A macroscopic crowd motion model of gradient flow type
aa r X i v : . [ m a t h . A P ] F e b A MACROSCOPIC CROWD MOTION MODELOF GRADIENT FLOW TYPE
BERTRAND MAURY, AUDE ROUDNEFF-CHUPIN, AND FILIPPO SANTAMBROGIO
Abstract.
A simple model to handle the flow of people in emergency evacuation situationsis considered: at every point x , the velocity U ( x ) that individuals at x would like to realize isgiven. Yet, the incompressibility constraint prevents this velocity field to be realized and theactual velocity is the projection of the desired one onto the set of admissible velocities. Insteadof looking at a microscopic setting (where individuals are represented by rigid discs), here themacroscopic approach is investigated, where the unknown is a density ρ ( t, x ) . If a gradientstructure is given, say U = −∇ D where D is, for instance, the distance to the exit door, theproblem is presented as a Gradient Flow in the Wasserstein space of probability measures. Thefunctional which gives the Gradient Flow is neither finitely valued (since it takes into accountthe constraints on the density), nor geodesically convex, which requires for an ad-hoc study ofthe convergence of a discrete scheme. Crowd motion; Gradient Flow; Wasserstein distance; Continuity equation.1.
Introduction
In the last two decades, several strategies have been proposed to model the motion of pedes-trians. Most of them rely on a microscopic approach: the degrees of freedom are the positionsof individuals, and their evolution depends on a balance between selfish behaviour, congestionconstraints, and possibily social factors (politeness, gregariousness). Among those microscopicmodels, some are based on a stochastic description of the individual behaviour (see e.g. [27]),whereas others are purely deterministic (see [24, 28, 29]).An essential ingredient in those models lies in the way interactions between individuals arehandled, in particular in the case of high density (congestion phenomena). Following the clas-sification which holds in the modelling of granular flows, one can differentiate the MolecularDynamics (MD) approach (the non overlapping constraint between rigid grains is relaxed, andhandled by a short range repulsive force) and the Contact Dynamics (CD) one (the collisionsare explicitely taken into account). In the context of pedestrians, MD strategy has proved to bequite efficient to model congestion. In particular Helbing[22, 24, 26] introduced the concept ofsocial forces, which are designed in such a way that individuals tend to repel each other whentheir distance drops below a certain value. The model proposed in [34] relies on the alternativestrategy: individuals do not interact with each other as soon as they are not in contact, and thenon overlapping constraint is treated in a strong (non relaxed) way. Although it is natural toexpect some link between the two approaches (MD models are likely to converge in some wayto their CD counterparts as the repulsive force stiffness goes to infinity), it is to be noticed thatthe mathematical structures of the two classes of models are quite different. In the first case,Cauchy-Lipschitz theory for ODE’s applies, whereas CD models present some analogies with theso-called sweeping process introduced by Moreau[36] in the 70’, for which a dedicated frameworkhas been developped (see [20], [21], [35]).In the case of macroscopic models, the first strategy (congestion is treated in a relaxed way)is favoured, as it allows to use classical methods for studying PDE. For example, crowd motionmodels inspired from traffic flow models have been developped (see Refs. [14, 12, 13]). Theytake the form of hyperbolic conservation laws, and they are essentially monodimensional in space. In higher dimension, Bellomo and Dogbe[4, 19] proposed second order models, where aphenomenological relation describes how the crowd modifies its own speed: (cid:26) ∂ t ρ + ∇ x · ( ρ v ) = 0 ∂ t v + ( v · ∇ x ) v = F ( ρ, v ) . Typically, the motion is governed by F , which has two parts: a relaxation term toward a definitespeed, and a repulsive term to take into account that pedestrians tend to avoid high densityareas. Degond[18] uses the same approach to model sheep herds. In this model, the term F depends on a pressure which blows up when the density approaches a given congestion density(barrier method). There also exist first order models, where the velocity field is directly definedas a function of the density (see e.g. [30, 31, 15]). Another class of models is described byPiccoli and Tosin in Refs. [38], [39]. They propose a time-evolving measures framework, wherethe velocity of the pedestrian is composed by two terms: a desired velocity and an interactionvelocity. The last one models the reaction of the pedestrian to the other surrounding pedestrians(namely, people can deviate from their preferred path if they enter a crowded area).To our knowledge, as the ones presented above, all macroscopic models rely on a relaxed ex-pression of the congestion. Let us mention however the work of Buttazzo, Jimenez and Oudet in[9], where the optimal transportation between two given densities is computed under constraints(obstacles, congestion, ...) which can be strongly expressed. Yet, this approach is very differentfrom the model we describe later, since its goal is to find an optimal transport between den-sities as in the work of Benamou and Brenier[5] (which is the classical reference for dynamicalformulations of transport problems), whereas optimal transportation is in our case a very suit-able tool. Moreover, we will mainly make use of the distance that optimal transport induces onprobability measures rather than looking at the optimal maps themselves, as we will see after abrief description of the model we consider.The macroscopic model we present here is based on a strong expression of the congestionconstraint. It is a natural extension of the microscopic approach proposed in Refs. [34, 35, 41],which we describe here in its simpler form. The crowd configuration is represented by theposition vector q = ( q , . . . , q N ) . Each of the N individuals whishes to have a velocity U i which depends on its position only: U i = U ( q i ) , where U ( · ) is some given velocity field over R (typically U = −∇ D , where D is the geodesic distance to the exit). To account for non-overlapping, it is assumed that the actual velocity u = ( u , . . . , u N ) is the ℓ -projection of ˜ U = ( U , . . . , U N ) = ( U ( q ) , . . . , U ( q N )) onto the cone of feasible velocities C q (i.e. the set ofvelocities which do not lead to a violation of the non-overlapping constraint). The model takesthe form(1) d q dt = uu = P C q ˜ U . In the spirit of this microscopic approach, the model we propose here rests on the two followingprinciples(1) the pedestrian population is described by a density ρ which is subject to remain belowa certain maximal value (equal to in what follows), this density follows an advectionequation,(2) the advecting field u is the closest, among admissible fields (i.e. which do not lead toa violation of the constraint), to some spontaneous field U , which corresponds to thestrategy people would follow in the absence of others.If we denote by C ρ the cone of admissible velocities (i.e. set of velocities which do not increasedensity in already saturated zones, see next section for a proper definition), the model takes the MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE 3 following form(2) ( ∂ t ρ + ∇ · ρ u = 0 u = P C ρ U , where the projection is meant in the L sense. As a matter of fact, in the same way as Cauchy-Lipschitz theory for ODE’s no longer applies for CD in the microscopic case, we cannot useclassical methods to study Equation (2), as well as most of the PDE’s we could encounter inthe CD macroscopic models. This is due in particular to the lack of regularity of the velocity u (whose natural regularity is L ), which prevents us to apply the characteristic method or evenDiPerna-Lions theory. The non-continuous dependence of the operator P C ρ with respect to ρ isanother source of problems.Instead, we will see that this PDE corresponds to a Gradient Flow in the Wasserstein space(i.e. the space of probability measures endowed with the distance W induced by the optimaltransport under quadratric cost), provided that the spontaneous velocity field has a gradientstructure: U = −∇ D . This means that we consider the functional Φ( ρ ) = Z Ω D ( x ) ρ ( x ) dx if ρ ≤ ∞ otherwiseand we look for the curve of measures ρ ( t, . ) which follows the steepest descent direction of Φ starting from a given datum ρ . This curve will happen to solve equation (2). This is a generaland very efficient method to find solutions to certain evolution PDE’s which been made possibleby the theory of optimal transportation. This theory owes its origin to Kantorovich[33], but hasbeen widely developped thereafter (see the books by Villani[42, 43]). Several equations have beenapproached by this method, for instance the classical heat equation, as well as the Fokker-Planckor the porous media equations (see Refs. [32, 37, 11]). Notice that, as the functional which isused to produce the porous media equation as a Gradient Flow is ρ Z ρ ( x ) m dx, our case can be considered as its formal limit when m tends to infinity. All the theory of GradientFlow in Wasserstein Spaces is treated in the reference book by Ambrosio-Gigli-Savaré[2] and oneof the key assumptions is the λ –convexity of the functional, which ensures better estimates. Onthe other hand, some existence results can be obtained without this assumption, but they haveto be treated carefully by hand, as it happens in Ref. [7]. In our case, even if we suppose D tobe λ –convex, we face the same kind of difficulties if we want to add the presence of an exit dooron the boundary of Ω where the measure can concentrate (see Section 2).The paper is organized as follows: In Section 2 we present the model in the Eulerian settingand a related discrete minimizing movement scheme (MMS). We explain how a straightforwarduse of a convergence theorem in [2] asserts a convergence of the trajectories for the discrete MMSto some continuous pathline. Identification of this limit with a solution to the initial problem canbe done unformally. Yet some technical obstacles (in particular the handling of walls) prevent usfrom obtaining a fully rigorous proof based on this approach. The actual proof of convergence toa solution of the crowd motion model is based on alternative arguments. The end of this sectiondescribes this convergence results. As the presence of an exit raises some very specific technicaldifficulties, we propose in Section 3 a proof in the case there is no exit. The proof in the generalcase in given in Section 4. To illustrate the convergence theorem, we present in Section 5 anidealized (yet non trivial) situation where both eulerian solutions and discrete MMS trajectoriescan be described with accuracy. Finally, we discuss in Section 6 the limitations of this modeland its possible extensions to other fields of natural sciences. In particular, we explain why we BERTRAND MAURY, AUDE ROUDNEFF-CHUPIN, AND FILIPPO SANTAMBROGIO developped the theory in any dimension although dimensions greater than two do not make clearsense as far as crowd motion is concerned.2.
The eulerian model and its gradient flow formulation
Eulerian model.
