Maria Laura

Delle Monache

Paola Goatin
email: paola.goatin@inria.fr.

Scalar Conservation Laws with Moving Density Constraints arising in Traffic Flow Modeling

Keywords: Scalar conservation laws with constraints, Traffic flow modeling, PDE-ODE coupling, Wave-front tracking approximations Lois de conservation scalaires avec contraintes, Modélisation du trafic routier, Couplage EDP-EDO, Méthode du suivi de fronts

We prove the existence of solutions of a coupled PDE-ODE system modeling the interaction of a large slow moving vehicle with the surrounding traffic flow. The model consists in a scalar conservation law with moving density constraint describing traffic evolution coupled with an ODE for the slow vehicle trajectory. The constraint location moves due to the surrounding traffic conditions, which in turn are affected by the presence of the slower vehicle, thus resulting in a strong non-trivial coupling.

Introduction

A slow moving large vehicle, like a bus or a truck, reduces the road capacity and thus generates a moving bottleneck for the surrounding traffic flow. From the macroscopic point of view this can be modeled by a PDE-ODE coupled system consisting in a scalar conservation law with moving density constraint and an ODE describing the slower vehicle motion, i.e.,

           ∂ t ρ + ∂ x f (ρ) = 0, (t, x) ∈ R + × R, ρ(0, x) = ρ 0 (x),
x ∈ R, ρ(t, y(t)) ≤ αR, t ∈ R + , ẏ(t) = ω(ρ(t, y(t)+)), t ∈ R + , y(0) = y 0 .

(1.1) Above, ρ = ρ(t, x) ∈ [0, R] is the scalar conserved quantity representing the mean traffic density, R is the maximal density allowed on the road and the flux function f : [0, R] → R + is a strictly concave function such that f (0) = f (R) = 0. It is given by the following flux-density relation

f (ρ) = ρv(ρ),
where v is a smooth decreasing function denoting the mean traffic speed and here set to be v(ρ) = V (1 -ρ R ), V being the maximal velocity allowed on the road. The time-dependent variable y denotes the slower vehicle position, that moves with a traffic density dependent speed of the form

ω(ρ) = V b if ρ ≤ ρ * . = R(1 -V b V ), v(ρ) otherwise, (1.2)
that is, it moves with constant speed V b < V as long as it is not slowed down by downstream traffic conditions. When this happens, it moves with the mean traffic speed. Finally, the constant coefficient α ∈ ]0, 1[ gives the reduction rate of the road capacity due to the presence of this large vehicle. For our analytical purposes, it is not restrictive to assume that R = V = 1, so that the model becomes

           ∂ t ρ + ∂ x (ρ(1 -ρ)) = 0, (t, x) ∈ R + × R, ρ(0, x) = ρ 0 (x), x ∈ R, ρ(t, y(t)) ≤ α, t ∈ R + , ẏ(t) = ω(ρ(t, y(t)+)), t ∈ R + , y(0) = y 0 . (1.3)
The above model was introduced in [START_REF] Giorgi | Prise en compte des transports en commun de surface dans la modélisation macroscopique de l'écoulement du trafic[END_REF] to model the effect of urban transport systems, such as buses, in a road network. Other macroscopic models for moving bottlenecks in road traffic were recently proposed by [START_REF] Borsche | Mixed systems: ODEs-balance laws[END_REF][START_REF] Lattanzio | Moving bottlenecks in car traffic flow: a PDE-ODE coupled model[END_REF]. Compared to those approaches, the model described by (1.1) offers a more realistic definition of the slower vehicle speed and a description of its impact on traffic conditions which is simpler to handle both from the analytical and the numerical point of view.

From the analytical point of view, model (1.1) can be viewed as a generalization to moving constraints of the problem consisting in a scalar conservation law with a (fixed in space) constraint on the flux, introduced and studied in [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF][START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF]. In the present case, the constraint location moves due to the surrounding traffic conditions, which in turn are modified by the presence of the slower vehicle, thus resulting in a strong non-trivial coupling between the conservation equation and the trajectory of the vehicle.

The study of coupled PDE-ODE systems is not new in the conservation laws framework, we refer the reader to [START_REF] Borsche | On the coupling of systems of hyperbolic conservation laws with ordinary differential equations[END_REF][START_REF] Bressan | Uniqueness for discontinuous ODE and conservation laws[END_REF][START_REF] Colombo | A Hölder continuous ODE related to traffic flow[END_REF][START_REF] Lattanzio | Moving bottlenecks in car traffic flow: a PDE-ODE coupled model[END_REF]. Nevertheless, the problem posed here is slightly different. On one side, we deal with a strong coupling with the PDE and the ODE affecting each other, unlike [START_REF] Bressan | Uniqueness for discontinuous ODE and conservation laws[END_REF][START_REF] Colombo | A Hölder continuous ODE related to traffic flow[END_REF], where the PDE solution does not depend on the ODE. On the other side, even if the ODE has discontinuous right-hand side, the particular definition of the model allows us to consider classical Carathéodory solutions as in [START_REF] Borsche | On the coupling of systems of hyperbolic conservation laws with ordinary differential equations[END_REF][START_REF] Borsche | Mixed systems: ODEs-balance laws[END_REF][START_REF] Bressan | Unique solutions for a class of discontinuous differential equations[END_REF][START_REF] Bressan | Uniqueness for discontinuous ODE and conservation laws[END_REF] instead of the weaker Filippov's generalized solutions needed in [START_REF] Colombo | A Hölder continuous ODE related to traffic flow[END_REF][START_REF] Lattanzio | Moving bottlenecks in car traffic flow: a PDE-ODE coupled model[END_REF].

