) = 0 in (0, T ) × Ω ρ t ≤ 1, ρ| t=0 = ρ 0 v t = P adm(ρt) [u t ],
where the first equation is meant in the weak sense and the minimal hypothesis in order to have a well defined projection is u ∈ L 2 ([0, T ] × Ω). In the following lemma we characterize the projection of the velocity field:

Lemma 1.1. Let ρ ∈ P(Ω) such that ρ ≤ 1 a.e. and let u ∈ L 2 (Ω; R d ). Then there exists p ∈ H 1 ρ (Ω) such that P adm(ρ) [u] = u -∇p. Furthermore p is characterized by (i) ˆΩ ∇p • (u -∇p) dx = 0; (ii) ˆΩ ∇q • (u -∇p) dx ≤ 0, for all q ∈ H 1 ρ (Ω). Proof. Let us set K = {∇p : p ∈ H 1 ρ (Ω)}. It is easy to see that K is a closed cone in L 2 (Ω; R d ).
Let us recall that the polar cone to K is defined as

K o := v ∈ L 2 (Ω; R d ) : ˆΩ v • ∇q dx ≤ 0, ∀ ∇q ∈ K .
By the definition of the admissible velocities we have adm(ρ) = K o . Moreau decomposition applied to K and K o gives

u = P K [u] + P K o [u] ∀u ∈ L 2 (Ω; R d ).
This proves the claims (i) and (ii), since these are precisely the conditions of being the projection onto a cone.

Corollary 1.2. Lemma 1.1 (i) implies that ˆΩ |u| 2 dx = ˆΩ |∇p| 2 dx + ˆΩ |u -∇p| 2 dx
, and so in particular we get

ˆΩ |∇p| 2 dx ≤ ˆΩ |u| 2 dx, ˆΩ |P adm(ρ) [u]| 2 dx ≤ ˆΩ |u| 2 dx.
Using Lemma 1.1 we get that if (ρ, v) is a solution to (1.5) then v t = u t -∇p t . Now, using that ρ t ∇p t = ∇p t , we have that (ρ, p) is a solution to

(1.5) ∂ t ρ t + ∇ • (ρ t u t ) = ∆p t , in (0, T ) × Ω ρ t ≤ 1, p t ∈ H 1 ρt (Ω).
Thus we found again equation (1.2), where we imposed also the no-flux boundary condition, which is the one expected when we model a crowd in a closed room Ω. Now we are ready to give the formal definition of solution to our problem:

Definition 1.1. Let u ∈ L 2 ([0, T ] × Ω) and let ρ 0 ∈ K 1 . Then we define a solution to (1.5) to be a couple (ρ, p) ∈ L ∞ ([0, T ] × Ω) ∩ AC([0, T ]; (P(Ω), W 2 )) × L 2 ([0, T ]; H 1 (Ω))
with ρ| t=0 = ρ 0 and such that:

• for all φ ∈ C ∞ c (R d
) and for all 0 ≤ r < s ≤ T we have

ˆs r ˆΩ(u t (x) -∇p(x)) • ∇φ(x)ρ t (x) dx dt = ˆΩ φ(x)ρ s (x) dx -ˆΩ φ(x)ρ r (x) dx; • we have 0 ≤ ρ ≤ 1 for L 1 [0, T ]⊗L d Ω-a.e. (t, x) and p t ∈ H 1 ρt (Ω) for L 1 [0, T ]-a.e. t.
Remark 1.2. We remark that the boundary condition implied by our definition is homogeneous Neumann. Note that we assume for the density ρ to be also an absolutely continuous curve in the space of probability measures equipped with the Wasserstein distance1 . Like this it is meaningful to assume ρ| t=0 = ρ 0 .

In [START_REF] Maury | Handling congestion in crowd motion modeling[END_REF][START_REF] Maury | A macroscopic crowd motion model of gradient flow type[END_REF] and [START_REF] Roudneff-Chupin | Modélisation macroscopique de mouvements de foule[END_REF] the following regularity hypotheses have been assumed to show the existence result: u ∈ C 1 or u = -∇D for a λ-convex potential D and in both cases no dependence on time.

1.2. The diffusive case. Recently (see [START_REF] Mészáros | Advection-diffusion equations with density constraints[END_REF]) a second order model for crowd motion has been proposed. In a nutshell, it consists of adding a non-degenerate diffusion to the movement and imposing the density constraint. This leads to a modified Fokker-Planck equation and, with the notations previously introduced, it reads as

(1.6)      ∂ t ρ t -∆ρ t + ∇ • (ρ t v t ) = 0 ρ t ≤ 1, ρ| t=0 = ρ 0 v t = P adm(ρt) [u t ],
where u t is -as before -the desired given velocity field of the crowd. Introducing the pressure gradient in the characterization of the projection, one can write system (1.6) as

(1.7) ∂ t ρ t + ∇ • (ρ t u t ) = ∆(p t + ρ t ) ρ t ≤ 1, ρ| t=0 = ρ 0 , p t ∈ H 1 ρt (Ω)
where, as before we equip the equation with the natural no-flux boundary condition on ∂Ω; the rigorous definition of solution is similar to that of Definition 1.1. We notice also the fact that we have in particular p t + ρ t ∈ β 2 (ρ t ) and so we are in fact solving equation (1.1) with β = β 2 .

Under the assumption that Ω is convex and

u ∈ L ∞ ([0, T ] × Ω) it has been shown (see [31, Theorem 3.1]) that the system (1.7) admits a solution (ρ, p) ∈ L ∞ ([0, T ]×Ω)×L 2 ([0, T ]; H 1 (Ω)).
In addition [0, T ] ∋ t → ρ t is an absolutely continuous curve in the 2-Wasserstein space (see Subsection 1.3). We direct the reader to [START_REF] Mészáros | Advection-diffusion equations with density constraints[END_REF] for further details on this model.

Optimal trasport.

Here we collect some facts about optimal transport that will be needed in the sequel. Given X 1 and X 2 two measurable spaces and T a measurable map between them, we say that the Borel measure µ 2 is the push forward of the Borel measure µ 1 through T and we write

µ 2 = T ♯ µ 1 , if µ 2 (A) = µ 1 (T -1 (A)) for every measurable set A ⊆ X 2 .
Given two measures µ ∈ P(X 1 ) and ν ∈ P(X 2 ), we define Π(µ, ν) as the set of γ ∈ P(X 1 × X 2 ) such that (π 1 ) ♯ γ = µ and (π 2 ) ♯ γ = ν, where π i is the projection to the i-th coordinate: these measures are called transport plans between µ and ν.

A particular example of transport plans is given by the transport maps: whenever we have a measurable T : X 1 → X 2 such that T ♯ µ = ν we have that the induced plan γ T = (id, T ) ♯ µ belongs to Π(µ, ν). Let us summarize some well-known facts about optimal transport in the following theorem (see for instance [START_REF] Villani | Topics in Optimal Transportation[END_REF] or [START_REF] Santambrogio | Optimal Transport for Applied Mathematicians[END_REF]).

Theorem 1.3. Let Ω ⊂ R d be an open bounded set and let µ, ν ∈ P(Ω). Let us consider the following quantities

(P) A(µ, ν) = inf ˆΩ×Ω |x -y| 2 dγ : γ ∈ Π(µ, ν) (D) B(µ, ν) = sup ˆΩ ϕ(x) dµ + ˆΩ ψ(x) dν : ϕ, ψ ∈ C b (Ω), ϕ(x) + ψ(y) ≤ 1 2 |x -y| 2 ∀ x, y ∈ Ω ,
where C b (Ω) denotes the space of bounded continuous functions on Ω. We will call (P) the primal and (D) the dual problem.