The model we propose is designed to handle emergency evacuation situ-ations : the behaviour of individuals is based on optimizing their very own trajectory, regardlessof others, but the fulfillment of individual strategies is made impossible because of congestion.The model takes the following form: given a domain Ω (the building), whose boundary Γ iscomposed of Γ out (the exit) and Γ w (the walls), we describe the current distribution of people bya measure ρ of given mass (say without loss of generality) supported within Ω . To model thefact that people getting through the door are out of danger, yet keeping a constant total masswithout having to model the exterior of the building, we shall assume that ρ may concentrateon Γ out . In this spirit, we denote by K the set of all those probablity measures over R that aresupported in Ω , and that are the sum of a diffuse part, with density between and , in Ω , anda singular part carried by Γ out . ΩΓ out Γ w Figure 1.
Geometry.We shall denote by U the spontaneous velocity field: U ( x ) represents the velocity that anindividual at x would have if he were alone. It is taken equal to outside Ω . The set C ρ offeasible velocities corresponds to all those fields which do not increase ρ on the saturated zone(unformally, ∇ · u ≥ in [ ρ = 1] ), and which account for walls (people do not walk throughthem). As we plan to define C ρ as a closed convex set in L (Ω) , those constraints do not makesense as they are, and we shall favor a dual definition of this set. Let us introduce the “pressure”space H ρ = { q ∈ H (Ω) , q ≥ a.e. in Ω , q ( x ) = 0 a.e. on [ ρ < , q | Γ out = 0 } . The proper definition of C ρ reads(3) C ρ = { v ∈ L (Ω) , Z Ω v · ∇ q ≤ ∀ q ∈ H ρ } . The model is based on the assumption that the actual instantaneous velocity field is the feasiblefield which is the closest to U in the least-square sense, i.e. it is defined as the L -projection of U onto the closed convex cone C ρ . Finally the problem consists in finding a trajectory t ρ ( t ) ∈ K which is advected by u , i.e. such that ( ρ, u ) is a (weak) solution of the transport equation in R (4) ∂ t ρ + ∇ · ( ρ u ) = 0 , where u verifies, for almost every t ,(5) u = P C ρ U . MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE 5
Remark . The fact that Γ out is likely to carry some mass calls for some proper definition ofthe velocity on this zero-measure set. As the exit plays the role of a reservoir in our model, weshall actually consider that all feasible fields vanish on Γ out , so that velocity u given by (5) willbe considered as defined Lebesgue-a.e. in Ω and vanishing on Γ out . Remark . Boundary conditions (walls and exit) .The unilateral divergence constraint and the behaviour at walls and exit are implicitly containedin the dual expression of C ρ , as illustrated by the following considerations. We assume in thisremark that [ ρ = 1] = ω where ω ⊂ Ω is a smooth subdomain, and that all fields are smooth.First of all, by taking tests pressures which are smooth and compactly supported in ω , we obtain ∇ · u ≥ in the saturated zone. As the pressure vanishes on Γ out , the velocity is free on thatpart of the boundary (free outlet condition, as in Darcy flows). Consider now a situation wherethe saturated zone covers the wall Γ w . For any smooth function ϕ defined on Γ w consider asequence of extensions ϕ ε supported within ω ∪ Γ w , which converges to in L (Ω) . Then Z Ω u · ∇ ϕ ε ≤ ∀ ε > implies − Z Ω ϕ ε ∇ · u + Z Γ w ϕ ε u · n ≤ ∀ ε > . As the first term goes to with ε we obtain that the velocity necessarily enters the domain onthe saturated wall (what we adressed before as “people do not walk through walls”).2.2. Gradient flow formulation.
In this section we introduce a discrete evolution problemin the Wasserstein space, whose limit will be the gradient flow of a suitable functional, and weestablish unformally the link between this new problem and the crowd motion model. The formalequivalence, which will be proved rigorously in the following sections, will be satisfied in the casewhere U = −∇ D is the opposite of a gradient.Let us denote by P the set of probablity measures over R endowed with the Wassersteindistance, and by(6) K = { ρ ∈ P , supp ( ρ ) ⊂ Ω , ρ = ρ out + ρ Ω , ρ Ω ( x ) ≤ a.e. , supp ( ρ out ) ⊂ Γ out } the set of feasible densities. Let an initial density ρ be given, and τ > a time step. We build ρ τ = ρ , ρ τ , . . . as follows(7) ρ kτ = argmin P ( R d ) (cid:26) J ( ρ ) + I K ( ρ ) + 12 τ W ( ρ, ρ k − τ ) (cid:27) , where W is the Wasserstein distance, J is the dissatisfaction functional defined as(8) J ( ρ ) := Z Ω D ( x ) ρ ( x ) dx, and I K is the indicatrix of K : I K ( ρ ) = (cid:12)(cid:12)(cid:12)(cid:12) if ρ ∈ K + ∞ if ρ / ∈ K. The function D is typically the distance to the door Γ out , and to D we associate a vector field U = −∇ D . It is important in order to have vanishing velocities on the door that D is minimaland constant on Γ out .We admit here that under reasonable assumptions this process is indeed an algorithm (i.e. ρ k +1 τ is uniquely defined as the minimizer of the function above), and we denote by ρ τ the piecewiseconstant interpolate of ρ τ , ρ τ , . . . . BERTRAND MAURY, AUDE ROUDNEFF-CHUPIN, AND FILIPPO SANTAMBROGIO As τ goes to , by Prop. 2.2.3, Th. 2.3.1, and Th. 11.1.3 in [2], ρ τ converges to some trajectory t ρ in K , which is a (weak) solution to ∂ t ρ + ∇ · ( ρ u ) = 0 , where u is such that, for almost every t , u ∈ − ∂ ( J + I K ) ( ρ ) , where ∂ Ψ denotes the strong subdifferential of Ψ . Furthermore u minimizes the L norm amongall those fields in the subdifferential above.Let us now prove unformally that this characterizes the instantaneous velocity as the projectionof U = −∇ D onto C ρ . The subdifferential of a function Ψ at ρ in the Wasserstein setting isdefined as the set of fields u such that Ψ( ρ ) + Z Ω h u , t ( x ) − x i dρ ( x ) ≤ Ψ( t ρ ) + o ( || t − i || ) , where t denotes a transport map acting on ρ . Note that the previous inequality does not provideany information as soon as t ρ is not feasible (in that case the right-hand side is + ∞ ). Letus consider a feasible field v ∈ C ρ , and let us assume that, for ε small enough, t ε = i + ε v pushes forward ρ onto a measure in K (this is not true in general, see Remark 2.3). Note that t ε is defined ρ -almost everywhere, with Γ out carrying some mass, but as it vanishes on Γ out (seeRemark 2.1), the singular part of ρ remains unchanged. Having ε go to in the subdifferentialinequality, we obtain Z ∇ D · v dρ ( x ) + Z u · v dρ ( x ) ≤ , so that u + ∇ D = u − U belongs to C ◦ ρ , the polar cone to C ρ . As u minimizes the L norm over U + C ρ , u identifies with the projection of U onto C ρ , which ends this unformal proof. Remark . In general, there exist feasible densities ρ ∈ K (defined by (6)) and fields v ∈ C ρ (defined by (3)) such that ( i + ε v ) ρ exits K for any ε > , this is why the considerationsabove do not make a rigorous proof. Consider for example ω a dense open subset in Ω , with asmall measure, and define ρ as ω c . The pressure space is { } , and C ρ is L (Ω) : any field isfeasible. If one considers now a strictly contractant field (with negative divergence), it is clearthat ( i + ε v ) ρ / ∈ K for any ε > . Notice also that this kind of paradox does not depend on thefact that we chose a “linear” perturbation ( i + ε v ) , since the same would happen if one, instead,perturbs the identity by following the flow of the vector field v for a time ε (which is classicallya better choice in order to satisfy the density constraint).As explained in the previous remark, the approach carried out in this section is not a rigorousproof that the advecting field is actually the projection of U onto C ρ . We conjecture thatprojecting ( i + ε v ) ρ onto K (for the Wasserstein distance) introduces a perturbation which isnegligible compared to ε , so that v may actually be used as a test-function, but this conjectureraises some technical issues which we were not able to solve. In what follows we give an alternateproof which circumvents the necessity to characterize ∂ ( J + I K ) .2.3. Notations and statement of the main result.
We first recall some results on thecontinuity equation: let ( ρ ( t, . )) t> be a family of density measures on R d , and v : ( t, x ) ∈ R + × R d v ( t, x ) ∈ R d be a Borel velocity field such that(9) Z T Z R d | v ( t, x ) | ρ ( t, x ) dx < + ∞ . MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE 7
We say that ( ρ, v ) satisfies the continuity equation with initial condition ρ (10) ( ∂ t ρ + ∇ · ( ρ v ) = 0 ρ (0 , . ) = ρ if for all ϕ ∈ C ∞ c ([0 , T [ × R d ) we have(11) Z T Z R d ( ∂ t ϕ ( t, x ) + ∇ ϕ ( t, x ) · v ( t, x )) ρ ( t, x ) dx + Z R d ϕ (0 , x ) ρ ( x ) dx = 0 . Let us recall that if ρ ( t, . ) is a solution of the continuity equation, there exists a narrowly con-tinuous curve ˜ ρ ( t, . ) such that ρ ( t, . ) = ˜ ρ ( t, . ) for a.e. t. In general, we will always focus on thiscontinuous representation.We now detail the construction of a discrete family of densities ( ρ kτ ) that approches in a sensewe will precise later the solution of the continuity equation we are interested with. For a fixedtime step τ > , we define the sequence ( ρ kτ ) of density measures on Ω using the recursive scheme:(12) ρ τ = ρ ρ kτ ∈ argmin P ( R d ) (cid:26) J ( ρ ) + I K ( ρ ) + 12 τ W ( ρ, ρ k − τ ) (cid:27) , where W is the Wasserstein distance, and J is the dissatisfaction functional defined in (8).This construction is a minimizing movement scheme as described by DeGiorgi and Ambrosioin [17, 1] and then - in the framework of probability measures - in [2, 3] with functional Φ( ρ ) = J ( ρ ) + I K ( ρ ) .We define on ˚Ω the discrete velocities: v kτ = i − t kτ τ , where t kτ is the unique optimal transportfunction from ρ kτ to ρ k − τ , which is well defined on ˚Ω (but not necessarely on Γ out , due to thesingular part of ρ kτ ). We also define E k τ = ρ kτ v kτ on ˚Ω (by abuse of notation, we will write ˚Ω insteadof Ω when we want to stress that we are not considering the boundary). We can interpolate thesediscrete values ( ρ kτ , v kτ , E k τ ) k ≥ by the piecewise constant functions defined by:(13) ρ τ ( t, . ) = ρ kτ v τ ( t, . ) = v kτ E τ ( t, . ) = E k τ if t ∈ ]( k − τ, kτ ] . Our goal is to prove that ρ τ converges when τ → to a solution of the continuity equation (10).Here is our main result: Theorem 2.4.