The present paper provides an existence result for solutions of (1.1) constructed by wave-front tracking approximations, as stated by Theorem 1 in Section 3. The continuous dependence on initial data is currently under investigation.

The Riemann problem with moving density constraint

Consider (1.3) with the particular choice

y 0 = 0 and ρ 0 (x) = ρ L if x < 0, ρ R if x > 0. (2.1)
We aim at defining a Riemann solver for the conservation law with moving density constraint. Therefore we consider the following Riemann problem

     ∂ t ρ + ∂ x f (ρ) = 0, ρ(0, x) = ρ L if x < 0, ρ R if x > 0, (2.2)
under the constraint ρ(t, V b t) ≤ α.

(2.3)

Let f α : [0, α] → R + be the function describing the constrained flow at x = y(t), i.e.,

f α (ρ) = ρ 1 - ρ α ,
and

ρ α ∈ ]0, α/2[ such that f ′ (ρ α ) = V b , i.e., ρ α = α 2 (1 -V b ) .
Problem (2.2), (2.3) can be recasted in the framework of conservation laws with flux constraint studied in [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF][START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF]. Rewriting the equations in the bus reference frame (setting

X = x -V b t), we get      ∂ t ρ + ∂ X (f (ρ) -V b ρ) = 0, ρ(0, x) = ρ L if X < 0, ρ R if X > 0, (2.4)
under the constraint ρ(t, 0) ≤ α.

(2.5)

Remark that solving problem (2.4), (2.5) is equivalent to solving (2.4) under the corresponding constraint on the flux

f (ρ(t, 0)) -V b ρ(t, 0) ≤ f α (ρ α ) -V b ρ α . = F α .

Inria

We are now ready to define the Riemann solver for (1.3), (2.1) following [10, §V]. Denote by R the standard Riemann solver (i.e., without the constraint (2.3)) for (2.2), i.e., the (right continuous) map

(t, x) → R(ρ L , ρ R )( x t )
is the standard weak entropy solution to (2.2). Moreover, let ρα and ρα , with ρα ≤ ρα , be the intersections of the flux function f (ρ) and the line

f α (ρ α ) + V b (ρ -ρ α ) (see Figure 1). ρα f (ρ) ρ V b ρα ρ * Figure 1: Flux function Definition 1 The constrained Riemann solver R α for (1.3), (2.1) is defined as follows. 1. If f (R(ρ L , ρ R )(V b )) > F α + V b R(ρ L , ρ R )(V b ), then R α (ρ L , ρ R )(x) = R(ρ L , ρα ) if x < V b t, R(ρ α , ρ R ) if x ≥ V b t, and
y(t) = V b t. 2. If V b R(ρ L , ρ R )(V b ) ≤ f (R(ρ L , ρ R )(V b )) ≤ F α + V b R(ρ L , ρ R )(V b ), then R α (ρ L , ρ R ) = R(ρ L , ρ R ) and y(t) = V b t. 3. If f (R(ρ L , ρ R )(V b )) < V b R(ρ L , ρ R )(V b ), then R α (ρ L , ρ R ) = R(ρ L , ρ R ) and y(t) = v(ρ R )t.
Note that, when the constraint is enforced (point 1. in the above definition), a nonclassical shock arises, which satisfies the Rankine-Hugoniot condition but violates the Lax entropy condition.

Remark 1

The above definition is well-posed even if the classical solution R(ρ L , ρ R )(x/t) displays a shock at x = V b t. In fact, due to Rankine-Hugoniot equation, we have

f (ρ L ) = f (ρ R ) + V b (ρ L -ρ R ) and hence f (ρ L ) > f α (ρ α ) + V b (ρ L -ρ α ) ⇐⇒ f (ρ R ) > f α (ρ α ) + V b (ρ R -ρ α ).
Remark 2 The density constraint ρ(t, y(t)) ≤ α does not appear explicitly in Definition 1, and in the following Definition 2. It is handled by the corresponding condition on the flux f (ρ(t, y(t))) -ω(ρ(t, y(t)))ρ(t, y(t)) ≤ F α .