(i) There exists at least a minimizer for the primal problem (the set of minimizers is denoted by Π o (µ, ν)) and there exists also a maximizer (ϕ, ψ) in the dual problem. (ii) A(µ, ν) = 2B(µ, ν) and we will call W 2 2 (µ, ν) the common value. (iii) We can choose a maximizer (ϕ, ψ) of (D) such that ϕ and ψ are Lipschitz in Ω and also such that 1 2 |x| 2 -ϕ(x) is a convex function on the convex hull of Ω. If µ ≪ L d then its gradient is a map T (x) = x -∇ϕ(x) such that T ♯ µ = ν and whose associated plan is the unique optimal plan, that is Π o (µ, ν) = {γ T }.

(iv) If Ω is convex, P(Ω) endowed with the Wasserstein distance W 2 is a geodesic space and if µ ≪ L d the geodesic between µ and ν is unique and it is described by

[0, 1] ∋ t → µ t := (id + t(T -id)) ♯ µ.
The space of probability measures equipped with the Wasserstein distance W 2 is called the Wasserstein space. We denote it by W 2 := (P(Ω), W 2 ).

Structurally, the following two sections contain our main results. Section 2 is devoted to the uniqueness question for first order models where the main tool is the theory of optimal transport. In Section 3 we investigate the uniqueness issue for second order models, using PDE techniques. Finally, in Section 4 we discuss an approach that could lead to an L 1 -contraction result in the first order case as well.

Monotone vector fields in the first order case

Let Ω ⊂ R d be a bounded convex domain. In this section we suppose that the desired velocity field u : [0, T ] × Ω → R d of the crowd is a monotone vector field in L 2 ([0, T ] × Ω; R d ), i.e. the following assumption is fulfilled: there exists λ ∈ R such that for all t ∈ [0, T ] there exists a Borel measurable set A t ⊆ Ω (possibly depending on t) with full measure, i.e. L d (Ω \ A t ) = 0 and (H1)

(u t (x) -u t (y)) • (x -y) ≤ λ|x -y| 2 , ∀ x, y ∈ A t .
The following contractivity results are not very surprising in the Wasserstein context. In practice we show that (Lemma 2.1)

{-∇p : p ∈ H 1 ρ (Ω)} ⊆ ∂ W 2 I 1 (ρ),
where I 1 the indicatrix function of K 1 and ∂ W 2 denotes the Wasserstein subdifferential (see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]): then we exploit the geodesic convexity of I 1 in order to get the contraction properties. However, in order to let the reader understand clearly the proofs we will omit the W 2 technical language. It would be interesting to adapt these tools to more general Hele-Shaw problems.

Although a first written version of these results is essentially contained in [START_REF] Mészáros | Mean Field Games with density constraints[END_REF] (Section 4.3.1), here we simplified and clarified some of the proofs. A key observation is the following lemma (see also Lemma 4.3.13 in [START_REF] Mészáros | Mean Field Games with density constraints[END_REF]): Lemma 2.1. Let Ω be a convex bounded domain of R d and let ρ 0 , ρ 1 ∈ P(Ω) two absolutely continuous measures such that ρ 0 ≤ 1 and ρ 1 ≤ 1 a.e. Take a Kantorovich potential ϕ from ρ 0 to ρ 1 and p ∈ H 1 ρ 0 (Ω). Then

ˆΩ ∇ϕ • ∇p dx = ˆΩ ∇ϕ • ∇p dρ 0 ≥ 0.
To prove this result we consider the following extra lemma:

Lemma 2.2.
Let Ω be a convex bounded domain of R d and let ρ 0 , ρ 1 ∈ P(Ω) two absolutely continuous measures such that ρ 0 ≤ 1 and ρ 1 ≤ 1 a.e. Take a Kantorovich potential ϕ from ρ 0 to ρ 1 and p ∈ H 1 (Ω). Let [0, 1] ∋ t → ρ t be the geodesic connecting ρ 0 to ρ 1 , with respect to the 2-Wasserstein distance W 2 . Then we have that

d dt {t=0} ˆΩ p dρ t = -ˆΩ ∇ϕ • ∇p dρ 0 .
Proof. We know (using the interpolation introduced by R. McCann, see [START_REF] Mccann | A convexity principle for interacting gases[END_REF] or Theorem 1.3 (iv)) that ρ t = (x -t∇ϕ(x)) ♯ ρ 0 for all t ∈ [0, 1] and so we have

d dt {t=0} ˆΩ p dρ t = lim t→0 ˆΩ p(x -t∇ϕ(x)) -p(x) t dρ 0 (x) = -lim t→0 ˆΩ 1 t ˆt 0 ∇p(x -s∇ϕ(x)) • ∇ϕ(x) ds dρ 0 (x) = -lim t→0 ˆΩ A t (∇p) • ∇ϕ dρ 0 (x),
where the second equality is easy to prove, for fixed t, by approximation via smooth functions and, for t ∈ [0, 1], we denoted by

A t : L 2 (Ω; R d ) → L 2 ρ 0 (Ω; R d ) the linear operator A t (h)(x) = 1 t ˆt 0 h(x -s∇ϕ(x)) ds.
Now as a general fact we will prove that

A t (h) → h strongly in L 2 ρ 0 (Ω; R d ) as t → 0, for every h ∈ L 2 (Ω; R d ).
First of all it is easy to see that A t ≤ 1: indeed

ˆΩ |A t (h)| 2 dρ 0 ≤ 1 t ˆΩ ˆt 0 |h(x -s∇ϕ(x))| 2 ds dρ 0 (x) = 1 t ˆt 0 ˆΩ |h| 2 dρ s (x) ds ≤ ˆΩ |h| 2 dx.
Here we used the fact that since ρ 0 , ρ 1 ≤ 1 a.e we have also ρ t ≤ 1 a.e. for all t ∈ [0, 1]. Now it is sufficient to note that for every ε > 0 there exists a Lipschitz function h ε such that h ε -h L 2 ≤ ε, and so we have

A t (h) -h L 2 ρ 0 ≤ A t (h -h ε ) L 2 ρ 0 + h -h ε L 2 ρ 0 + A t (h ε ) -h ε L 2 ρ 0 ≤ 2ε + tL ∇ϕ L 2 ρ 0
, where L is the Lipschitz constant of h ε . Taking now the limit as t goes to 0 we obtain lim sup t→0 A t (h) -h L 2 ρ 0 ≤ 2ε; by the arbitrariness of ε > 0 we conclude. Now it is easy to finish the proof, since ∇p ∈ L 2 (Ω) and so

d dt {t=0} ˆΩ p dρ t = -lim t→0 ˆΩ A t (∇p) • ∇ϕ dρ 0 = -ˆΩ ∇p • ∇ϕ dρ 0 .
Proof of Lemma 2.1. Let [0, 1] ∋ t → ρ t be the Wasserstein geodesic between ρ 0 and ρ 1 . We know that ρ t ≤ 1 a.e. for all t ∈ [0, 1] and in particular it is true that ˆΩ p dρ t ≤ ˆΩ p dx = ˆΩ p dρ 0 , which means that the function [0, 1] ∋ t → ˆΩ p dρ t has a local maximum in t = 0, hence its derivative at 0 is non-positive.

Given this, the claim is a consequence of Lemma 2.2.