Let Ω be a convex bounded set of R d , D : R d → R a continuous λ -convex function, ρ a probability density, and ( ρ kτ ) constructed following the recursive scheme (12).Then there exists a family of probability densities ( ρ ( t, . )) t> , and a family of velocities ( u ( t, . )) t> such that ( ρ τ ( t, . ) , E τ ( t, . )) narrowly converges to ( ρ ( t, . ) , ρ ( t, . ) u ( t, . )) for a.e. t . Moreover, ( ρ, u ) satisfies the continuity equation: (14) ∂ t ρ + ∇ . ( ρ u ) = 0 u ( t, . ) = P C ρ ( t,. ) U for a.e. tρ (0 , . ) = ρ where U = −∇ D , and C ρ ( t,. ) is defined in (3). We will first prove this theorem in the particular case where there is no exit. In the followingsection, we thus assume that Γ out = ∅ , which will imply that all the measures are absolutelycontinuous with respect to the Lebesgue measure. Then we will extend the proof to the generalcase. BERTRAND MAURY, AUDE ROUDNEFF-CHUPIN, AND FILIPPO SANTAMBROGIO
Remark . We chose to assume here a λ − convexity hypothesis on D both in order to clarifysome statements, which are easier to state and prove under this assumption (see for instanceLemma 3.1 and the subsequent Remark 3.2) and because the typical case we think of is D = d ( · , Γ out ) , where Γ out is a flat part of the boundary of the convex set Ω . This implies that D isconvex as well. It would be interesting to study the case of non-convex domains Ω (for instancewith obstacles), and use the geodesic distance for computing D , which would lead to a non- λ − convex function, but this is not yet possible by means of our techniques, since one shouldwork with the Wasserstein distance W computed w.r.t. the geodesic distance itself, which is notmuch studied.Anyway, it can be checked that the only point throughout the paper where λ − convexity isused is the proof of Lemma 3.1, but Remark 3.2 explains how to get rid of this assumption: thismeans that, for existence purposes, this assumption may be withdrawn. On the other hand, the λ − convexity assumption is typical in this gradient flow problems, because it allows for uniquenessand stability results, and we think that similar results could be achieved in our case as well.3. Existence result in a domain with no exit
Technical lemmas.
Since we will make a strong use of optimality conditions in terms ofthe dual problem in Monge-Kantorovitch theory, let us briefly recall what we need.Given the two probabilities µ and ν on Ω we always have W ( µ, ν ) = max (cid:26)Z Ω ϕ dµ + Z Ω ψ dν, φ, ψ ∈ C (Ω) : φ ( x ) + ψ ( y ) ≤ | x − y | (cid:27) , the maximum being always realized by a pair of c − concave conjugate functions ( ϕ, ψ ) with ϕ = ψ c and ψ = ϕ c , where the c − transform of a function χ is defined through χ c ( y ) = inf x ∈ Ω | x − y | − χ ( x ) (with generalizations to other costs c rather than the square of the distance). We will callKantorovitch potential from µ to ν (resp., from ν to µ ) any c − concave function ϕ (resp., ψ ) suchthat ( ϕ, ϕ c ) (resp., ( ψ c , ψ ) ) realizes such a maximum. We have uniqueness of the optimal pairas soon as one of the support of one of the two measures is the whole domain Ω . Lemma 3.1.
Let D : R d R λ –convex, and ¯ ρ ∈ K . Then, there exists τ ∗ such that for all τ < τ ∗ :(i) The functional φ ( ρ ) = Φ( ρ ) + 12 τ W ( ρ, ¯ ρ ) admits a unique minimizer ρ m .(ii) There exists a Kantorovitch potential ¯ ϕ from ρ m to ¯ ρ , such that: (15) Z Ω (cid:16) D + ¯ ϕτ (cid:17) ρ ≥ Z Ω (cid:16) D + ¯ ϕτ (cid:17) ρ m for all ρ ≤ a.e. . Proof. (i) The existence of a minimizer can easily be proved using a minimizing sequence of φ ( ρ ) .Let ρ , ρ be two different minimizers, and r i the optimal transport between ¯ ρ and ρ i . We define r t := (1 − t ) r + t r and ρ t := r t ρ , for t ∈ ]0 , . We know that ρ t = ρ | det ∇ r t | ◦ ( r t ) − .As M (det M ) − is convex on the set of positive definite matrices S ++ d , and ∇ r i ∈ S ++ d , wehave : ρ t ( x ) ≤ (cid:18) − t det ∇ r + t det ∇ r (cid:19) ¯ ρ ◦ ( r t ) − ( x ) . MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE 9
We also know that ρ and ρ are admissible, therefore: ¯ ρ det ∇ r i ≤ a.e., and we obtain: ρ t ≤ .We have then φ ( ρ t ) = Z Ω D ((1 − t ) r ( x ) + t r ( x ))¯ ρ ( x ) dx + 12 τ W ( ρ t , ¯ ρ ) . Since D is λ –convex D ((1 − t ) r ( x ) + t r ( x )) ≤ (1 − t ) D ( r ( x )) + tD ( r ( x )) − λ t (1 − t ) | r ( x ) − r ( x ) | . Moreover, W ( ., ¯ ρ ) is –convex along the interpolation ρ t (see lemma 9.2.1 p. 206 in Ref [2]),therefore, for τ small enough, we have φ ( ρ t ) < (1 − t ) φ ( ρ ) + tφ ( ρ ) = inf K φ ( ρ ) , which is absurd.(ii) We first assume that ¯ ρ > a.e., which implies that the Kantorovich potential ¯ ϕ from ρ m to ¯ ρ ,satisfying ¯ ϕ ( x ) = 0 (with x any fixed point in Ω ), is unique. Let us define a small perturbationof ρ m : let ρ ≤ be a probability density, ε > and ρ ε := ρ m + ε ( ρ − ρ m ) . As ρ m minimizes φ ( ρ ) , we have:(16) J ( ρ ε ) − J ( ρ m ) + 12 τ ( W ( ρ ε , ¯ ρ ) − W ( ρ m , ¯ ρ )) ≥ . The first part of the left side of the inequality can easily be calculated: J ( ρ ε ) − J ( ρ m ) = Z Ω D ( x )( ρ ε − ρ m )( x ) dx = ε Z Ω D ( x )( ρ − ρ m )( x ) dx. Let us estimate the second part: we denote by ( ϕ ε , ψ ε ) some Kantorovich potentials associatedto ¯ ρ and ρ ε . We have W ( ρ ε , ¯ ρ ) = Z Ω ϕ ε ( x ) ρ ε ( x ) dx + Z Ω ψ ε ( y )¯ ρ ( y ) dy W ( ρ m , ¯ ρ ) ≥ Z Ω ϕ ε ( x ) ρ m ( x ) dx + Z Ω ψ ε ( y )¯ ρ ( y ) dy, where φ ε is a Kantorovitch potential from ρ ε to ¯ ρ . Thus:
12 ( W ( ρ ε , ¯ ρ ) − W ( ρ m , ¯ ρ )) ≤ Z Ω ϕ ε ( x )( ρ ε − ρ m )( x ) dx = ε Z Ω ϕ ε ( x )( ρ − ρ m )( x ) dx, and we can deduce from inequality (16) that: Z Ω D ( x )( ρ − ρ m )( x ) dx + 1 τ Z Ω ϕ ε ( x )( ρ − ρ m )( x ) dx ≥ for all admissible ρ. Let ε tend to : ϕ ε converges to the unique Kantorovich potential ¯ ϕ from ρ m to ¯ ρ . This gives Z Ω D ( x )( ρ − ρ m )( x ) dx + 1 τ Z Ω ψ c ( x )( ρ − ρ m )( x ) dx ≥ for all admissible ρ. We now prove the general case: let ¯ ρ δ > a.e., ¯ ρ δ ≤ a.e., such that ¯ ρ δ converges to ¯ ρ when δ tends to . Using (i), there exists a unique minimizer ρ m,δ of φ δ ( ρ ) := Z Ω Dρ + I K + 12 τ W ( ρ, ¯ ρ δ ) ,and it converges to ρ m as δ tends to . Moreover, we have proved that: Z Ω D ( x )( ρ − ρ m,δ )( x ) dx + 1 τ Z Ω ¯ ϕ δ ( x )( ρ − ρ m,δ )( x ) dx ≥ for all admissible ρ, with ¯ ϕ δ that converges to a Kantorovitch potential ¯ ϕ . Taking the limit δ → , we obtain thedesired inequality. For this kind of arguments concerning optimality for transport costs andother functionals, see for instance Ref. [10]. (cid:3) Remark . if D is not λ –convex, we cannot prove uniqueness of the minimizer of φ . However,if ρ m is a minimizer, it still satisfies inequality (ii). Indeed, in the second part of the proof of (ii),we can define ρ m,δ as a minimizer of φ δ ( ρ ) + c δ W ( ρ, ρ m ) , where c δ → (so that the optimalitycondition that we see at the limit δ → disregards this term), but slowly (so that it makes ρ m,δ converge to ρ m ). Obviously this kind of argument was not necessary if one only wanted to provethis optimality condition for one minimizer ρ m , and not for every minimizer. Lemma 3.3. (Decomposition of the spontaneous velocity):The spontaneous velocity U = −∇ D can be written as follows: (17) U = v kτ + ∇ p kτ with p kτ ∈ H ρ kτ . Proof.
