UNIQUENESS ISSUES FOR EVOLUTIVE EQUATIONS WITH DENSITY CONSTRAINTS

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1 UNIQUENESS ISSUES FOR EVOLUTIVE EQUATIONS WITH DENSITY CONSTRAINTS SIMONE DI MARINO AND ALPÁR RICHÁRD MÉSZÁROS 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 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 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. In this case, by the regularization effect of the non-degenerate diffusion, the uniqueness 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, [2, 2, 22]). For more details on these models we direct the reader to the non-exhaustive list of works [4, 5, 6, 7,,, 2, 3, 4]. 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 work, we limit ourselves to models where so-called hard congestion effects are studied. It is worthwhile to notice that this problem has many features in common with the Hele-Shaw model and the Stefan problem (see for instance [6, 7]); the only difference is that in those models, instead of a velocity field, we have a source term for the mass, but both are regulated by the density constraint. Moreover our equations can be seen also as a quasilinear elliptic-parabolic system with very degenerate nonlinearity (see [3] ), for which the issue of uniqueness is still an open problem when we add a driving vector field without imposing an entropy condition (see [5, 3]). A very powerful tool to attack macroscopic hard-congestion problems where we impose a density constraint on the density of the population is the theory of optimal transport (see [36, 35]), as we can see in the recent works [23, 24, 25, 29, ]. 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 for uniqueness in this setting. As far as we are aware of, this question is a missing puzzle in full generality in the models studied in [23, 24, 25, 29]. 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 [33]). Date: July, 25. Keywords and phrases: crowd motion models; uniqueness results; density constraints; optimal transport; quasilinear elliptic-parabolic system. Mathematics Subject Classification: 35A2; 35K6; 35F25; 49J45. in our case ρ b(p) is maximal monotone, b() = [, ], b(p) = for p > and b(p) = for p <

2 2 S. DI MARINO AND A.R. MÉSZÁROS We will treat two different cases, with very different approaches: in the first one we simply consider a crowd driven by a 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 contraction result for the solutions, that will imply uniqueness. In the second case we add a diffusive term, which is modeling some randomness in the crowd movement (see [29] for recent developments and existence results in this setting); in this case we prove uniqueness 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 uniqueness even if the velocity is merely bounded. We remark also that in both cases we expect also a L contraction result that is true in the generic quasilinear elliptic-parabolic case [3]; this type of results will be the objective of future research in this field... Admissible velocities and pressures. In order to model crowd movement in the macroscopic setting with hard congestion, we work in a convex bounded domain R d with Lipschitz boundary such that >. The evolution of the crowd will be analyzed by the evolution of its density, which is assumed to be a probability measure on. The condition we impose is simply a bound on the density of the crowd, which is considered to be always less than. In particular the set of admissible measures will be denoted by K := {ρ P() : ρ 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 if v in the set {ρ = } and v n on where n is the outward normal. In order to make a rigorous definition we have to introduce the set of pressures: H ρ() := {p H () : p, p( ρ) = a.e.}. Then, using the integration by parts formula, informally we should have that p v dx pv n dh d = v p dx and so we can define adm(ρ) := { v L 2 (; R d ) : } v p dx for every p Hρ(). In the sequel we will denote by P adm(ρ) the L 2 (, L d ; R d ) -projection on the cone adm(ρ). Now, in order to preserve the constraint ρ, 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 ) = ρ t, v t adm(ρ t ). One of the simplest such model is when we have a prescribed 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 t ρ t + (ρ t v t ) = (.) ρ t, ρ t= = ρ 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 ([, T ] R d ). Nevertheless in [25, 23] and [34] the following regularity hypotheses have been assumed to show the existence result: u C or u = D for