We will also need a regularity lemma on the continuity equation: by definition a curve ρ t satisfies a continuity equation with velocity v t if for every φ ∈ C 1 0 (R d ) 2 the application [0, T ] ∋ 2 we recall that for us Proof. Let us prove the following general statement: for a given separable Banach space B and a curve x * ∈ L 1 ([0, T ]; B * ), there exists Υ ⊂ [0, T ] such that Υ is a set of Lebesgue points for the map t → x * t (b) for every b ∈ B, and moreover L 1 ([0, T ] \ Υ) = 0. This can be proven easily by choosing a dense subset (b n ) ⊂ B and then taking Υ n as the set of Lebesgue points of t → x * t (b n ) and Υ 0 as the Lebesgue points of t → x * t , and then we take Υ = n≥0 Υ n . For every b ∈ B and ε > 0 let us consider i ∈ N such that b i -b ≤ ε and then taking t ∈ Υ we have 1 2δ

C 1 0 (R d ) is the closure of C ∞ c (R d ) with respect to the norm φ 1 = φ ∞ + ∇φ ∞ or, equivalently, the set of φ ∈ C 1 (R d ) such that lim |x|→∞ |φ(x)| + |∇φ(x)| → 0 t → ˆRd φ dρ t is
ˆt+δ t-δ |x * s (b) -x * t (b)| ds ≤ ε x * t + 1 2δ ˆt+δ t-δ x * s ds + 1 2δ ˆt+δ t-δ |x * s (b i ) -x * t (b i )| ds.
Now, taking the limit as δ → 0 and using the properties of Υ we get lim sup is the Banach space C 1 0 (R d ) in the first case and H 1 (R d ) in the second case. Notice that if we follow the construction of Υ for the case B = C 1 0 (R d ), this set of times will work also for H 1 (R d ) since C 1 0 is dense in H 1 . Now we are in position to prove the main theorem of this section, namely: Theorem 2.4. Suppose Ω ⊂ R d is a bounded convex domain, u is a vector field satisfying Assumption (H1) and let ρ 0 ∈ K 1 be an admissible initial density. Let us suppose that there exist (ρ 1 , p 1 ), (ρ 2 , p 2 ) two solutions to the system

(2.2)        ∂ t ρ t + ∇ • (ρ t (u t -∇p t )) = 0 in (0, T ) × Ω ρ t ≤ 1, p t ≥ 0, (1 -ρ t )p t = 0 a.e. in [0, T ] × Ω, ρ| t=0 = ρ 0 , p i ∈ L 2 ([0, T ]; H 1 (Ω)
) and ρ i ∈ P(Ω) for i ∈ {1, 2}, where the first equation is supposed to be satisfied in duality with C ∞ c (R d ) (see Definition 1.1) in order to take into account the boundary conditions. Then, ρ 1 = ρ 2 and p 1 = p 2 a.e. In particular, under the same assumptions, we can say that there exists a unique pair (ρ, v) that solves (1.4).

Proof. We associate to the two curves ρ 1 t and ρ 2 t a continuity equation (1.5) with the corresponding vector fields v 1 t = u 1 t -∇p [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]Theorem 8.4.7] for a more general statement, but we prefer to include a simpler proof in this case. We know that t → W 2 2 (ρ 1 t , ρ 2 t ) is absolutely continuous: let us consider a time t for which its derivative exists and also such that t ∈ Υ 1 ∩ Υ 2 , where Υ i is the set for which (2.1) is satisfied for the continuity equation for ρ i . Then we know that, for all s ∈ [0, T ] we have

(2.3) 1 2 W 2 2 (ρ 1 s , ρ 2 s ) ≥ ˆΩ ϕ t dρ 1 s + ˆΩ ψ t dρ 2 s ,
where (ϕ t , ψ t ) is a pair of Kantorovich potentials for ρ 1 t and ρ 2 t . In particular we have equality in (2.3) for t = s and so, since both sides are differentiable by hypothesis, their derivatives are equal. Hence we get d dt

1 2 W 2 2 (ρ 1 t , ρ 2 t ) = ˆΩ v 1 t • ∇ϕ t dρ 1 t + ˆΩ v 2 t • ∇ψ t dρ 2 t ,
where we used the fact that, since Ω is bounded, we can assume ϕ t and ψ t to be Lipschitz and in particular they belong to H 1 , which allows to use Lemma 2.3.

We also know that thanks to Theorem 1.3 there is a pair of optimal transport maps T t (x) = x -∇ϕ t (x) and S t (y) = y -∇ψ t (y) such that (T t ) ♯ ρ 1 t = ρ 2 t and (S t ) ♯ ρ 2 t = ρ 1 t for all t ∈ [0, T ]; moreover T t is the inverse of S t (in the appropriate almost everywhere sense). Using this, we have ∇ψ t (T t (x)) = -∇ϕ t (x) and in particular using the change of variable formula y = T t (x) we get

ˆΩ ∇ψ t (y) • u t (y) dρ 2 t (y) = -ˆΩ ∇ϕ t (x) • u t (T t (x)) dρ 1 t (x).
We can use this to split the formula for the derivative of W 2 2 (ρ 1 t , ρ 2 t )/2. We use the result of Lemma 2.1 and then rewrite the term regarding u t in terms of transport maps and see:

d dt 1 2 W 2 2 (ρ 1 t , ρ 2 t ) = ˆΩ v 1 t • ∇ϕ t dρ 1 t + ˆΩ v 2 t • ∇ψ t dρ 2 t = ˆΩ u t • ∇ϕ t dρ 1 t + ˆΩ u t • ∇ψ t dρ 2 t -ˆΩ ∇p 1 t • ∇ϕ t dρ 1 t -ˆΩ ∇p 2 t • ∇ψ t dρ 2 t ≤ ˆΩ ∇ϕ t (x) • u t (x) -u t (T t (x)) dρ 1 t ≤ ˆΩ(x -T t (x)) • u t (x) -u t (T t (x)) dρ 1 t ≤ λ ˆΩ |x -T t (x)| 2 dρ 1 t ≤ λW 2 2 (ρ 1 t , ρ 2 t ).
Grönwall's lemma implies that

W 2 2 (ρ 1 t , ρ 2 t ) ≤ e 2λt W 2 2 (ρ 1 0 , ρ 2 0 ). Since ρ 1 0 = ρ 2 0 = ρ 0 a.e.
, the above property implies that ρ 1 = ρ 2 a.e. in [0, T ] × Ω. From this fact we can easily deduce that ∆(p 1 t -p 2 t ) = 0, for a.e. t ∈ [0, T ] in the sense of distributions. In particular p 1 t -p 2 t is analytic in the interior of Ω. Moreover, both p 1 t and p 2 t vanish a.e. in the set {ρ 1 t < 1} which has a positive Lebesgue measure greater than |Ω| -1 > 0. Thus, p 1 = p 2 a.e. in [0, T ] × Ω. The claim follows.

Remark 2.1. The existence result for system (2.2) was obtained in different settings in the literature. On the one hand, if u = -∇D (for a reasonably regular potential D), the existence and uniqueness of a pair (ρ, p) can be obtained by gradient flow techniques in W 2 (Ω) (see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Maury | A macroscopic crowd motion model of gradient flow type[END_REF][START_REF] Roudneff-Chupin | Modélisation macroscopique de mouvements de foule[END_REF]), under the assumption that u is monotone that translates into a λ-convexity for D. On the other hand, if u is a general field with C 1 regularity, the existence result is proven with the help of a well-chosen splitting algorithm (see [START_REF] Maury | Handling congestion in crowd motion modeling[END_REF][START_REF] Roudneff-Chupin | Modélisation macroscopique de mouvements de foule[END_REF]).