Using the previous lemma, we know that there exists a Kantorovich potential ¯ ϕ from ρ kτ to ρ k − τ such that ρ kτ is a solution of the minimizing problem: ρ kτ ∈ argmin ρ ∈ K (cid:26)Z Ω D ( x ) ρ ( x ) dx + 1 τ Z Ω ¯ ϕ ( x ) ρ ( x ) dx (cid:27) , which imposes: ρ kτ = 1 on [ F < l ] ρ kτ ≤ on [ F = l ] ρ kτ = 0 on [ F > l ] , with F : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ω → R x D ( x ) + ¯ ϕ ( x ) τ , and l ∈ R chosen such that ρ kτ satisfies: Z Ω ρ kτ dx = 1 .We can then define a pressure like function:(18) p kτ ( x ) := ( l − F ( x )) + = (cid:18) l − D ( x ) − ¯ ϕ ( x ) τ (cid:19) + which satisfies: p kτ ≥ , and p kτ = 0 on [ ρ kτ < , therefore p kτ ∈ H ρ kτ .Moreover, on [ ρ kτ > , we have: ∇ p kτ = −∇ D − ∇ ¯ ϕτ (where the density vanishes v kτ may bemodified at will, so that we can keep the same formula). Since we have v kτ = i − t kτ τ = ∇ ¯ ϕτ , we get the desired decomposition for the spontaneous velocity : U = v kτ + ∇ p kτ . (cid:3) Let us now define the densities ˜ ρ τ ( t ) that interpolate the discrete values ( ρ kτ ) along geodesics:(19) ˜ ρ τ ( t ) = (cid:18) t − ( k − ττ ( id − t kτ ) + t kτ (cid:19) ρ kτ . We also define ˜v τ ( t, . ) as the unique velocity field such that ˜v τ ( t, . ) ∈ Tan ˜ ρ t P ( R d ) and (˜ ρ τ , ˜v τ ) satisfy the continuity equation. As before, we define: ˜E τ = ˜ ρ τ ˜v τ .After these definitions we will give some a priori bounds on the curves, the pressures and thevelocities that we defined. In order to get these bounds, we need to start from some estimateswhich are standard in the framework of Minimizing Movements. The sequence ( ρ kτ ) k satisfiesan estimate on its variation which gives a Hölder and H behavior. From the minimality of ρ kτ ,compared to ρ k − τ , one gets W ( ρ kτ , ρ k − τ ) ≤ τ (Φ( ρ kτ ) − Φ( ρ k − τ )) . MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE 11
Since Φ coincides with J , which is bounded, on the sequence ( ρ kτ ) k , then we have W ( ρ kτ , ρ k − τ ) ≤ Cτ (discrete Hölder behavior), but we also have, if we sum up over k (20) X k τ (cid:18) W ( ρ kτ , ρ k − τ ) τ (cid:19) ≤ ρ ) , which is the discrete version of an H estimate. As for ˜ ρ τ ( t ) , it is an absolutely continuous curvein the Wasserstein space and its velocity on the time interval [( k − τ, kτ ] is given by the ratio W ( ρ k − τ , ρ kτ ) /τ . Hence, the L norm of its velocity on [0 , T ] is given by(21) Z T | ˜ ρ ′ τ | W ( t ) dt = X k W ( ρ kτ , ρ k − τ ) τ , and, thanks to (20), it admits a uniform bound independent of τ (here we use the notation | σ ′ | ( t ) for the metric derivative of a curve σ and | σ ′ | W ( t ) means that this metric derivative is computedaccording to the distance W ). This gives compactness of the curves ˜ ρ τ , as well as an Hölderestimate on their variations (since H ⊂ C , / ). Lemma 3.4. (A priori estimates):We have the following a priori estimates:(i) v τ is τ -uniformly bounded in L ((0 , T ) , L ρ τ (Ω)) .(ii) p τ is τ -uniformly bounded in L ((0 , T ) , H (Ω)) .(iii) E τ and ˜E τ are τ -uniformly bounded measures.Proof. (i) We have the following equalities: Z T Z Ω ρ τ | v τ | = X k Z kτ ( k − τ Z Ω ρ kτ | v kτ | = X k Z kτ ( k − τ dt ! (cid:18)Z Ω ρ kτ ( x ) | x − t kτ ( x ) | τ dx (cid:19) = X k τ W ( ρ k − τ , ρ kτ ) τ = 1 τ X k W ( ρ k − τ , ρ kτ ) . Thanks to the general estimate (20) we get Z T Z Ω ρ τ | v τ | ≤ ρ ) .(ii) Since we have shown the following decomposition: ∇ p τ = −∇ D − v τ , we have: Z T Z Ω ρ τ |∇ p τ | ≤ Z T Z Ω ρ τ | v τ | + 2 Z T Z Ω ρ τ |∇ D | ≤ C. But p τ = 0 on [ ρ τ < , therefore Z T Z Ω |∇ p τ | = Z T Z Ω ρ τ |∇ p τ | ≤ C .(iii) We look at ˜E τ and we use the estimates (20) and (21). Z T Z Ω | ˜E τ | = Z T Z Ω ˜ ρ τ | ˜v τ | ≤ Z T (cid:18)Z Ω ˜ ρ τ | ˜v τ | (cid:19) (cid:18)Z Ω ρ τ (cid:19) | {z } =1 ≤ Z T (cid:18)Z Ω ˜ ρ τ | ˜v τ | (cid:19) ≤ √ T (cid:18)Z T Z Ω ρ τ | v τ | (cid:19) ≤ C. Therefore, ˜E τ is a τ -uniformly bounded measure. The proof for E τ is almost the same, estimating L norms with L norms by Cauchy-Schwartz. (cid:3) Lemma 3.5.
Assume that µ and ν are absolutely continuous measures, whose densities arebounded by a same constant C. Then, for all function f ∈ H (Ω) , we have the following inequality: (22) Z Ω f d ( µ − ν ) ≤ √ C ||∇ f || L (Ω) W ( µ, ν ) . Proof.
Let µ t be the constant speed geodesic between µ and ν , and w t the velocity field such that ( µ, w ) satisfies the continuity equation, and || w t || L ( µ t ) = W ( µ, ν ) . For all t , µ t is absolutelycontinuous, and its density is bounded by the same constant C a.e.. Therefore: Z Ω f d ( µ − ν ) = Z ddt (cid:18)Z Ω f ( x ) dµ t ( x ) (cid:19) dt = Z Z Ω ∇ f · w t dµ t dt ≤ (cid:18)Z Z Ω |∇ f | dµ t dt (cid:19) / (cid:18)Z Z Ω | w t | dµ t dt (cid:19) / ≤ √ C ||∇ f || L (Ω) W ( µ, ν ) . (cid:3) Remark:
With the same method, we can also prove: Z Ω f d ( µ − ν ) ≤ C p ||∇ f || L p (Ω) W q ( µ, ν ) for all f ∈ W ,p and q such that p + q = 1 . More generally, if µ, ν ∈ L r (Ω) and || µ || L r , || ν || L r ≤ C ,one has Z Ω f d ( µ − ν ) ≤ C q ′ ||∇ f || L p (Ω) W q ( µ, ν ) , provided p + q + r = 1 + qr . Proof of the theorem in a domain with no exit.
Step 1: convergence of (˜ ρ τ , ˜E τ ) and ( ρ τ , E τ ) . We have proved that ˜ ρ τ and ˜E τ are τ -uniformlybounded measures, thus there exists ( ρ, E ) such that (˜ ρ τ , ˜E τ ) converges narrowly to ( ρ, E ) . Letus prove that ( ρ τ , E τ ) converges to the same limit as (˜ ρ τ , ˜E τ ) .We start from the ρ − part. The curves ˜ ρ τ actually converge uniformly in [0 , T ] with respectto the W − distance. The curves ρ τ and ˜ ρ τ coincide on every time of the form kτ . The formeris constant on every interval ]( k − τ, kτ ] , whereas the latter is uniformly Hölder continuous ofexponent / , which implies W (˜ ρ τ ( t ) , ρ τ ( t )) ≤ Cτ . This proves that ρ τ converges uniformlyto the same limit as ˜ ρ τ .We now consider a function f ∈ C ∞ c ([0 , T ] × Ω) , and prove that Z T Z Ω f (cid:0) ˜E τ − E τ (cid:1) convergesto as τ tends to . We have: ˜ ρ τ ( t, . ) = T t ρ kτ where T t = ( t − ( k − τ ) v kτ + t k τ . Therefore ˜ ρ τ ( t + h, . ) = ( T t + h v kτ ) ρ kτ = (( id + h v kτ ◦ T t − ) ◦ T t ) ρ kτ = ( id + h v kτ ◦ T t − ) ρ τ ( t, . ) , which implies that: t ˜ ρ τ ( t + h ,. ) ˜ ρ τ ( t ,. ) = id + h v kτ ◦ T t − . We can then express ˜v τ explicitely : ˜v τ ( t, . ) = lim h → t ˜ ρ τ ( t + h ,. ) ˜ ρ τ ( t ,. ) − id h = lim h → h v kτ ◦ T t − h = v kτ ◦ T t − , MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE 13 and obtain Z Ω f ( t, x ) ˜ ρ τ ( t, x ) ˜v τ ( t, x ) dx = Z Ω f ( t, T t ( x )) ρ kτ ( x ) ˜v τ ( t, T t ( x )) dx = Z Ω f ( t, T t ( x )) ρ kτ ( x ) v kτ ( x ) dx. Hence Z T Z Ω f (cid:0) ˜E τ − E τ (cid:1) ≤ X k Z τ k +1 τ k Z Ω | f ( t, x ) − f ( t, T t ( x )) | | v kτ ( x ) | ρ kτ ( x ) dx dt ≤ X k Z τ k +1 τ k Z Ω Lip f | x − T t ( x ) || v kτ ( x ) | ρ kτ ( x ) dx dt ≤ X k Z τ k +1 τ k Z Ω Lip f τ | v kτ ( x ) | ρ kτ ( x ) dx dt ≤ C Lip f τ.