3 a λ convex potential D and in both cases no dependence on time. In the following lemma we characterize the projection: Lemma.. Let ρ P() such that ρ a.e. and let u L 2 (; R d ). Then there exists p press(ρ) such that P adm(ρ) [u] = u p. Furthermore p is characterized by (i) p (u p) dx = ; (ii) q (u p) dx, for all q press(ρ); Proof. Let us set K = { p : p press(ρ)}. It is easy to see that K is a closed cone in L 2 (; R d ). Let us recall that the polar cone to K is defined as { } K o := v L 2 (; R d ) : v q dx, 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 (; R d ). This proves the claims (i) and (ii), since these are precisely the conditions of being the projection on a cone. Corollary.2. Lemma. (i) implies in particular that p 2 dx u 2 dx..2. The diffusive case. Recently (see [29]) a second order model for crowd motion has been proposed. 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 (.2) t ρ t ρ t + (ρ t v t ) = ρ t, ρ t= = ρ 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 (.2) as t ρ t ρ t + (ρ t (u t p t )) = (.3) ρ t, ρ t= = ρ p t, p t ( ρ t ) = a.e., and in both cases the systems are equipped with natural no-flux boundary condition on. Under the assumption u L ([, T ] ) it has been shown (see [29, Theorem 3..]) that the system (.3) admits a solution (ρ, p) L ([, T ] ) L 2 ([, T ]; H ()). In addition [, T ] t ρ t is a continuous curve in the 2-Wasserstein space (see Subsection.3)..3. Optimal trasport. Here we collect some facts about optimal transport that will be needed in the sequel. Given X 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 of the Borel measure µ through T and we write µ 2 = T µ, if µ 2 (A) = µ (T (A)) for every measurable set A X 2. Given two measures µ P(X ) and ν P(X 2 ), we define Π(µ, ν) as the set of γ P(X X 2 ) such that (π ) 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. 3

4 4 S. DI MARINO AND A.R. MÉSZÁROS A particular example of transport plans is given by the transport maps: whenever we have a measurable T : X 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 [36] or [35]). Theorem.3. Let R d be an open bounded set and let µ, ν P(). Let us consider the following quantities { } (P) A(µ, ν) = inf x y 2 dγ : γ Π(µ, ν) (D) { B(µ, ν) = sup ϕ(x) dµ + We will call (P) the primal and (D) the dual problem. ψ(x) dν : ϕ, ψ C (), ϕ(x) + ψ(y) x y 2 2 } x, y. (i) There exists a minimizer for the primal problem and 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 W2 2 (µ, ν) the common value. (iii) We can choose a maximizer (ϕ, ψ) of (D) such that ϕ and ψ are Lipschitz in and also such that 2 x 2 ϕ(x) is a convex function on the convex hull of. 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 is convex, P() 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 [, ] 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(), 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. 2. Monotone vector fields and the first order case Let R d be a bounded convex domain. In this section we suppose that the desired velocity field u : [, T ] R d of the crowd is a Borel monotone vector field, i.e. the following assumption is fulfilled: There exists λ R such that for a.e. x, y (H) (u t (x) u t (y)) (x y) λ x y 2, t [, T ]. The following results are well-known in the community of researchers working with the above mentioned models, but seems to be less known in the Hele-Shaw language. A first written version is essentially contained in [28] (Section 4.3.), nevertheless we simplified and clarified some of the proofs, hence we present them here. A key observation is the following lemma (see also Lemma in [28]): Lemma 2.. Let be a convex bounded domain of R d and let ρ, ρ P() two absolutely continuous measures such that ρ and ρ a.e. Take a Kantorovich potential ϕ from ρ to ρ and p H () such that p and p( ρ ) = a.e.

5 5 Then ϕ p dx = ϕ p dρ. To prove this result we consider the following extra lemma: Lemma 2.2. Let be a convex bounded domain of R d and let ρ, ρ P() two absolutely continuous measures such that ρ and ρ a.e. Take a Kantorovich potential ϕ from ρ to ρ and p H (). Let [, ] t ρ t be the geodesic connecting ρ to ρ, with respect to the 2-Wasserstein distance W 2. Then we have that d p dρ t = ϕ p dρ. dt {t=} Proof. We know (using the interpolation introduced by R. McCann, see [27] or Theorem.3 (iv)) that ρ t = (x t ϕ(x)) ρ for all t [, ] and so we have d p(x t ϕ(x)) p(x) p dρ t = lim dρ (x) dt t t {t=} = lim t t t = lim A t ( p) ϕ dρ (x), t p(x s ϕ(x)) ϕ(x) ds dρ (x) where the second equality is easy to prove, for fixed t, by approximation via smooth functions and, for t [, ], we denoted by A t : L 2 (; R d ) L 2 ρ (; R d ) the linear operator A t (h)(x) = t t h(x s ϕ(x)) ds. Now as a general fact we will prove that A t (h) h strongly in L 2 ρ (; R d ) as t, for every h L 2 (; R d ). First of all it is easy to see that A t : indeed A t (h) 2 dρ t = t t t h(x s ϕ(x)) 2 ds dρ (x) h 2 dρ s (x) ds h 2 dx. Here we used the fact that since ρ, ρ a.e we have also ρ t a.e. for all t [, ]. Now it is sufficient to note that for every ε > 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 A t (h h ε ) L 2 ρ + h h ε L 2 ρ + A t (h ε ) h ε L 2 ρ 2ε + tl ϕ L 2 ρ, where L is the Lipschitz constant of h ε. Taking now the limit as t goes to we obtain lim sup t A t (h) h L 2 ρ 2ε; by the arbitrariness of ε > we conclude. Now it is easy to finish the proof, since p L 2 () and so d p dρ t = lim A t ( p) ϕ dρ = p ϕ dρ. dt t {t=}