Nevertheless, combining the techniques developed in [START_REF] Mészáros | Advection-diffusion equations with density constraints[END_REF] on the one hand, and the well-known DiPerna-Lions-Ambrosio theory on the other hand, we expect to obtain existence result for (2.2) for more general vector fields with merely Sobolev regularity and suitable divergence bounds.

Remark 2.2. The monotonicity assumption (H1) is not surprising in this setting. We remark that the same assumption was required in [START_REF] Natile | Contraction of general transportation costs along solutions to Fokker-Planck equations with monotone drifts[END_REF] to prove contraction properties for a general class of transport costs along the solution of the Fokker-Planck equation in R d

∂ t ρ -∆ρ + ∇ • (Bρ) = 0, ρ| t=0 = ρ 0 ,
where the velocity field B : R d → R d was supposed to satisfy the monotonicity property (H1).

We note also the fact that we can allow for moving domains Ω t (considering always a no-flux boundary condition): in fact our proof never uses that the domains are fixed but uses only convexity at any fixed time. This generalization has been used in [START_REF] Marino | Measure sweeping processes[END_REF] for proving uniqueness for an evolution equation with density constraint driven only by the boundary of the moving sets.

Bounded vector field in the diffusive case

We use Hilbert space techniques (similarly to the one developed in [START_REF] Crowley | On the weak solution of moving boundary problems[END_REF][START_REF] Bouillet | Nonuniqueness in L ∞ : an example[END_REF][START_REF] Perthame | The Hele-Shaw asymptotics for mechanical models of tumor growth[END_REF]; see also Section 3.1. from [START_REF] Porretta | Weak Solutions to Fokker-Planck Equations and Mean Field Games[END_REF]) to study the uniqueness of a solution of the diffusive crowd motion model with density constraints described in Subsection 1.2 (see also [START_REF] Mészáros | Advection-diffusion equations with density constraints[END_REF]). Moreover we can expect that this holds under more general assumptions in the presence of a non-degenerate diffusion in the model.

Let u : [0, T ] × Ω → R d be a given vector field, which represents again the desired velocity field of the crowd, Ω ⊂ R d a bounded open set with C 1 boundary, ρ 0 ∈ P(Ω) the initial density of the population such that 0 ≤ ρ 0 ≤ 1 a.e. in Ω and let us consider the following problem

(3.1) ∂ t ρ t -∆ρ t + ∇ • (P adm(ρt) [u t ]ρ t ) = 0, in (0, T ) × Ω; ρ| t=0 = ρ 0 , 0 ≤ ρ t ≤ 1, a.e. in Ω,
equipped with the natural no flux boundary condition. Introducing the pressure variable, equivalently the above system can be written as

(3.2)          ∂ t ρ t -∆ρ t -∆p t + ∇ • (u t ρ t ) = 0, in (0, T ) × Ω, ρ| t=0 = ρ 0 , in Ω (∇ρ t + ∇p t -u t ρ t ) • n = 0, on ∂Ω, for a.e. t ∈ [0, T ],
for a pressure field p t ∈ H 1 ρt (Ω). It has been shown in [START_REF] Mészáros | Advection-diffusion equations with density constraints[END_REF] that under the assumption that

(H2) u ∈ L ∞ ([0, T ] × Ω; R d )
the systems (3.1) and (3.2) have a solution. More precisely there exist an absolutely continuous curve

[0, T ] ∋ t → ρ t ∈ W 2 and p t ∈ H 1 ρt (Ω) for a.e. t ∈ [0, T ] (in particular ρ ∈ L ∞ ([0, T ] × Ω) and p ∈ L 2 ([0, T ]; H 1 (Ω)) such that (p, ρ) solves (3.2) in weak sense (see (3.3)).
Our aim in this section is to show the following theorem:

Theorem 3.1. Let u satisfy (H2). Then there exists a unique pair

(ρ, p) ∈ L ∞ ([0, T ] × Ω) × L 2 ([0, T ]; H 1 (Ω)) that solves (3.
2) in the weak sense (3.3). Moreover for every solution (ρ 1 , p 1 ) and (ρ 2 , p 2 ) to (3.2) we have the L 1 -contraction property

ˆΩ |ρ 1 (T, x) -ρ 2 (T, x)| dx ≤ ˆΩ |ρ 1 (0, x) -ρ 2 (0, x)| dx.
Proof. The existence of a solution (ρ, p) was obtained in [START_REF] Mészáros | Advection-diffusion equations with density constraints[END_REF]. Now let us show the uniqueness of the solution via an L

I(φ, T ) = I(φ, 0) + ˆT 0 ˆΩ (ρ 1 -ρ 2 )∂ t φ + (ρ 1 -ρ 2 + p 1 -p 2 )∆φ + (ρ 1 -ρ 2 )u • ∇φ dx dt,
where I(φ, t) = ˆΩ φ(t, x)[ρ 1 (t, x) -ρ 2 (t, x)] dx. We introduce the following quantities

A := ρ 1 -ρ 2 (ρ 1 -ρ 2 ) + (p 1 -p 2 )
and B :=

p 1 -p 2 (ρ 1 -ρ 2 ) + (p 1 -p 2 )
.

Note that 0 ≤ A ≤ 1 and 0 ≤ B ≤ 1 a.e. in [0, T ] × Ω and A + B = 1. To be consistent with these bounds, we set A = 0 when ρ 1 = ρ 2 , even if p 1 = p 2 and B = 0 when p 1 = p 2 , even if ρ 1 = ρ 2 . With these notations the weak formulation for the difference gives Let us remark that if we are able to find a (reasonably regular) solution φ for this problem for any θ smooth then, using the maximum principle and then optimizing in θ, we would get an L 1 -contraction result for ρ: in particular we get uniqueness for the initial value problem for ρ and hence also for p (as done in the end of the proof of Theorem 2.4). However, since the coefficients in (3.6) are not regular, we study a regularized problem. For ε > 0 let us consider A ε , B ε , u ε to be smooth approximations of A, B and u such that

(3.7) A -A ε L r ([0,T ]×Ω) + B -B ε L r ([0,T ]×Ω) + u -u ε L r ([0,T ]×Ω) < ε, , ε < A ε , B ε ≤ 1,
for some 1 ≤ r < +∞, the value of which to be chosen later. The regularized problem reads as follows

(3.8) ∂ t φ ε + (1 + B ε /A ε )∆φ ε + u ε • ∇φ ε = 0, in (0, T ) × Ω, ∇φ ε • n = 0 a.e. on [0, T ] × ∂Ω, φ ε (T, •) = θ in Ω.
For all ε > 0 the above problem is uniformly parabolic and B ε /A ε is continuous and positive. Moreover θ is smooth, thus by classical results (see for instance [START_REF] Ladyzenskaja | Linear and quasilinear equations of parabolic type, Translated from the Russian[END_REF][START_REF] Krylov | Lectures on elliptic and parabolic equations in Sobolev spaces[END_REF]) the problem has a (unique) solution φ ε ∈ C 1 ([0, T ] × Ω). In particular φ ε can be used as test function in (3.3). In the followings we shall use some standard uniform estimates (in ε) on φ ε given in Lemma 3.2. In particular, using φ ε as test function in (3.5) one has