Step 2: existence of the limit velocity.
Let us prove that E is absolutely continuous with respectto ρ . Let θ be a scalar measure, and F a vectorial measure: the function Θ : ( θ, F ) Z T Z Ω | F | θ if F << θ a.e. t ∈ [0 , T ]+ ∞ otherwiseis l.s.c. for the weak– ⋆ convergence of measures. Since we have shown the τ -uniform bound: Z T Z Ω | E τ | ρ τ ≤ C, we have Θ( ρ, E ) < + ∞ . Therefore E is absolutely continuous with respect ρ , and there exists u ( t, . ) ∈ L ( ρ ( t, . )) such that E = ρ u . Moreover, ( ρ, ρ u ) satisfies the (linear) continuity equation,as limit of (˜ ρ τ , ˜E τ ) .Let us now prove that u ( t ) ∈ C ρ ( t ) . Let t ∈ (0 , T ) , h > , and q ∈ H ρ ( t ,. ) . By the continuityequation, we have Z t + ht Z Ω ∇ q ( x ) · u t ( x ) ρ ( t, x ) dx = Z Ω [ ρ ( t , x ) − ρ ( t + h, x )] q ( x ) dx. Since ρ ( t , . ) = 1 wherever q > , and ρ ( t + h, . ) ≤ a.e., Z Ω [ ρ ( t , x ) − ρ ( t + h, x )] q ( x ) dx ≤ ,and we have for a.e. t ≥ h Z t + ht Z Ω ∇ q ( x ) . u t ( x ) ρ ( t, x ) dx −→ h → Z Ω ∇ q ( x ) · u ( t , x ) ρ ( t , . )( x ) dx = Z Ω ∇ q ( x ) . u ( t , x ) dx. Using the same method between t − h and t , we also obtain the converse inequality. Finally,we have for a.e. t (23) Z Ω ∇ q ( x ) · u ( t , x ) dx = 0 for all q ∈ H ρ ( t ,. ) . Step 3: the limit velocity satisfies: u = P C ρ U . We first prove the decomposition: U = u ( t, . ) + ∇ p ( t, . ) for a.e. t. We have E τ = ρ τ v τ = − ρ τ ( ∇ D + ∇ p τ ) = − ρ τ ∇ D − ∇ p τ since p τ = 0 on [ ρ τ < . Let us prove that p τ converges to p ∈ H ρ : as p τ ∈ L ([0 , T ] , H (Ω)) , there exists p such that p τ weakly converges to p in L ([0 , T ] , H (Ω)) . We have obviously: p ≥ a.e., but it ismore difficult to show that p ( t, . ) = 0 on [ ρ ( t ) < . We consider the average functions: p a,bτ = 1 b − a Z ba p τ ( t, . ) dt and p a,b = 1 b − a Z ba p ( t, . ) dt. Since p τ = 0 on [ ρ τ < , we have Z ba Z Ω p τ ( t, x )(1 − ρ τ ( t, x )) dx dt = 1 b − a Z ba Z Ω p τ ( t, x )(1 − ρ τ ( a, x )) dx dt + 1 b − a Z ba Z Ω p τ ( t, x )( ρ τ ( a, x ) − ρ τ ( t, x )) dx dt. The first integral reads: Z Ω p a,bτ ( x )(1 − ρ τ ( a, x )) dx −→ τ → Z Ω p a,b ( x )(1 − ρ ( a, x )) dx , as p a,bτ weaklyconverges in H (Ω) – therefore strongly in L (Ω) – to p a,b , and ρ τ ( a, . ) weakly– ⋆ converges in L ∞ (Ω) to ρ ( a, . ) . Moreover, for every Lebesgue point a of p ( ., x ) , we have: p a,b −→ b → a p ( a, . ) ,therefore, for all these a , we have Z Ω p a,b ( x )(1 − ρ ( a, x )) dx dt −→ b → a Z Ω p ( a, x )(1 − ρ ( a, x )) dx. Using lemma 3.5, we obtain for the second integral: Z ba Z Ω p τ ( t, x ) (cid:0) ρ τ ( a, x ) − ρ τ ( t, x ) (cid:1) dx dt ≤ Z ba ||∇ p τ ( t, . ) || L (Ω) W ( ρ τ ( a, . ) , ρ τ ( t, . )) | {z } ≤ C √ b − a dt ≤ C √ b − a (cid:18)Z ba ||∇ p τ ( t, . ) || L (Ω) dt (cid:19) (cid:18)Z ba dt (cid:19) ≤ C ( b − a ) (cid:18)Z ba ||∇ p τ ( t, . ) || L (Ω) dt (cid:19) . As Z T ||∇ p τ ( t, . ) || L (Ω) dt is τ -uniformly bounded, ||∇ p τ ( t, . ) || L (Ω) weakly converges to a mea-sure µ . Therefore, beyond a zero measure set of points a , we have lim τ → b − a Z ba Z Ω p τ ( t, x )( ρ τ ( a, x ) − ρ τ ( t, x )) dx dt ≤ C p µ ([ a, b ]) −→ b → a . We finally obtain: Z Ω p ( a, x )(1 − ρ ( a, x )) dx = 0 for almost every a .Hence E = − ρ ∇ D − ∇ p , with p = 0 on [ ρ < , so: E = − ρ ( ∇ D + ∇ p ) . Since: E = ρ u , wehave shown the following decomposition: u = −∇ D − ∇ p i.e. U = ∇ p + u . MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE 15
Moreover, by Equality (23), u and ∇ p satisfy the complementarity relation Z Ω ∇ p ( t, x ) · u ( t, x ) dx = 0 for a.e. t, which implies that we have exactly: u ( t ) = P C ρ ( t ) U .4. Proof of the theorem in the general case
We consider here the general case where Γ out = ∅ .4.1. Lack of geodesic convexity.
The main problem we encounter when we want to generalizethe previous proof is the fact that the classical geodesics no longer belong to the admissiblespace K , which is no more a geodesically convex set. Indeed, if we consider a density ρ whichis constant equal to on a subset of Ω , a measure ρ which is concentrated on Γ out , and thegeodesic ρ ( t, . ) between them, the density of ρ ( t, . ) will be of the order of / (1 − t ) where it ispositive, and therefore ρ ( t, . ) K for all t ∈ ]0 , .This is one of the main reasons that prevent from using the standard theory of gradient flowfor geodesically convex functionals in the Wasserstein space (see [2]).In this section we will investigate the connectedness properties of the set K . For the sake ofthis work, we will see that we need to estimate the length to connect two measures in K at avery single point of the proof. Yet, we think that these estimates are interesting in themselvesand this is why we try to present them so that they will be valid in any dimension d .We define a new distance, coming from a minimal length approach, on K : Proposition 4.1. (Continuity of the length L) For µ, ν ∈ K , we define the length (24) L ( µ, ν ) = inf (cid:26)Z | σ ′ | W ( t ) dt : σ ( t ) ∈ K, σ (0) = µ, σ (1) = ν (cid:27) . This length is finite, and it is a distance on K which is continuous for the narrow convergence:if ( µ n ) , ( ν n ) are sequences that narrowly converge in K to µ and ν , then L ( µ n , ν n ) converges to L ( µ, ν ) . To prove this proposition, we will first analyze the case were the domain Ω is the unit cubeand the door is one of the sides. We set Q = ]0 , d − × ] − , , Q = [0 , d − × [ − , and S = [0 , d − × { } . We will still denote by K the set of admissible measures, i.e. those who arecomposed by a density less than in Q and by a possibly singular part on S . We will denote by y the last component of a point ( x, y ) ∈ R d = R d − × R . When integrating over S , we write dx instead of H d − ( dx ) or similar expressions.Let us start from a simpler case.A first useful lemma is the following: Lemma 4.2.
Let ρ , ρ be two probability measures on Q of the form ρ i = ρ iQ + ρ iS , where ρ iQ has a density on Q bounded by k and ρ iS is concentrated on S . Set ℓ = W ( ρ , ρ ) . Then, forany Lipschitz continuous function j we have Z S jd ( ρ S − ρ S ) ≤ Lip ( j ) ℓ + c ( k ) || j || L ∞ ℓ / , Z Q j ( ρ Q − ρ Q ) ≤ Lip ( j ) ℓ + c ( k ) ||| j || L ∞ ℓ / . Proof.
We start from the first estimate: consider a function χ δ : Q → [0 , such that χ δ = 1 on S , χ δ = 0 outside a strip of width δ from S , and |∇ χ δ | ≤ δ − (as a matter of fact, it defines this function as χ δ ( x, y ) = (1 + δ − y ) + ). We may write Z S jd ( ρ S − ρ S ) = Z Q jχ δ d ( ρ − ρ ) − Z Q jχ δ d ( ρ Q − ρ Q ) ≤ Lip ( jχ δ ) ℓ + kδ || j || L ∞ . Then we use
Lip ( jχ δ ) ≤ Lip ( j ) + || j || L ∞ δ − and we get Z S jd ( ρ S − ρ S ) ≤ (cid:18) Lip ( j ) + || j || L ∞ δ (cid:19) ℓ + kδ || j || L ∞ , which implies, by choosing δ = ℓ / , Z S jd ( ρ S − ρ S ) ≤ Lip ( j ) ℓ + c ( k ) || j || L ∞ ℓ / . As far as the second estimate is concerned, just write Z Q j ( ρ Q − ρ Q ) = Z Q j ( ρ − ρ ) − Z S jd ( ρ S − ρ S ) and use Z Q j ( ρ − ρ ) ≤ Lip ( j ) ℓ and the previous inequality. (cid:3) It is important to notice in the above inequality that, once we fix ρ iQ or ρ iS , the two estimatesseparately make ℓ appear, where ℓ may be the W distance between any pair of measures,satisfying the constraints, having ρ iQ or ρ iS as an internal or boundary part, respectively. Thepair of measures we use need not to be the same in the two estimates. Lemma 4.3.