6 6 S. DI MARINO AND A.R. MÉSZÁROS Proof of Lemma 2.. Let [, ] t ρ t be the Wasserstein geodesic between ρ and ρ. We know that ρ t a.e. for all t [, ] and in particular it is true that p dρ t p dx = p dρ, which means that the function [, ] t derivative at is non-positive. Given this, the claim is a consequence of Lemma 2.2. p dρ t has a local maximum in t =, hence its 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 everyφ C (Rd ) the application [, T ] t φ dρ t is absolutely continuous and its derivative is v t φ dρ t for almost every time R d R t. We will prove that under some integrability assumption there d exists a universal full-measure set of differentiability, even when φ H (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 ) = holds with a velocity field v t such that T v t dρ t dt <. Then there exists a set Υ [, T ] such that L ([, T ] \ Υ) = and R d ( ) (2.) lim φ dρ t+h φ dρ t = φ v t dρ t t Υ h h R d R d R d T for every φ C (Rd ). Moreover if we further have that ρ t a.e. and v t 2 dρ t dt < R d then we can require that (2.) also holds for every φ H (R d ). Proof. Let us prove the following general statement: given a separable Banach space B and a curve x L ([, T ]; B ), then there exists Υ [, T ] such that Υ is a set of Lebesgue points for the map t x t (b) for every b B, and moreover L ([, T ] \ Υ). 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 as the Lebesgue points of t x t, and then we take Υ = n Υ n. For every b B and ε > let us consider i N such that b i b ε and then taking t Υ we have 2δ t+δ t δ ( x s(b) x t (b) ds ε x t + 2δ t+δ t δ ) x s ds + 2δ Now, taking the limit as δ and using the properties of Υ we get lim sup δ 2δ t+δ t δ t+δ t δ x s(b) x t (b) ds 2ε x t x s(b i ) x t (b i ) ds. 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 t+h φ dρ t+h φ dρ t = φ v s dρ s, R d R d t R d and noticing that x s : φ φ v s dρ s satisfies the assumption x L ([, T ]; B ), when B R d is the Banach space C (Rd ) in the first case and H (R d ) in the second case. Notice that if we follow the construction of Υ for the case B = C (Rd ), this set of times will work also for H (R d ) since C is dense in H. Now we are in position to prove the main theorem of this section, namely:

7 Theorem 2.4. Suppose R d is a bounded convex domain, u is a vector field satisfying Assumption (H) and let ρ a.e. be an admissible initial density. Let us suppose that there exist (ρ, p ), (ρ 2, p 2 ) two solutions to the system t ρ t + (ρ t (u t p t )) = in ], T ] (2.2) ρ t, p t, ( ρ t )p t = a.e. in, t [, T ] ρ t= = ρ, p i L 2 ([, T ]; H ()) for i {, 2}. Then, ρ = ρ 2 and p = p 2 a.e. In particular, under the same assumption, we can say that there exists a unique pair (ρ, v) that solves (.). Proof. We associate to the two curves ρ t and ρ 2 t a continuity equation (.) with the corresponding vector fields v t = u t p t and v 2 t = u 2 t p 2 t. Let us compute and estimate d dt 2 W 2 2 (ρ t, ρ 2 t ); we refer also to [2, Theorem 8.4.7] for a more general statement, but we prefer to include a simpler proof in this case. We know that t W2 2(ρ t, ρ 2 t ) is absolutely continuous: let us consider a time t for which its derivative exists and also such that t Υ Υ 2, where Υ i is the set for which (2.) is satisfied for the continuity equation for ρ i. Then we know that, for all s [, T ] we have (2.3) 2 W 2 2 (ρ s, ρ 2 s) ϕ t dρ s + ψ t dρ 2 s, R d R d where (ϕ t, ψ t ) is a pair of Kathorovich potentials for ρ 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 2 W 2 2 (ρ t, ρ 2 t ) = v t ϕ t dρ t + v 2 t ψ t dρ 2 t, where we used the fact that, since is bounded, we can assume ϕ t and ψ t to be Lipschitz and in particular they belong to H, which allows to use Lemma 2.3. We also know that thanks to Theorem.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 t = ρ 2 t and (S t ) ρ 2 t = ρ t for all t [, 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 of variable formula y = T t (x) we get ψ t (y) u t (y) dρ 2 t (y) = ϕ t (x) u t (T t (x)) dρ t (x). We can use this to split the formula for the derivative of W2 2(ρ t, ρ 2 t )/2. We use the result of Lemma 2. and then rewrite the term regarding u t in terms of transport maps and see: d dt 2 W 2 2 (ρ t, ρ 2 t ) = v t ϕ t dρ t + v 2 t ψ t dρ 2 t = u t ϕ t dρ t + u t ψ t dρ 2 t p t ϕ t dρ t p 2 t ψ t dρ 2 t ϕ t (x) [u t (x) u t (T t (x)) ] dρ t (x T t (x)) [u t (x) u t (T t (x)) ] dρ t λ x T t (x) 2 dρ t λw2 2 (ρ t, ρ 2 t ). 7

8 8 S. DI MARINO AND A.R. MÉSZÁROS Grönwall s lemma implies that W 2 2 (ρ t, ρ 2 t ) e 2λt W 2 2 (ρ, ρ 2 ). Since ρ = ρ2 = ρ a.e., the above property implies that ρ = ρ 2 a.e. in [, T ]. From this fact we can easily deduce that (p t p 2 t ) =, for a.e. t [, T ] in the sense of distributions. In particular p t p 2 t is analytic in the interior of. Moreover, both p t and p 2 t vanish a.e. in the set {ρ t = } which has a positive Lebesgue measure greater than >. Thus, p = p 2 a.e. in [, T ]. The claim follows. Remark 2.. 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 () (see [2, 23, 34]), 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 regularity, the existence result is proven with the help of a well-chosen splitting algorithm (see [25, 34]). Nevertheless, combining the techniques developed in [29] 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 (H) is not surprising in this setting. We remark that the same assumption was required in [3] 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) =, ρ(t =, ) = ρ, where the velocity field B : R d R d was supposed to satisfy the monotonicity property (H). We note also the fact that we can allow for moving domains 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 [9] for proving uniqueness for an evolution equation with density constraint driven only by the boundary of the moving sets. 3. The general diffusive case We use Hilbert space techniques (similarly to the one developed in [8, 32]; see also Section 3.. from [33]) to study the uniqueness of a solution of the diffusive crowd motion model with density constraints described in Subsection.2 (see also [29]). Since our objective is only to obtain uniqueness results (not necessarily via a contraction property), we can expect that this holds under more general assumptions in the presence of a non-degenerate diffusion in the model. Let u : [, T ] R d be a given vector field, which represents again the desired velocity field of the crowd, R d a convex, bounded open set with Lipschitz boundary, ρ P() the initial density of the population such that ρ a.e. in and let us consider the following problem { t ρ t ρ t + (P adm(ρt)[u t ]ρ t ) =, in ], T ] ; (3.) ρ t= = ρ, ρ t, a.e. in, equipped with the natural no flux boundary condition. Equivalently the above system can be written as t ρ t ρ t p t + (u t ρ t ) =, in ], T ], (3.2) ρ t= = ρ, in ( ρ t + p t u t ρ t ) n =, on, for a.e. t [, T ],