I(φ ε , T ) -I(φ ε , 0) = ˆT 0 ˆΩ(ρ 1 -ρ 2 + p 1 -p 2 ) [A∂ t φ ε + (A + B)∆φ ε + Au • ∇φ ε ] dx dt = ˆT 0 ˆΩ(ρ 1 -ρ 2 + p 1 -p 2 ) [A∂ t φ ε + (A + B)∆φ ε + Au • ∇φ ε ] dx dt - ˆT 0 ˆΩ(ρ 1 -ρ 2 + p 1 -p 2 )A [∂ t φ ε + (1 + B ε /A ε )∆φ ε + u ε • ∇φ ε ] dx dt = ˆT 0 ˆΩ(ρ 1 -ρ 2 + p 1 -p 2 )(B ε /A ε )(A ε -A)∆φ ε dx dt + ˆT 0 ˆΩ(ρ 1 -ρ 2 + p 1 -p 2 )(B -B ε )∆φ ε dx dt + ˆT 0 ˆΩ(ρ 1 -ρ 2 + p 1 -p 2 )A(u ε -u) • ∇φ ε dx dt := I 1 ε + I 2 ε + I 3 ε Let us show that |I 1 ε | → 0, |I 2 ε | → 0 and |I 3 ε | → 0 as ε → 0. More precisely, first let us recall that 0 ≤ ρ 1 , ρ 2 ≤ 1 a.e. in [0, T ] × Ω, hence ρ 1 , ρ 2 ∈ L ∞ ([0, T ] × Ω). On the other hand p 1 , p 2 ∈ L 2 ([0, T ]; H 1 (Ω)) and
p i ∈ L ∞ ([0, T ]; L q (Ω)), i ∈ {1, 2} for all 1 ≤ q ≤ 2 * . Let us fix q ∈ (2, 2 * ), where 2 * = 2d/(d -2) if d ≥ 3 and 2 * = ∞ if d = 2.
This implies the following estimates

|I 1 ε | ≤ ρ 1 -ρ 2 L ∞ ([0,T ]×Ω) • (B ε /A ε ) 1/2 (A -A ε ) L 2 ([0,T ]×Ω) • (B ε /A ε ) 1/2 ∆φ ε L 2 ([0,T ]×Ω) + ˆT 0 p 1 -p 2 L q (Ω) • (B ε /A ε ) 1/2 (A -A ε ) L r (Ω) • (B ε /A ε ) 1/2 ∆φ ε L 2 (Ω) dt ≤ C(1/ε) 1/2 ε + p 1 -p 2 L ∞ (L q ) • (B ε /A ε ) 1/2 (A -A ε ) L 2 (L r ) • (B ε /A ε ) 1/2 ∆φ ε L 2 (L 2 ) ≤ Cε 1/2 → 0, as ε → 0 and similarly |I 2 ε | ≤ ρ 1 -ρ 2 L ∞ ([0,T ]×Ω) • (A ε /B ε ) 1/2 (B -B ε ) L 2 ([0,T ]×Ω) • (B ε /A ε ) 1/2 ∆φ ε L 2 ([0,T ]×Ω) + p 1 -p 2 L ∞ (L q ) • (A ε /B ε ) 1/2 (B -B ε ) L 2 (L r ) • (B ε /A ε ) 1/2 ∆φ ε L 2 (L 2 ) ≤ C(1/ε) 1/2 ε = Cε 1/2 → 0, as ε → 0; finally we have |I 3 ε | ≤ ρ 1 -ρ 2 L ∞ (L q ) • u -u ε L 1 (L r ) • ∇φ ε L ∞ (L 2 ) ≤ Cε → 0,
where r > 1 is the exponent such that 1 2 + 1 r + 1 q = 1, i.e. r = 2q q -2 (recall that q ∈ (2, 2 * ). We used the approximations (3.7) with this specific r. Hence we obtained that

I(φ ε , T ) = I(φ ε , 0) + o(1);
now we use the fact that we fixed the final condition θ for all ε, and also that |φ ε | ≤ 1, that is implied by the strong maximum principle (see for example Theorem 2.9 in [START_REF] Lieberman | Second order parabolic differential equation[END_REF]); in particular we can write

ˆΩ θ(x) ρ 1 (T, x) -ρ 2 (T, x) dx = ˆΩ φ ε (0, x)(ρ 1 0 -ρ 2 0 ) dx + o(1) ≤ ˆΩ |ρ 1 0 -ρ 2 0 | dx + o(1). Letting ε → 0 we finally find ˆΩ θ(ρ 1 T -ρ 2 T ) dx ≤ ρ 1 0 -ρ 2 0 L 1 ,
and so, optimizing in |θ| ≤ 1, we proved the L 1 -contraction. In particular this implies that ρ 1 = ρ 2 a.e. in [0, T ] × Ω whenever ρ 1 0 = ρ 2 0 . Then one obtains also that p 1 = p 2 a.e. in [0, T ] × Ω, as in the end of the proof of Theorem 2.4. The result follows. Lemma 3.2. Let φ ε be a solution of (3.8). Then there exists a constant

C = C(T, u L ∞ , ∇θ L 2 (Ω) ) > 0
such that we have the following estimates, uniformly in ε > 0:

(i) sup

t∈[0,T ] ∇φ ε (t) L 2 (Ω) ≤ C; (ii) (B ε /A ε ) 1 2 ∆φ ε L 2 ([0,T ]×Ω) ≤ C; (iii) ∆φ ε L 2 ([0,T ]×Ω) ≤ C.
Proof. Let us multiply the first equation in (3.8) by ∆φ ε and integrate over [t, T ] × Ω for 0 ≤ t < T . We obtain

1 2 ∇φ ε (t) 2 L 2 (Ω) + ˆT t ˆΩ(1 + B ε /A ε )|∆φ ε | 2 dx dt = 1 2 ˆΩ |∇θ| 2 dx - ˆT t ˆΩ u ε • ∇φ ε ∆φ ε dx dt (3.9)
Hence by Young's inequality we have

1 2 ∇φ ε (t) 2 L 2 (Ω) ≤ u L ∞ 2δ ˆT t ∇φ ε (s) 2 L 2 (Ω) ds + 1 2 ∇θ 2 L 2 (Ω) ≤ C + C 2 ˆT t ∇φ ε (s) 2 L 2 (Ω) ds
where the term in |∆φ ε | 2 has been absorbed by the left hand side, and 0 < δ ≤ 2/ u L ∞ is a fixed constant and the constant C > 0 is depending just on ∇θ L 2 (Ω) and u L ∞ . Hence by Grönwall's inequality we obtain

1 2 ∇φ ε (t) 2 L 2 (Ω) ≤ Ce C(T -t) ,
which implies in particular that sup

t∈[0,T ] ∇φ ε (t) L 2 (Ω) ≤ C. Thus (i) follows.
On the other hand choosing δ := 2/ u L ∞ in Young's inequality used in (3.9) and using (i), we obtain

ˆT t ˆΩ(B ε /A ε )|∆φ ε | 2 dx dt ≤ C hence (B ε /A ε )
1 2 ∆φ ε L 2 ([0,T ]×Ω) ≤ C, and thus (ii) follows. By (3.9), (i) and (ii) easily imply (iii).

4. About the L 1 -contraction in the first order case

In the previous section we proved uniqueness in the second order case using an L 1 -contraction result. We expect this result to be true also in the first order case; however we expect the treating of this L 1 -contraction problem in the most general framework (let us say when u ∈ W 1,1 ) to be difficult, since this approach should "include" (in the proof) also the well-posedness in the Di Perna-Lions theory as a special case. In fact, when one has ∇ • u = 0 and if one starts with 0 ≤ 1 then this condition is preserved without adding the pressure term.