Let ρ , ρ ∈ K be two admissible probability measures on Q and L, M ≥ . Supposethat ρ and ρ are of the following form: ρ i = ρ iQ + ρ iS , ρ iQ ≪ L d , ρ iS = h i · H d − , h i ≤ M, Lip ( h i ) ≤ L, i = 0 , . Then, there exists a curve ρ t from ρ to ρ , contained in K (the set of admissible measures) andsuch that its W − length does not exceed C ( d ) M / p Lℓ + M ℓ / , where we set ℓ := W ( ρ , ρ ) .Moreover, the same stays true if ℓ stands for a number such that there exist “extensions” of ρ iQ on S and of ρ iS on Q that belong to K and such that for both extensions the new W − distance issmaller than ℓ (but the two extensions may be different). If instead of staying in K the constrainton the density in Q is relaxed to “being smaller than k ” with k > , the constant will also dependon k , as in Lemma 4.2.Proof. It is possible to replace the two probabilities on Q with probabilities ˜ ρ i on R = [0 , d − × [ − , M ] so that ˜ ρ i is absolutely continuous with density less than and ( π Q ) ˜ ρ i = ρ i (where π Q is the projection on Q ). We will take ˜ ρ i = ρ iQ + y To estimate W , take a function f ∈ Lip ( R ) . What follows will be easier to justify in case f is regular but everything will work (by density, or instance), for any f whose Lipschitz constantdoes not exceed . Let us define, for x ∈ [0 , d − and a, b ∈ [0 , M ] , g ( x, a, b ) = R ba f ( x, t ) dtb − a . We denote by g x , g a and g b the partial derivatives of g . We can verify that | g x ( x, a, b ) | = (cid:12)(cid:12)(cid:12)R ba f x ( x, t ) dt (cid:12)(cid:12)(cid:12) | b − a | ≤ Lip ( f ) = 1 , then we compute g b and we get | g b ( x, a, b ) | = (cid:12)(cid:12)(cid:12)(cid:12) f ( x, b ) b − a − g ( x, a, b ) b − a (cid:12)(cid:12)(cid:12)(cid:12) ≤ Lip ( f ) | b − a | | b − a | = 1 , and, analogously, | g a | ≤ .In particular, if one takes two Lipschitz functions a ( x ) and b ( x ) , one has Lip ( g ( x, a ( x ) , b ( x ))) ≤ Lip ( a ) + Lip ( b ) .Now we write Z R f d (˜ ρ − ˜ ρ ) = Z Q f d ( ρ Q − ρ Q ) + Z S Z h ( x )0 f ( x, t ) dt − Z h ( x )0 f ( x, t ) dt ! dx. We estimate both terms thanks to the previous lemma. The first term in the right hand side isless than ℓ , while for the second we may write Z S Z h ( x )0 f ( x, t ) dt − Z h ( x )0 f ( x, t ) dt ! dx = Z S g ( x, h ( x ) , h ( x ))( h − h )( x ) dx. Hence we are in the case of the previous lemma with j ( x ) = g ( x, h ( x ) , h ( x )) , and hence Lip ( j ) ≤ L and || j || L ∞ ≤ M + √ d (the first estimate comes from our study of g , for thesecond just suppose that f vanishes somewhere on S ).Hence we get, using the arbitrariness of the function fW (˜ ρ , ˜ ρ ) ≤ ℓ + (1 + 2 L ) ℓ + 2 M ℓ / . To simplify the computations we use ≤ M, L and get W (˜ ρ , ˜ ρ ) ≤ C ( d ) M / W (˜ ρ , ˜ ρ ) / ≤ C ( d ) M / p Lℓ + M ℓ / . The last part of the statement is an easy consequence of the technique we used and of Lemma4.2. (cid:3) Theorem 4.4. Let µ and µ be two probabilities in K . Then there exists a curve ( µ t ) t connecting µ to µ , such that its W − length does not exceed C ( d ) W ( µ , µ ) / (4 d ) and that µ t ∈ K for every t .Proof. Take ε > and modify µ i into a new measure ρ i ∈ K by regularizing in the directionof x : it is sufficient to take the convolution of µ iS with a kernel of the form C ( ε − d − ε − d | x | ) + .This ensures that the W distance has not increased and that the new measures on S will haveLipschitz and bounded densities on S , with M ≤ Cε − d and L ≤ Cε − d , and on Q the constraintis kept as well. Yet, there is a problem: these measures may exit the domain. There are twopossible ways for solving this problem, and both will be useful.One possibility is rescaling of a factor (1 + 2 ε ) − , so that all the mass is pushed again into thedomain. This does not change significantly the values of M and L but the densities inside willbe no more bounded by . They will be bounded by a constant k close to . In this case too theWasserstein distance has not increased, since the rescaling was a contraction. The other possibility is composing with a contracting transport T : Q ε → Q ( Q ε being the ε − neighborhood of Q ), which is chosen so that the convolution of the constant function is sentonto the constant function (this is possible thanks to the fact the convolution keeps the massunchanged). This construction ensures that the constraint inside Q will be satisfied but unluckily,since the inverse of T is not Lipschitz continuous (due to the fact that the densities vanished onthe boundary of Q ε ), it is not suitable for S . Anyway, in this case too, the Wasserstein distancewas not increased.Hence we do a mixed procedure: we use the second possibility in Q and the first on S . It isclear that in this way we have good densities both in Q and on S , and we can apply Lemma 4.2and the last statement of Lemma 4.3. Notice that the W − distance between the two measures ρ i ∈ K that we constructed could be larger than ℓ . It is easily estimated by something like ℓ + ε but this is not sufficient for the following estimates.Now, to connect µ to µ , one can first connect each µ i to ρ i , and the cost is no more than ε ,since it is sufficient to spread every particle on a ball of radius ε (i.e. the radius of the supportof the previous kernel) when we do convolution, and then to move it no more than ε when wecompose with a contraction. After that, one uses the previous Lemma to estimate the length forconnecting ρ to ρ and gets min length ≤ ε + C ( d ) ε (1 − d ) / p ε − d ℓ + ε − d ℓ / = 2 ε + C ( d ) ε − d r ℓε + ℓ / ε , where ℓ denotes the W distance between µ and µ . If one supposes that ε is chosen so that ℓε − is smaller than , one can estimate the last sum in the square root and get min length ≤ ε + C ( d ) ε − d ℓ / . Choosing ε = ℓ / (4 d ) gives at the same time that ℓε − is small and that the minimal length maybe estimated by ℓ / (4 d ) . (cid:3) To approach the general case one needs to use the following theorem, which has alreadybeen used in a transport-related setting with density constraints in the variational theory ofincompressible Euler equations by Y. Brenier and provides a useful tool for reducing to the cube(see Refs [16] and [40] for the applications to fluid mechanics). Theorem 4.5. For any sufficiently good domain Ω ⊂ R d which is homeomorphic to the cube,there exists a bi-lipschitz homeomorphism φ : Ω → Q such that φ ( L d | Ω ) = c L d | Q . Moreover, thebehavior of φ on the boundary may be prescribed at will. Generalization of the technical lemmas. In this section, we briefly explain how togeneralize the technical lemmas we used in the first proof (with Γ out = ∅ ).Conditions on the minimizer for the discrete problem. • First of all, in lemma 3.1, we can’t prove the uniqueness of the minimizer ρ m with thesame method. Indeed, exactly as we explained in the previous section for geodesics,the interpolation ρ t between two possible minimizers does not necessarily belong to K .Therefore, we will have to apply the “selection” method explained in Remark 3.2 in orderto prove inequality (15). More precisely, the case where ¯ ρ > remains unchanged, butin the general case, we fix a minimizer ρ m of φ , and we define ρ m,δ as a minimizer of φ δ ( ρ ) := Z Ω Dρ + I K ( ρ ) + 12 τ W ( ρ, ¯ ρ δ ) + c δ W ( ρ, ρ m ) , MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE 19 with c δ that converges to slower than W (¯ ρ, ¯ ρ δ ) . Since ρ m and ρ m,δ are minimizers of J and φ δ , we have the following inequalities: Z Ω Dρ m,δ + 12 τ W ( ρ m,δ , ¯ ρ δ ) + c δ W ( ρ, ρ m ) ≤ Z Ω Dρ m + 12 τ W ( ρ m , ¯ ρ δ ) Z Ω Dρ m + 12 τ W ( ρ m , ¯ ρ ) ≤ R Ω Dρ m,δ + 12 τ W ( ρ m,δ , ¯ ρ ) which implies, using the triangular inequality: W ( ρ m,δ , ρ m ) ≤ τ c δ (cid:2) W ( ρ m,δ , ¯ ρ ) − W ( ρ m,δ , ¯ ρ δ ) + W ( ρ m , ¯ ρ δ ) − W ( ρ m , ¯ ρ ) (cid:3) ≤ C τ c δ W (¯ ρ, ¯ ρ δ ) −→ δ → . Therefore, ρ m,δ converges to the fixed minimizer ρ m . We can then pass to the limit δ → , and obtain the same inequality for ρ m . • It is also useful to notice another feature of the problem with an exit Γ out : once somemass arrives to the exit, it does not move anymore. This precisely means the following:if γ kτ is an optimal transport plan from the selected measure ρ kτ to the previous one, ρ k − τ ,and ( x, y ) ∈ supp( γ kτ ) with y ∈ Γ out , then y = x . This means that all the mass whichwas already on the door for ρ k − τ will not move. To prove it, it is sufficient to considerthe map F : Ω × Ω → Ω × Ω defined by F ( x, y ) = ( y, y ) if y ∈ Γ out and F ( x, y ) = ( x, y ) if y / ∈ Γ out . The measure F γ kτ is a transport