9 for a pressure field p t Hρ t. It has been shown in [29] that under the assumption that u L ([, T ] ; R d ) the systems (3.) and (3.2) have a solution. More precisely there exist a continuous curve [, T ] t ρ t W 2 and p t Hρ t for all t [, T ] (in particular ρ L ([, T ] ) and p L 2 ([, T ]; H ()) such that (p, ρ) solves (3.2) in weak sense (see (3.3)). Our aim in this section is to show that the solution (ρ, p) of (3.2) is unique. For a smooth test function φ : [, T ] R with φ n = on [, T ] and φ(t, ) = a.e. in, let us write the weak formulation of (3.2): T (3.3) [ρ t φ + (ρ + p) φ + ρu φ] dx dt + ρ (x)φ(, x) dx =. By density arguments the above formulation holds for φ W, ([, T ]; L ()) L 2 ([, T ]; H 2 ()). Now, let us suppose that Problem 3.2 has two solutions (ρ, p ) and (ρ 2, p 2 ) with ρ = ρ2 = ρ. Writing the weak formulation (3.3) for both of them and taking the difference we obtain T [ (3.4) (ρ ρ 2 ) t φ + (ρ ρ 2 + p p 2 ) φ + (ρ ρ 2 )u φ ] dx dt =. We introduce the following notations A := ρ ρ 2 (ρ ρ 2 ) + (p p 2 ) and B := p p 2 (ρ ρ 2 ) + (p p 2 ). Note that A and B a.e. in [, T ] and A + B =. To be consistent with these bounds, we set A = when ρ = ρ 2, even if p = p 2 and B = when p = p 2, even if ρ = ρ 2. With these notations the weak formulation for the difference gives T (3.5) ((ρ ρ 2 ) + (p p 2 )) [A t φ + (A + B) φ + Au φ] dx dt =. For a smooth function G : [, T ] R let us consider the dual problem { A t φ + (A + B) φ + Au φ = AG, in [, T [, (3.6) φ n = on [, T ], φ(t, ) = a.e. in. Let us remark that if we are able to find a (reasonably regular) solution φ for this problem for any G smooth we would get uniqueness for ρ, and hence also for p (as done in the end of the proof of Theorem 2.4). Since the coefficients in (3.6) are not regular, we study a regularized problem. For ε > let us consider A ε and B ε to be smooth approximations of A and B such that A A ε L r ([,T ] ) < C(, r)ε, ε < A ε and B B ε L r ([,T ] ) < C(, r)ε, ε < B ε, for r < +, the value of which to be chosen later. Here C(, r) > is a constant depending only on and r. The regularized problem reads as follows { t φ ε + ( + B ε /A ε ) φ ε + u φ ε = G, in [, T [, (3.7) φ ε n = a.e. on [, T ], φ ε (T, ) = a.e. in. For all ε > the above problem is uniformly parabolic and B ε /A ε is continuous. Moreover G is smooth and u L ([, T ] ), thus by classical results (see for instance [9, 8]) the problem has a (unique) solution φ ε H ([, T ]; L 2 ()) L 2 ([, T ]; H 2 ()). In particular φ ε can be used as test function in (3.3). Now let us establish some standard uniform estimates (in ε) on φ ε. 9

10 S. DI MARINO AND A.R. MÉSZÁROS Lemma 3.. Let φ ε be a solution of (3.7). Then there exists a constant C = C(T, u L, G L 2 ([,T ] )) > such that we have the following estimates, uniformly in ε > : (i) sup φ ε (t) L 2 () C; t [,T ] (ii) (B ε /A ε ) 2 φ ε L 2 ([,T ] ) C; (iii) φ ε L 2 ([,T ] ) C. Proof. Let us multiply the first equation in (3.7) by φ ε and integrate over [t, T ] for t < T. We obtain (3.8) T 2 φ ε(t) 2 L 2 () + = t T t ( + B ε /A ε ) φ ε 2 dx dt T φ ε G dx dt u φ ε φ ε dx dt t Hence by Young s inequality we have ( 2 φ ε(t) 2 L 2 () u L 2δ C + C 2 T t + ) T φ ε (s) 2 L 2 2 () ds + 2 G 2 L 2 ([,T ] ) t φ ε (s) 2 L 2 () ds where the term in φ ε 2 has been absorbed by the left hand side, and < δ 2/ u L is a fixed constant and the constant C > is depending just on G L 2 ([,T ] ) and u L. Hence by Grönwall s inequality we obtain 2 φ ε(t) 2 L 2 () CeC(T t), which implies in particular that sup φ ε (t) L 2 () C. Thus (i) follows. t [,T ] On the other hand choosing δ := 2/ u L in Young s inequality used in (3.8) and using (i), we obtain T t (B ε /A ε ) φ ε 2 dx dt C hence (B ε /A ε ) 2 φ ε L 2 ([,T ] ) C, and thus (ii) follows. By (3.8), (i) and (ii) easily imply (iii).