However in the case treated in Section 2 (i.e. with a monotone velocity field) we can try to sketch a proof of the L 1 contraction result using the uniqueness already proved: let us approximate the solution discretizing in time using the splitting discrete scheme "continuity equation + Wasserstein projection onto the set {ρ ≤ 1}" (similarly to the scheme introduced in [START_REF] Mészáros | Advection-diffusion equations with density constraints[END_REF]). Since the vector field is monotone there exists a unique flow, that implies also that there exists a unique solution to the initial value problem ∂ t ρ t + ∇ • (u t ρ t ) = 0, ρ t 0 = ρ; in particular one can define a function Ψ t 1 t 0 (ρ) := ρ t 1 , that will satisfy also the semi-group rule Ψ s ′ t • Ψ t s = Ψ s ′ s . Let τ be a time step and define recursively ρ τ 0 = ρ 0 and then

ρ τ n+1 = Ψ (n+1)τ nτ (ρ τ n ) if n is even P(ρ n )
if n is odd.

Then the L 1 distance is preserved through the continuity equation step while it decreases after the projection thanks to Lemma 4.1. So in particular the L 1 -contraction is true in the discrete scheme; once we have that the scheme converges as τ → 0 to our equation, the uniqueness result gives that this property is preserved in the limit. In order to guarantee convergence (see [START_REF] Marino | Measure sweeping processes[END_REF] for general conditions) the crucial quantity to estimate is W 2 (Ψ t+τ t (ρ), ρ); using the Benamou-Brenier formula, two conditions that guarantee a good estimate are:

• u ∈ L 2 ([0, T ]; L ∞ (R d )): in this case we would have W L 2 ds. We believe that this general scheme (uniqueness in the Wasserstein framework and approximation with L 1 -contractive time discrete approximations) could be adapted also with some other convection terms Φ. Proof. First we see that formula (4.1) let us extend P to measures in M + (R d ) with mass less than |Ω|. In this context a monotonicity property is true (see for example Theorem 5.1 in [START_REF] Alexander | Quasi-static evolution and congested crowd transport[END_REF]), that is we have P(ρ) ≤ P(η) almost everywhere if ρ ≤ η. Now we can derive the L 1 contraction: let us denote ρ = min{ρ 1 , ρ 2 }. First of all we have that |ρ 1 -ρ 2 | = max{ρ 1 , ρ 2 } -min{ρ 1 , ρ 2 } = (ρ 1 + ρ 2 -min{ρ 1 , ρ 2 }) -min{ρ 1 , ρ 2 }; in particular ρ 1 -ρ 2 L 1 = 2 -2 ´ρ dx. By the monotonicity we have P(ρ i ) ≥ P(ρ) and in particular we have min{P(ρ 1 ), P(ρ 2 )} ≥ P(ρ) and so P(ρ 1 )-P(ρ 2 ) L 1 = 2-2 ˆmin{P(ρ 1 ), P(ρ 2 )} dx ≤ 2-2 ˆP(ρ) dx = 2-2 ˆρ dx = ρ 1 -ρ 2 L 1 , which proves the claim.

  ) -x * t (b)| ds ≤ 2ε x * t and by the arbitrariness of ε we conclude that t is a Lebesgue point for x * s (b). Now it is easy to conclude thanks to the fact that we know that ˆRd φ dρ t+h -ˆRd φ dρ t = ˆt+h t ˆRd ∇φ • v s dρ s , and noticing that x * s : φ → ˆRd ∇φ • v s dρ s satisfies the assumption x * ∈ L 1 ([0, T ]; B * ), when B

( 3 . 5 )

 35 I(φ, T ) = I(φ, 0) + ˆT 0 ˆΩ((ρ 1 -ρ 2 ) + (p 1 -p 2 )) [A∂ t φ + (A + B)∆φ + Au • ∇φ] dx dt. For a smooth function θ : Ω → R such that |θ| ≤ 1, let us consider the dual problem (3.6) A∂ t φ + (A + B)∆φ + Au • ∇φ = 0, in [0, T ) × Ω, ∇φ • n = 0 on [0, T ] × ∂Ω, φ(T, •) = θ a.e. in Ω.

u s 2 Lu s 2

 22 2 2 (Ψ t+τ t (ρ), ρ) ≤ τ ´t+τ t ∞ ds; • (∇ • u) -∈ L 1 ([0, T ]; L ∞ (R d )): in this case we would have Ψ s t (ρ) L ∞ ≤ C ρ L ∞ for some universal C and in particular we have W 2 2 (Ψ t+τ t (ρ), ρ) ≤ C ρ L ∞ τ ´t+τ t

Lemma 4 . 1 .

 41 Let us consider the projection operator P :P(R d ) → P(Ω) (4.1) P(ρ) = argmin{W 2 2 (ρ, η) : η ∈ P(Ω), η ≤ 1}.Then, when ρ 1 , ρ 2 are probability densities we have P(ρ 1 ) -P(ρ 2 ) L 1 ≤ ρ 1 -ρ 2 L 1 .

  absolutely continuous and its derivative is ˆRd v t • ∇φ dρ t for almost every time t. We will prove that under some integrability assumption there exists a universal full-measure set of differentiability, even when φ ∈ H 1 (R d ).Lemma 2.3. Let ρ t be a weakly continuous curve of probability measures on R d ; let us suppose that the continuity equation ∂ t ρ t + ∇ • (ρ t v t ) = 0 holds with a velocity field v t such that ˆT 0 ˆRd |v t | dρ t dt < +∞. Then there exists a set Υ ⊂ [0, T ] such that L 1 ([0, T ] \ Υ) = 0 and Moreover if we further have that ρ t ≤ 1 a.e. and ˆT 0 ˆRd |v t | 2 dρ t dt < +∞ then we can require that (2.1) also holds for every φ ∈ H 1 (R d ).

	(2.1)	lim h→0	1 h ˆRd	φ dρ t+h -	ˆRd	φ dρ t =	ˆRd	∇φ • v t dρ t	∀t ∈ Υ
	for every φ ∈ C 1 0 (R

d

).

  1 -contraction property; notice that this contraction is valid for a general open set Ω with C 1 boundary.Let us write the weak formulation of (3.2): for every smooth test function φ: [0, T ] × Ω → R with ∇φ • n = 0 on [0, T ] × ∂Ω we have (3.3) ˆT 0 ˆΩ [ρ∂ t φ + (ρ + p)∆φ + ρu • ∇φ] dx dt + ˆΩ ρ 0 (x)φ(0, x) dx = ˆΩ ρ(T, x)φ(T, x) dx.By density arguments the above formulation holds for φ ∈ W 1,1 ([0, T ]; L 1 (Ω))∩L 2 ([0, T ]; H 2 (Ω)). Now, let us consider two solutions (ρ 1 , p 1 ) and (ρ 2 , p 2 ) of Problem 3.2, with initial conditions

	respectively ρ 1 0 and ρ 2 0 . Writing the weak formulation (3.3) for both of them and taking the difference we obtain
	(3.4)

  by Corollary 1.2 we have that In addition, p i 's being pressures one has|{p i t = 0}| ≥ |{ρ i t < 1}| ≥ |Ω| -1 > 0 for a.e. t ∈ [0, T ],and so by a suitable version of Poincaré's inequality (since p i t vanishes on a set of positive Lebesgue measure) one obtains p i ∈ L ∞ ([0, T ]; H 1 (Ω)). In particular, by the Sobolev embedding this implies that

	ˆΩ |∇p i

t | 2 dx ≤ ˆΩ |u t | 2 dt, for almost every t ∈ [0, T ]. This implies that (since u is bounded) ess -sup t∈[0,T ] ∇p i t L 2 (Ω) ≤ C.

see the subsection on optimal transport below and[START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] for further details.