plan between a new measure ρ and ρ k − τ ,which reduces the transport cost and the functional J (since D is minimal on the exit).Moreover, since ρ is obtained from ρ kτ by moving some mass onto the door, we have ρ ∈ K as well. This would contradict the optimality of ρ kτ unless F γ kτ = γ kτ , which isthe thesis.This also proves uniqueness of the optimal transport plan between ρ kτ and ρ k − τ since,if we look it the other way around (from ρ k − τ to ρ kτ ), we can decompose the problemin one part which will not move (corresponding to ρ k − τ Γ out ) and one part which is thetransport of an absolutely continuous density ( ρ k − τ ˚Ω ).We will also denote by E kτ the excess mass of ρ kτ with respect to ρ k − τ on the exit, i.e. E kτ := ρ kτ (Γ out ) − ρ k − τ (Γ out ) ≥ . • In lemma 3.3, the solution of ρ kτ ∈ argmin ρ ∈ K (cid:26)Z Ω D ( x ) ρ ( x ) dx + 1 τ Z Ω ¯ ϕ ( x ) ρ ( x ) dx (cid:27) is not necessarily the same in the general case, as there exists no limit density on Γ out .Let us define: l := inf x ∈ Γ out F ( x ) , and Γ min = { x ∈ Γ out : F ( x ) = l } . If | [ F < l ] | ≥ , thenthe solution is the same as in the previous proof. However, if | [ F < l ] | < , it costs lessto put a part of the density onto Γ out . The solution is therefore given by: ρ kτ = 1 on [ F < l ] ,ρ kτ > on Γ min , with ρ kτ (Γ min ) = 1 − | [ F < l ] | ,ρ kτ ≤ on [ F = l ] \ Γ min ,ρ kτ = 0 on [ F > l ] . The pressure p kτ defined by p kτ ( x ) := ( l − F ( x )) + = (cid:18) l − D ( x ) − ¯ ϕ ( x ) τ (cid:19) + then belongs to H ρ kτ , and we prove the decomposition U = v kτ + ∇ p kτ as before.In order to prove the a priori estimates of lemma 3.4, we have to take into account thesingularity part of the densities on Γ out . Notice that, to avoid any ambiguity where the transportdoes not exist, we only defined a discrete velocity vector field inside ˚Ω . To be clearer, we wantto spend some disambiguation words on what ˜E τ and E τ are in this case. • The measure ˜E τ is as usual defined as the vector measure satisfying the continuity equa-tion with the curve ˜ ρ τ . We also have an explicit formula in terms of the optimal transportplans γ kτ from ρ kτ to ρ k − τ : for any t ∈ [ k − τ, kτ [ take ˜E τ ( t ) := (cid:0) π ( kτ − t ) /τ (cid:1) (cid:18) x − yτ · γ kτ (cid:19) , where π s ( x, y ) = (1 − s ) x + sy . • The measure E τ is simply defined as the product of ρ τ ˚Ω times the velocity vector fielddefined in Section 2.3, on the non-singular part only (again we use ˚Ω instead of Ω tostress that the boundary is excluded). As before, the idea is that this vector measuresatisfies good properties from optimality conditions, while the previous one satisfies thecontinuity equation. We need to compare them. • There is also in this case a third vector measure, that we can call ˆE τ , which is definedexactly as ˜E τ but ignoring the part on Γ out : ˆE τ ( t ) := (cid:0) π ( kτ − t ) /τ (cid:1) (cid:18) x − yτ · x ∈ ˚Ω γ kτ (cid:19) . The utility of ˆE τ is that it is more easily comparable to E τ .We come back to the proof of lemma 3.4: as a matter of fact, we now have: W ( ρ k − τ , ρ kτ ) = τ Z Ω ρ kτ | v k τ | + Z Γ out × Ω | x − y | dγ kτ ( x, y ) , where γ kτ is the optimal transport plan between ρ kτ and ρ k − τ . Therefore, we have Z Ω ρ τ | v τ | ≤ τ − W ( ρ k − τ , ρ kτ ) , and the a priori estimates (i) and (ii) are still satisfied. The proof of (iii) is unchanged, but letus remark that we have no longer the equality: Z Ω ˜ ρ kτ | ˜v k τ | = Z Ω ρ kτ | v k τ | , and that the geodesic ˜ ρ τ does not belong to K .Lemma 3.5 is no longer true for densities that are not absolutely continuous with respect tothe Lebesgue measure. Indeed, as we have seen before, the geodesic between two densities of K does not belong to K . We prove instead the following lemma: Lemma 4.6. Let µ, ν ∈ K . Then, for all function f ∈ H with f = 0 on Γ out , we have thefollowing inequality: Z Ω f d ( µ − ν ) ≤ ||∇ f || L (Ω) L ( µ, ν ) where L ( µ, ν ) is the length of the shortest path in K joining µ and ν (see (24)). MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE 21 Proof. The proof is an adaptation of the one of Lemma 3.5 : let σ t be a minimal length curve in K joining µ and ν , and let w t such that ( σ, w ) satisfies the continuity equation and || w t || L ( σ t ) = L ( µ, ν ) . Since f ∈ H with f = 0 on Γ out , then ∇ f does not see the part of σ w on the boundary),so that we have: Z Ω f d ( µ − ν ) = Z ddt (cid:18)Z Ω f dσ t (cid:19) = Z Z ˚Ω ∇ f · w t dσ t dt ≤ (cid:18)Z Z ˚Ω |∇ f | dσ t dt (cid:19) / (cid:18)Z Z ˚Ω | w t | dσ t dt (cid:19) / ≤ ||∇ f || L (Ω) L ( µ, ν ) since σ t ≤ in ˚Ω . (cid:3) Generalization of the proof. At step 1, we need again to prove that the limits of ˜E τ and E τ are the same. As far as the limits of ˜ ρ τ et ρ τ are concerned, everything works as inSection 3.2: this also proves that the limit curve ρ belongs to K , since this is the case for ρ τ (but not for ˜ ρ τ ).It is easy to check that the comparison we did in Step 1 of Section 3.2 may be performed againso as to obtain that the limit of ˆE τ and E τ are the same. What we need to do now is provingthat the limit of ˆE τ and ˜E τ are the same. We will prove that the mass of ˜E τ − ˆE τ is negligible,i.e. that Z T dt Z Ω d (cid:12)(cid:12) ˜E τ ( t ) − ˆE τ ( t ) (cid:12)(cid:12) −→ τ → . To do this, it is sufficient to estimate T/τ X k =0 Z kτ ( k − τ dt Z Ω × Ω | x − y | τ Γ out × Ω dγ kτ = T/τ X k =0 Z Ω × Ω | x − y | Γ out × Ω dγ kτ . Thanks to what we underlined before, namely that the mass which is on Γ out does not moveany more, we know that | x − y | Γ out × Ω dγ kτ = | x − y | Γ out × ˚Ω dγ kτ and the mass of Γ out × ˚Ω dγ kτ isexactly the excess mass E kτ . Thanks to the Lemma 4.7 below, we can go on and obtain T/τ X k =0 Z Ω × Ω | x − y | Γ out × Ω dγ kτ = T/τ X k =0 Z Ω × Ω | x − y | Γ out × ˚Ω dγ kτ ≤ T/τ X k =0 (cid:18)Z Γ out × ˚Ω | x − y | dγ kτ (cid:19) (cid:18)Z Γ out × ˚Ω dγ kτ (cid:19) ≤ T/τ X k =0 W ( ρ k − τ , ρ kτ ) / ≤ E ( T/τ ) X k =1 W ( ρ k − τ , ρ kτ ) E ( T/τ ) X k =1 ≤ ( Cτ ) (cid:18) Tτ (cid:19) = C τ −→ τ → . Lemma 4.7. Suppose µ, ν ∈ K and set E := | µ (Γ out ) − ν (Γ out ) | . Then we have E ≤ CW / ( µ, ν ) ,where the constant C depends on the geometry of Ω and Γ out .Proof. Suppose for simplicity ν (Γ out ) ≥ µ (Γ out ) . Take an optimal transport plan γ from µ to ν .Consider γ ′ = ˚Ω × Γ out γ . The mass of γ ′ is a number E ′ , larger than E . Let µ ′ be the projectionof γ ′ on the first variable ( x ): it is a measure with mass E ′ , dominated by ˚Ω µ (and hence it isabsolutely continuous with density smaller than ). We have W ( µ, ν ) = Z | x − y | dγ ≥ Z | x − y | dγ ′ ≥ Z d ( x, Γ out ) dγ ′ = Z d ( x, Γ out ) dµ ′ . It is sufficient to prove that this last integral is larger than c ( E ′ ) . Set d ( x ) := d ( x, Γ out ) : wewill use the fact that | [ d ≤ t ] | ≤ ct . We have Z d ( x ) dµ ′ = Z ∞ µ ′ (cid:0) [ d > t ]) dt = Z ∞ (cid:0) E ′ − µ ′ ([ d ≤ √ t ]) (cid:1) dt ≥ Z ∞ (cid:0) E ′ − | [ d ≤ √ t ] | (cid:1) + dt ≥ Z ( E ′ /c ) (cid:0) E ′ − c √ t (cid:1) + dt = c ( E ′ ) . (cid:3) At step 2, we prove with the same method that E is absolutely continuous with respect to thedensity ¯ ρ := lim τ → ( ρ τ ) Ω = ρ Ω (the decomposition of the measures into a part on Γ out and a parton ˚Ω passes to the limit, because of the density bound on ˚Ω ): there exists u such that E = ρ Ω u .Moreover ( ρ, E ) satisfies the continuity equation, and we can prove again the equality Z Ω ∇ q · u dx = 0 ∀ q ∈ H ρ . At step 3, the first estimates are still true, since we integrate over ˚Ω ( p τ = 0 on Γ out ). However,we can’t use lemma 3.5 anymore. Instead, we apply lemma 4.6 and get the inequality Z ba Z Ω p τ ( t, x ) (cid:0) ρ τ ( a, x ) − ρ τ ( t, x ) (cid:1) dx dt ≤ Z ba ||∇ p τ ( t, . ) || L (Ω) L ( ρ τ ( a, . ) , ρ τ ( t, . )) dt. Using proposition (4.1) and the same notation as in Section 3.2, step 3, the limit τ → reads: lim τ → b − a Z ba Z Ω p τ ( t, x ) (cid:0) ρ τ ( a, x ) − ρ τ ( t, x ) (cid:1) dx dt ≤ b − a p µ ([ a, b ]) (cid:18)Z ba L ( ρ ( a ) , ρ ( t )) dt (cid:19) . Since at the limit, the curve ρ ( t ) belongs to K for every t, we have also: L ( ρ ( a ) , ρ ( t )) dt ≤ Z ta | ρ ′ | W ( s ) ds ≤ (cid:18)Z ta | ρ ′ | W ( s ) ds (cid:19) / ( t − a ) / ≤ C ( b − a ) / . Therefore, we have the following inequality: lim τ → b − a Z ba Z Ω p τ ( t, x ) (cid:0) ρ τ ( a, x ) − ρ τ ( t, x ) (cid:1) dx dt ≤ b − a p µ ([ a, b ]) (cid:18)Z ba C ( b − a ) dt (cid:19) = C p µ ([ a, b ]) −→ b → a for a.e. a, and we conclude the proof as in the particular case Γ out = ∅ .5. Illustration: a convergent corridor We present here an example where both the transport equation and discrete process of thegradient-flow problem can be solved quasi-explicitely. We also give numerical estimations on theconvergence of the discrete scheme to the solution of the continuity equation.We want to model the displacement of a crowd throught a convergent corridor. We thus takefor Ω a portion of a cone, expressed in polar coordinates as (cid:2) r ∈ [ a, R ] , θ ∈ [ − α, α ] (cid:3) (see fig2), with a possible “exit” Γ out = { a } × [ − α, α ] , and we take for D the distance to the exit (orto the apex, which is equivalent): D ( r ) = r . We assume that the initial density is uniform: ρ ( r ) = ρ < . We will consider in this section two examples: the case a = 0 with no exit (sothat people will in the end concentrate on the neighborhood of the vertex) and the case a > with exit. MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE 23 aU Rb ( t )ΩΓ out Γ w Figure 2. Modeling of the displacement of a crowd throught a convergent corridor.Thanks to the symmetry of this problem, the minimizing movement scheme can be written asa minimization problem on the transport function: ρ τ = ρ ρ kτ = s k ρ k − τ , s k ∈ argmin t ρ k − τ ∈ K (cid:26)Z Ra (cid:18) D ( t ( r )) + 12 τ | r − t ( r ) | (cid:19) ρ k − τ ( r ) r dr (cid:27) . Let us first consider the case where a = 0 (and Γ out = ∅ ), where this problem can be explicitelysolved: ρ kτ is given by(25) ρ kτ ( r ) = on [ a, b kτ [ ρ (cid:18) kτr (cid:19) on [ b kτ , R − kτ [0 on [ R − kτ, R ] , where b kτ satisfies the recurrence relation(26) (cid:26) b τ = 0( b kτ ) − ρ ( b kτ + kτ ) = ( b k − τ ) − ρ ( b k − τ + ( k − τ ) , and the solution of the continuity equation can be easily calculated: ρ ( t, r ) = if r ∈ [ a, b ( t )[ ρ (cid:18) tr (cid:19) if r ∈ [ b ( t ) , R − t ]0 if r ∈ [ R − t, R ] , where b (0) = 0 b ′ ( t ) = ρ b ( t ) + tb ( t ) − ρ ( b ( t ) + t ) . In figure 3, we represent the discrete densities ρ kτ at different times for the numerical values τ = 0 . , a = 0 , R = 10 , and ρ = 0 . . Let us remark that the recurrence relation that satisfies b kτ is a numerical scheme for the ODE on b ( t ) . Indeed, it writes b kτ − b k − τ τ = F (cid:18) b kτ + b k − τ , k − τ (cid:19) , where F ( r, t ) = ρ b ( t ) + tb ( t ) − ρ ( b ( t ) + t ) . Using the conservation of the total amount of people, it is easy to prove that this scheme is exactat every time step kτ , and so is the discrete solution ρ kτ . Figure 3. Evolution of the solution of the minimizing movement scheme in thecase where Γ out = ∅ .We now consider the case a > with exit. The densities have then the same form, exceptthat the evolution of the interface in the continuous case is now given by (cid:26) b ′ ( t ) = Φ( t, b ( t )) b ( t ) = a, with : Φ( t, r ) = ρ (cid:18) tr (cid:19) − r − ar ln( r/a )1 − ρ (cid:18) tr (cid:19) if r ≤ R − t − r − ar ln( r/a ) if r > R − t, whereas in the discrete case, b kτ satisfies now the recurrence relation ( b kτ ) − a − ρ ( b kτ + kτ ) = ( b k − τ ) − r e − ρ ( b k − τ + ( k − τ ) if b k − τ < R − ( k − τ , and ( b kτ ) − a = ( b k − τ ) − r e if b k − τ ≥ R − ( k − τ , where r e is the (unknown) radius such that people who were between a and r e at step k − will exit the corridor (i.e. arrive at a ) at step k . This radius is given as theminimum of an integral expression that we will not develop here. In figure 4, we represent thediscrete densities for a = 1 and for the same numerical values as before.It is also interesting to estimate numerically the error between the solution of the continuityequation and the solution of the minimizing movement scheme. In this purpose, we considerthe case where the density is initially saturated ( ρ = 1 ), and we compute b and b τ with highaccuracy (high order method for the ODE on b , and precise quadrature and optimization methodsto estimate r e and b τ ), so that space discretization does not affect error estimation. We obtainnumerically that b τ converges to b when τ tends to with an error of order 1 (interpolationpolynom of τ 7→ | b ( T ) − b τ ( T ) | gives order 0.989 for T = 1 ), which gives also an order 1 error forthe Wasserstein distance between ρ and ρ τ .6. Modelling issues, extensions We would like to conclude this paper by some remarks on the limitations of the overall approachin terms of modelling, and on possible extensions to other domains.With its very macroscopic and eulerian nature, the model is designed to handle large popula-tions as a whole, and does not allow to localize and follow in their path individual pedestrians.As a direct consequence, the spontaneous (or desired) velocity of an individual may depend on itslocation only, so that differentiated individual strategies (e.g. avoidance of crowded zone, skirting MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE 25 Figure 4. Evolution of the solution of the minimizing movement scheme in thecase with exit.of obstacles) cannot be included straightforwardly. Besides, the macroscopic expression of thenon-overlapping constraint is less restrictive than its microcopic counterpart. In the microscopicsetting (people are identified with rigid discs), for highly packed situations, the non overlappingconstraints induce some kind of non-negative divergence constraint in the directions of contacts.Consider the example of a cartesian distribution of monodisperse discs (see Fig. 5, left), with auniform spontaneous velocity directed toward a wall. The actual velocity will be , whereas inthe macroscopic version it is not (see Fig. 5, center). Note that, if the microscopic distributionis modified, while mean density is preserved, the situation is no longer static (see Fig. 5, right).This example illustrates the deep difference between the microscopic approach (for which localstructure and orientation of contacts lines play an essential role), and the macroscopic one, whichonly considers local density. Figure 5. Differences between micro and macro approachesAs a consequence the model is unlikely to reproduce the formation of blocked archs near anexit, which are observed in practice in highly critical emergency situations, and which can berecovered by the microscopic model, even without friction. Note also that, together with thestructural anisotropy we just mentionned, individual anisotropy of pedestrians is not handled(whereas it may be in microscopic models by replacing discs by ellipses, for example).Yet, despite these limitations in the modelling of crowd motion, we believe that this newtype of evolution problem may be fruitful to model phenomena, in particular in the domain of cell dynamics. In this context, the spontaneous velocity would be replaced by some kind ofchemotaxis velocity. As an illustration, let us express what could be called a unilateral versionof Keller-Segel equation, in the spirit of what has been presented here: ∂ρ∂t + ∇ · ( ρ u ) = 0 u = P C ρ U , U = ∇ c , − ∆ c = ρ, where c denotes the concentration of some attracting agent, generated by the cells themselves.Notice that the congestion constraint prevents concentration of mass. As a matter of fact, thecharacteristic function of a single ball is a static solution to this system, in the whole space R , and it can be expected that any solution converges to such a configuration. Note also thatbacterial growth could be handled by adding an appropriate term in the right-hand side of thetransport equation.We also believe that it can be fruitful in the modelling of granular media. Bouchut et al.[8]propose a model of pressureless gas for which the density is subject to remain less than 1. Thismodel is essentially mono-dimensionnal (the construction of explicit solutions proposed in [6]uses extensively the one-dimensionality). As our model can be seen as a first order (in time)version of this second order pressureless gas model, we believe that the handling of the congestionconstraint we propose here, which applies in any dimension, might be used in the future for amacroscopic description of granular flows in higher dimension. The corresponding model, whichis a second order version of the model we considered here, could be written as follows ∂ρ∂t + ∇ · ( ρ u ) = 0 ,∂ ( ρ u ) ∂t + ∇ · ( ρ u ⊗ u ) + ∇ p = 0 , ≤ ρ ≤ , p (1 − ρ ) = 0 , u + = P C ρ u − , where C ρ is the cone of feasible velocities which we introduced, and u − (resp. u + ) is the velocitybefore (resp. after) the collision (both are equal in case there is no collision).Those extensions are not straightforward. 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