11 In particular, using φ ε as test function in (3.5) one has T T (ρ ρ 2 )G dx dt = (ρ ρ 2 + p p 2 )AG dx dt = = = + T T T T T := I ε + I 2 ε (ρ ρ 2 + p p 2 )A [ t φ ε + ( + B ε /A ε ) φ ε + u φ ε ] dx dt (ρ ρ 2 + p p 2 )A [ t φ ε + ( + B ε /A ε ) φ ε + u φ ε ] dx dt (ρ ρ 2 + p p 2 ) [A t φ ε + (A + B) φ ε + Au φ ε ] dx dt (ρ ρ 2 + p p 2 )(B ε /A ε )(A A ε ) φ ε dx dt (ρ ρ 2 + p p 2 )(B ε B) φ ε dx dt Now we shall prove that Iε and Iε 2 as ε, which will lead to the uniqueness of ρ. First, let us recall that ρ, ρ 2 a.e. in [, T ], hence ρ, ρ 2 L ([, T ] ). On the other hand p, p 2 L 2 ([, T ]; H ()) and by Corollary.2 we have that p i t 2 dx u t 2 dt, for almost every t [, T ]. This implies that (since u is bounded) ess sup t [,T ] p i t L 2 () C. In addition, p i s being pressures one has {p i t = } {ρ i t < } > for a.e. t [, T ], and so by a suitable version of Poincaré inequality one obtains p i L ([, T ]; H ()). By the Sobolev embedding this means that p i L ([, T ]; L 2 ()), i {, 2}, where 2 = 2d/(d 2). This implies the following estimates I ε ρ ρ 2 L ([,T ] ) (B ε /A ε ) /2 (A A ε ) L 2 ([,T ] ) (B ε /A ε ) /2 φ ε L 2 ([,T ] ) + T and similarly p p 2 L 2 () (B ε/a ε ) /2 (A A ε ) L r () (B ε /A ε ) /2 φ ε L 2 () dt C(/ε) /2 ε + p p 2 L (L 2 ) (B ε/a ε ) /2 (A A ε ) L 2 (L r ) (B ε /A ε ) /2 φ ε L 2 (L 2 ) Cε /2, as ε I 2 ε ρ ρ 2 L ([,T ] ) (A ε /B ε ) /2 (B B ε ) L 2 ([,T ] ) (B ε /A ε ) /2 φ ε L 2 ([,T ] ) + p p 2 L (L 2 ) (A ε/b ε ) /2 (B B ε ) L 2 (L r ) (B ε /A ε ) /2 φ ε L 2 (L 2 ) C(/ε) /2 ε = Cε /2, as ε, where r > is the exponent such that 2 + r + =, i.e. r = d. Hence we obtained that 2 T (ρ ρ 2 )G dx dt =, for all G smooth,