Acknowledgements. We would like to thank Filippo Santambrogio for the fruitful discussions during this project and for the careful reading of various versions of the manuscript. The authors warmly acknowledge the support of the ANR project ISOTACE (ANR-12-MONU-0013). We also thank the referee for her/his very useful comments and remarks on the manuscript.


W Scholl

Prof W Fasold

Impact Sound Insulation: The Standard Tapping Machine Shall Learn to Walk!

Previous research on the standard tapping machine has shown that the impact on a floor depends strongly on the combined system of floor and tapping hammers. This means that in general, a particular measure to reduce the impact sound pressure level can be advantageous with one impact source and disadvantageouswith another. Conversion from one impact source to another has to be made for each floor individually, provided there is a sufficient knowledge of the dynamic properties of the source and floor. So, if the standard tapping machine is to be used to characterise a floor with respect to walking noise, the only simple way out of the problem is to make it 'behave like a walker'. That means it should have the same source impedance. Investigations at the Fraunhofer-Institut für Bauphysik Stuttgart showed that this could be achieved by small modifications to the standard tapping machine. Comparisons using a concrete floor and a wooden joist floor have proved that results using such a modified tapping machine are much closer to walking noise than the results using the standard tapping machine.

INTRODUCTION

This article is the continuation of a former publication about "Impact sound insulation of timber floors: interaction between source, floor coverings and load bearing floor" [START_REF] Scholl | Impact Sound Insulation of Timber Floors: Interaction between Source, Floor Coverings and Load Bearing Floor[END_REF]. For measurements of the impact noise level of floors according to ISO 140-6 [START_REF]Acoustics -Measurement of sound insulation in buildings and of building elements -Part 6: Laboratory measurement of impact sound insulation of floors[END_REF], a standard tapping machine is used as an impact sound source, which drops 500 g steel hammers onto the floor ten times per second. The noise level thus produced in the room below the floor is believed to be representative of the noise produced by 'real life' sources such as people walking or jumping on the floor. When a walker is replaced by a tapping machine it is assumed that the classification of the acoustical performance of different floors is unchanged, and that the difference of impact input by the sources is a constant typical of those sources, but independent of the floor under test. Thus the noise of the standard hammer machine could be easily converted into walking noise just by adding a known correction to the measured (hammer) impact sound pressure levels.

However, this approach does not work. The reason is that both the impact source and the receiving structure have a mechanical impedance, and the actual exciting force acting on a structure depends on the ratio of the impedances of the source and structure, as well as the 'strength' of the source. There may be some limited areas, where there is an approximate simple relation between hammer and walking noise, that is when the floor structure can be represented as a linear system with a much greater impedance than the source. In general this condition is not fulfilled, which becomes obvious when trying to apply results from heavy and hard floors to lightweight or soft ones (cf. figure 14 in [START_REF] Scholl | Impact Sound Insulation of Timber Floors: Interaction between Source, Floor Coverings and Load Bearing Floor[END_REF]). Two solutions of the problem were proposed in [START_REF] Scholl | Impact Sound Insulation of Timber Floors: Interaction between Source, Floor Coverings and Load Bearing Floor[END_REF]:

1 to characterise the dynamic system of impact source and floor by its dynamic properties, such as impedances and transfer functions (radiated sound power to input force, e.g.) or 2 to define a typical impact sound sourcefor example jumping children or adults in high-heelsand to imitate it by a mechanical source, which is as simple as possible. Experiments using the second approach are reported here.

DEVELOPMENT OF A WALKER LIKE TAPPING MACHINE

The aim of the investigation was to construct a simple mechanical impact source with a mechanical impedance as close as possible to that of a walker. At the same time this source should resemble the ISO standard tapping machine to retain the ISO measurement method [START_REF]Acoustics -Measurement of sound insulation in buildings and of building elements -Part 6: Laboratory measurement of impact sound insulation of floors[END_REF] and to make it easier for people to get accustomed to a new source. Firstly, measurements of the impedance of two people, at their heels, were made. Secondly, the frequency spectrum of the impedance was reconstructed by a combination of springs and masses. Such a simplified walker system was then applied on two mechanically very different floorsa heavy concrete floor and a timber joist floorand the results compared with those using the ISO standard tapping machine, and a real walker to produce impact sound levels.

Impedance measurements of walkers

The input impedance of two adult males was measured by exciting their heels with an electrodynamic shaker (Figure 1). The moving shaft of the shaker was fixed to an accelerometer, a force transducer and a 45 mm diameter aluminium plate, to spread the input force on the heel. The walkers were tested with and without shoes and in sitting and standing positions. Figures 2 and3 show the resulting impedance level spectra for each of the two people, which is defined as where F is the force at, and v the velocity of the force transducer/aluminium plate interface, both are RMS values. The reference values were chosen such that 0 dB correspond to 100 Ns/m. The results show approximately a negative slope at low and Instrumentation to measure the mechanical source impedance of a walker. From the bottom: electrodynamical shaker, acceleration transducer, force transducer, aluminium disc (45 mm diameter), foot with shoe

Figure 2

Measured impedance level spectra for the first of the two people (JK), who were tested with and without shoes and in sitting and standing positions (0 dB = 100 Ns/m) a positive slope at high frequencies with the minimum in between shifted to higher frequencies when shoes are worn. Interestingly, the differences between the standing and sitting positions are rather small, indicating that only the lower part of the leg may be involved dynamically. When repeating the measurement with shoes, shown in Figure 2, the level differences in the third octave bands were about 2 dB or less.

Approximation of the walker impedance by a simple mechanical system

The requirement is shown in Figure 4. The source impedance of the walker is to be replaced by a spring-mass combination, which yields the same impedance when looking into the system from below. The bare-footed walker of Figure 2 was taken as a basis. Shoes were left off to prevent additional unknown influences and because this was considered to be the more critical case with regard to low frequency excitation.

The negative and positive slopes in Figure 2 display mass/spring characteristics. From this the mass-spring-mass system shown in Figure 5 was derived. Also shown is its calculated impedance level spectrum. As indicated, different approximations are possible which have not been investigated in detail. The spring needs a high loss factor of 0.8, to give a broad enough minimum. It should be noted, that the small mass at the bottom includes a portion of 25 g belonging to the aluminium plate and part of the force transducer. The mass on top represents the dynamically resting part of the human body.

It can be reduced to 500 g with only a small error at low frequencies, because it is dynamically decoupled from the input by the spring. The decoupling starts somewhere For the experiments described below the spring was changed to a stiffness s of 24x10 3 N/m and a loss factor h = 0.2 simply because a material with these characteristics was available. Because it was intended to carry out impact sound level tests on hard floor surfaces the small lower mass could be omitted as it merges with the much larger mass of the floor during the contact time, even when the floor it is a 'lightweight' timber floor. Omitting the small lower mass is not allowed of course, when exciting softly covered floors, or when the mass of the excited structure within half a bending wavelength is not large compared to the omitted mass. Finally the springs do not have to be fixed to the falling source masses but can be realised by a soft layer laid on the bottom. Such an arrangement is shown in Figure 6. It consists of a standard tapping machine acting upon five layers of a rubber cork mixture fixed with adhesive tape. The falling height can (but does not necessarily have to) be adjusted to the thickness of the soft layers. As the movement of the spherically shaped hammer heads of the tapping machine are controlled by the soft material, the contact stiffness between hammer and floor does not depend on the special floor under test and thus one artificial and misleading effect of the standard tapping machine is eliminated.

The possibility of using rubber heads instead of (at that time) brass heads for the tapping machine was discussed by Gerretsen in 1976 [3], who presumed that the force applied to different floors could more closely resemble that of footsteps. Gerretsen

Figure 6

The modified tapping machine, with the hammers acting upon five layers of a rubber cork mixture, fixed with adhesive tape

The two different floors types tested: a bare concrete floor (160 mm thick) and a timber joist bare floor (22 mm chipboard, 120 by 180 mm timber joists with 100 mm mineral wool in between, 24 mm laths and 12.5 mm gypsum board). Impact sources: walker, the modified and standard tapping machines. DL = difference of the normalised impact sound pressure levels produced by the respective source.

presented the tapping machine noise level difference between brass and rubber heads as an average for 25 floor constructions, as different as bare floors, carpeted floors, wooden top floors and floating floors. The spread of the differences (±2.6 dB at 63 Hz to ±12.9 dB at 2000 Hz for 90% confidence limits) seems to confirm the strong influence of the floor construction on the source input.

IMPACT SOUND LEVEL MEASUREMENTS WITH THE MODIFIED TAPPING MACHINE

A tapping machine, modified as mentioned above, was used for impact sound pressure level measurements according to ISO 140-6. Two different floors types were tested: a 160 mm thick concrete bare floor and a timber joist bare floor with 22 mm chipboard on top, 120 by 180 mm timber joists, 100 mm mineral wool in between, 24 mm laths and 12.5 mm gypsum board below.

The purpose of the tests was to determine whether the modified tapping machine would yield the same normalised impact sound level difference between the two floors as would a walking person, and to compare the results with those of an unmodified

Figure 8

Impact sound pressure level frequency spectra, using the modified and standard tapping machines, and a walker with and without shoes standard tapping machine (Figure 7). The absolute values of the impact sound pressure levels should not be compared directly, because the three sources are of different strength.

The resulting frequency spectra of the impact sound pressure level are presented in Figure 8. The most obvious effect of the tapping machine modification is the reduction of the higher frequency input, which makes the impact sound pressure level spectrum look more like a walker's spectrum than that of the standard tapping machine, especially on the hard surface of the concrete floor. A comparison of the corresponding single number values, the weighted impact sound pressure level L n,w and the Aweighted sound pressure level L A , is shown in Figures 9 and10. Regarding L n,w ,t h e

Figure 9

Difference of the weighted impact sound pressure level L n,w for the wooden and concrete floors, using the modified and standard tapping machines, and a walker with and without shoes

Figure 10

Difference of the A weighted impact sound pressure level for the wooden and concrete floors, using the modified and standard tapping machines, and a walker with and without shoes walker produced 16 to 18 dB higher levels on the wooden floor, and the modified tapping machine 12 dB, compared to those on the concrete floor. The standard tapping machine gave an opposite answer, 8 dB lower levels on the wooden floor compared to the concrete floor. That is at least 24 dB in the wrong direction.

The situation became somewhat better with the results expressed as A weighted levels. The walker revealed levels for the timber joist floor to be 15 to 17 dB louder, and the modified tapping machine 12 dB louder, compared to those for the concrete floor. The standard tapping now gives levels 5 dB higher for the timber joist floor, compared to the concrete floor. This is now in the right direction, at least.

It should be noted, that the A weighted level for the standard tapping machine results was calculated differently from normal A-weighting, that is by adding the normalised impact sound pressure level and the impact spectrum adaptation term C I,50-2500 according to ISO 717-2 [START_REF]Acoustics -Measurement of sound insulation in buildings and of building elements -Part 11: Laboratory measurement of the reduction of transmitted impact noise by floor coverings on light-weight framed standard floors[END_REF]:

L A =L n,w +C I,50-2500
(This is based on a kind of spontaneous A-weighting by the standard tapping machine with respect to walking noise on some floors.)

DISCUSSION OF THE RESULTS

The above example is based on only one walking person and two floor constructions with hard surfaces. There are uncertainties concerning the influences of the static load on the source impedance of the walker. Simply because of lack of resources the measured spectrum of the walker's impedance was only simulated roughly using an existing elastic material. Nevertheless, the results for the modified tapping machine are very promising. Impact sound level spectra and single number values are much closer to those for the the walker than those with the standard tapping machine.

One of the great advantages of simulating the source impedance of the walker is that the modified tapping machine will behave on all different structures like the walker. If an adjustment of the impact sound pressure level of the modified tapping machine to the walking noise is made once, this remains valid on all floors as long as the systems tested are linear. For sources with different impedance there is no such a correction. It is not possible to show this by comparing the test results from only two floors. Future research on other types of floors is therefore necessary, such as softly covered floors.

FREQUENTLY ASKED QUESTIONS (FAQS)

When discussing the matter with colleagues, several questions were repeatedly asked, which are answered in the following:

Will the reference curve for the single number rating in ISO 717-2 have to be changed when modifying the tapping machine? No, if the reference curve is intended to represent only the perception of noise. Yes, if the reference curve is to include adjustments of the modified tapping machine to walking noise. Yes, if the required performance of impact sound insulation is to be expressed by the same figures after replacing the standard by the modified tapping machine. This can be achieved exactly for one chosen reference floor construction, when the existing reference curve is corrected for the difference of impact sound pressure levels caused by the two different tapping machines.

Are nonlinearities taken into account with the modified tapping machine? Not completely. The modification mainly tackles the problem of impedance matching of source and receiving structure. However, some artificial nonlinear effects of the contact area of the spherically shaped hammers and the floor, are reduced by the soft interlayer of the modified machine. Including the static load of a walker could mean using hammers much heavier than 500

Is the input high enough with the modification? Suppose that the practical case of interest was not a bare concrete floor but a floor with a resilient covering. The additional softness introduced by the modification to the tapping machine might be expected to be small, so that the impact of the modified tapping machine would be similar to that of the standard tapping machine. For floating floors with hard surfaces, this should be tested.

CONCLUSIONS

Measurements of the foot input impedance of human beings, and theoretical considerations, show that the source impedance of a walker can be simulated by the standard tapping machine according to ISO 140-6 with only minor modifications, comprising an additional soft layer and a small mass of a few grams, positioned below the hammers. Such a modified tapping machine would act like a walker on any floor, be it soft or light-weight. That means that old tapping machines can continue to be used in the future. The modification can be implemented supplementary. Compared with other walker simulating devices, like dropping rubber balls, a continuous easy to measure signal is received (It should nevertheless not be forgotten that it is a line spectrum with 10 Hz distance between the excited frequencies!). The good agreement between results for a walker and the modified tapping machine, in contrast to those for the standard tapping machine, on two different floors, indicated that proceeding with the modified machine is recommended.

The modified tapping machine will be included as an alternative exciting device in the new standard ISO 140-11 [5], which is concerned with laboratory measurement of the impact sound improvement of floor coverings on lightweight floors. So, under well defined rules, experience with the 'new hammers' can be collected, before it is adopted as a new walking standard source. Other floors should be investigated, such as soft and floating floors, to prove the advantages of impedance simulation.

Figure 1

 1 Figure 1

Figure 3 Figure 4

 34 Figure 3Measured impedance level spectra for the second of the two people (AJ), who were tested with and without shoes and in sitting and standing positions (0 dB = 100 Ns/m)

ACKNOWLEDGEMENTS

This work was carried out at the Fraunhofer-Institut für Bauphysik Stuttgart (Director: Prof. Dr. Dr. h.c. mult. Dr. E.h. mult Karl Gertis) and financially supported by the German Minister für Raumordnung, Bauwesen und Städtebau.