12 2 S. DI MARINO AND A.R. MÉSZÁROS which in particular implies ρ = ρ 2 a.e. in [, T ]. Then one obtains also that p = p 2 a.e. in [, T ], as in the end of the proof of Theorem 2.4. The result follows. 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-2-MONU-3). References [] D. Alexander, I. Kim, Y. Yao, Quasi-static evolution and congested crowd transport, Nonlinearity, 27 (24), No. 4, [2] L. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, (28). [3] J. Carrillo, Entropy Solution for Nonlinear Degenerate Problems, Arch. Rational Mech. Anal., 47 (999), [4] C. Chalons, Numerical Approximation of a macroscopic model of pedestrian flows, SIAM J. Sci. Comput., Vol. 29 (27), Issue 2, [5] R.M. Colombo, M.D. Rosini, Pedestrian Flows and non-classical shocks, Math. Mod. Meth. Appl. Sci., 28 (25), [6] V. Coscia, C. Canavesio, First-Order macroscopic modelling of human crowd dynamics, Math. Mod. Meth. Appl. Sci., 8 (28), [7] E. Cristiani, B. Piccoli, A. Tosin, Multiscale Modeling of Pedestrian Dynamics, Springer, (24). [8] A.B. Crowley, On the weak solution of moving boundary problems, J. Inst. Math. Applics, (979) 24, [9] S. Di Marino, B. Maury, F. Santambrogio, Masure sweeping processes, (25), preprint available at [] D. Helbing, A fluid dynamic model for the movement of pedestrians, Complex Systems, 6 (992), [] D. Helbing, P. Molnár, Social force model for pedestrian dynamics, Phys. Rev E 5 (995), [2] L.F. Henderson, The statistics of crowd fluids, Nature, 229 (97), [3] R. L. Hughes, A continuum theory for the flow of pedestrian, Transportation research Part B, 36 (22), [4] R. L. Hughes, The flow of human crowds, Annual review of fluid mechanics, Vol. 35. (23), Annual Reviews: Palo Alto, CA, [5] N. Igbida, J. M. Urbano, Uniqueness for nonlinear degenerate problem, NoDEA Nonlinear Differential Equations Appl., (23), No. 3, [6] N. Igbida, Hele-Shaw type problems with dynamical boundary conditions, J. Math. Anal. Appl., 335 (27), No. 2, [7] N. Igbida, K. Sbihi, P. Wittbold, Renormalized solution for Stefan type problems: existence and uniqueness, NoDEA Nonlinear Differential Equations Appl., 7 (2), No., [8] N.V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Mathematics, 96, AMS, (28). [9] O. A. Ladyzenskaja, V. A. Solonnikov, N.N. Uralceva, Linear and quasilinear equations of parabolic type, Translated from the Russian, Translations of Mathematical Monographs, Vol. 23, (968). [2] J.-M. Lasry, P.-L. Lions, Jeux à champ moyen I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (26), No. 9, [2] J.-M. Lasry, P.-L. Lions, Jeux à champ moyen II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (26), No., [22] J.-M. Lasry, P.-L. Lions, Mean field games, Jpn. J. Math., 2 (27), No., [23] B. Maury, A. Roudneff-Chupin, F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Math. Models and Meth. in Appl. Sci., 2 (2), No., [24] B. Maury, A. Roudneff-Chupin, F. Santambrogio, Congestion-driven dendritic growth, Discrete Contin. Dyn. Syst., 34 (24), no. 4, [25] B. Maury, A. Roudneff-Chupin, F. Santambrogio, J. Venel, Handling congestion in crowd motion modeling, Netw. Heterog. Media, 6 (2), No. 3, [26] B. Maury, J. Venel, Handling of contacts in crowd motion simulations, Traffic and Granular Flow, Springer (27). [27] R.J. McCann, A convexity principle for interacting gases, Adv. Math., 28 (997), No.,

13 [28] A.R. Mészáros, Mean Field Games with density constraints, MSc thesis, École Polytechnique, Palaiseau, France, (22), available at pdf. [29] A.R. Mészáros, F. Santambrogio, A diffusive model for macroscopic crowd motion with density constraints, (25), preprint available at [3] L. Natile, M. Peletier, G. Savaré, Contraction of general transportation costs along solutions to Fokker- Planck equations with monotone drifts, J. Math. Pures Appl., 95 (2), [3] F. Otto, L -contraction and uniqueness for quasilinear elliptic-parabolic equations, J. of Diff. Eq., 3 (996), [32] B. Perthame, F. Quirós, J.L. Vázquez, The Hele-Shaw asymptotics for mechanical models of tumor growth, Arch. Rational Mech. Anal., 22 (24), [33] A. Porretta, Weak Solutions to Fokker-Planck Equations and Mean Field Games, Arch. Rational Mech. Anal., 26 (25), Issue, -62. [34] A. Roudneff-Chupin, Modélisation macroscopique de mouvements de foule, PhD Thesis, Université Paris- Sud, (2), available at [35] F. Santambrogio, Optimal Transport for Applied Mathematicians, To be published by Birkäuser, (25). [36] C. Villani Topics in Optimal Transportation. Graduate Studies in Mathematics, AMS, (23). Laboratoire de Mathématiques d Orsay, Université Paris-Sud, France address: simone.dimarino@math.u-psud.fr Laboratoire de Mathématiques d Orsay, Université Paris-Sud, France address: alpar.meszaros@math.u-psud.fr 3