(2.6)

The corresponding density on the reduced roadway at x = y(t) is found taking the solution to the equation

f (ρ y ) + ω(ρ y )(ρ -ρ y ) = ρ 1 - ρ α , closer to ρ y . = ρ(t, y(t))).
3 The Cauchy problem: existence of solutions

The aim of this section is to study the existence of the solutions of problem (1.1), (1.3). A bus travels along a road modeled by

   ∂ t ρ + ∂ x (ρ(1 -ρ)) = 0, ρ(0, x) = ρ 0 (x), ρ(t, y(t)) ≤ α. (3.1)
The bus influences the traffic along the road but it is also influenced by it. The bus position y = y(t) then solves

ẏ(t) = ω(ρ(t, y(t)+)), y(0) = y 0 . (3.2)
Solutions to (3.2) will be intended in Carathéodory sense, i.e., as absolutely continuous functions which satisfy (3.2) for a.e. t ≥ 0. In our setting, due to the strong PDE-ODE coupling, we will prove existence of both solutions to (3.1) and (3.2) at the same time. We start giving our definition of solution.

Definition 2 A couple (ρ, y) ∈ C 0 R + ; L 1 ∩ BV(R; [0, R]) × W 1,1 (R + ; R) is a solution to (1.3) if 1.
ρ is a weak solution of the conservation law, i.e., for all ϕ ∈ C 1 c (R 2 ; R)

R + R (ρ∂ t ϕ + f (ρ)∂ x ϕ) dx dt + R ρ 0 (x)ϕ(0, x) dx = 0 ; (3.3a)
moreover, ρ satisfies the Kružhkov entropy conditions [START_REF] Kružhkov | First order quasilinear equations with several independent variables[END_REF] on

(R + × R)\{(t, y(t)) : t ∈ R + }, i.e., for every k ∈ [0, 1] and for all ϕ ∈ C 1 c (R 2 ; R + ) and ϕ(t, y(t)) = 0, t > 0, R + R (|ρ -k|∂ t ϕ + sgn(ρ -k) (f (ρ) -f (k)) ∂ x ϕ) dx dt + R |ρ 0 -k|ϕ(0, x) dx ≥ 0 ; (3.3b)
(f (ρ) -ω(ρ)ρ) (t, x) ≤ F α . (3.3d)
Remark that the above traces exist because ρ(t, •) ∈ BV(R) for all t ∈ R + .

Remark 3 Our choice of a Carathéodory solution of the ODE is motivated by the particular bus velocity defined by (1.2). With this choice it is not possible for a bus to end up trapped in a queue unless its speed is equal to V b , in which case ω(ρ(t, y(t

)+)) = ω(ρ(t, y(t)-)) = V b .
Therefore Carathéodory solutions are always well defined.

We are now ready to state the main result of the paper.

Theorem 1 Let ρ 0 ∈ BV(R; [0, R]), then the problem (1.1) admits a solution in the sense of Definition 2.

The rest of the section is devoted to the proof of Theorem 1. In particular, we will construct a sequence of approximate solutions via the wave-front tracking method, and prove its convergence. Finally, we will check that the limit functions satisfy conditions (3.3a)-(3.3d) of Definition 2.

Wave-front tracking

To construct piecewise constant approximate solutions, we adapt the standard wave-front tracking method, see for example [5, §6]. Fix a positive n ∈ N, n > 0 and introduce in [0, 1] the mesh

M n = {ρ n i } 2 n i=0 defined by M n = 2 -n N ∩ [0, 1] ∪ {ρ α , ρα }.
In order to include the critical points ρα , ρα , we modify the above mesh as follows:

• if min i |ρ α -ρ n i | = 2 -n-1
, then we simply add the new point to the mesh:

M n = M n ∪ {ρ α }; • if |ρ α -ρ n l | = min i |ρ α -ρ n i | < 2 -n-1
, then we replace ρ n l by ρα :

M n = M n ∪ {ρ α } \ {ρ n l };
• we perform the same operations for ρα .

In this way, the distance between two points of the mesh

M n = {ρ n i } 2 n
i=0 satisfies the lower bound ρn i -ρn j ≥ 2 -n-1 . Let f n be the piecewise linear function which coincides with f on M n , and let ρ n 0 be a piecewise constant function defined by

ρ n 0 = j∈Z ρ n 0,j χ ]xj-1,xj ] with ρ n 0,j ∈ M n ,
which approximates ρ 0 in the sense of the strong L 1 topology, that is

lim n→∞ ρ n 0 -ρ 0 L 1 (R) = 0, RR n°8119
and such that TV(ρ n 0 ) ≤ TV(ρ 0 ). Above, we set x 0 = y 0 . For small times t > 0, a piecewise approximate solution (ρ n , y n ) to (1.3) is constructed piecing together the solutions to the Riemann problems The approximate solution ρ n constructed above can be prolonged up to the first time t > 0, where two discontinuities collide, or a discontinuity hits the bus trajectory. In both cases, a new Riemann problem arises and its solution, obtained in the former case with R and in the latter case with the constrained Riemann solver R α , allows to extend ρ n further in time.

       ∂ t ρ + ∂ x (f n (ρ)) = 0, ρ(0, x) = ρ 0 if x < y 0 , ρ 1 if x > y 0 , ρ(t, y n (t)) ≤ α,        ∂ t ρ + ∂ x (f n (ρ)) = 0, ρ(0, x) = ρ j if x < x j , ρ j+1 if x > x j , j = 0,

Bounds on the total variation

Given an approximate solution ρ n = ρ n (t, •) constructed by the wave-front tracking method, we define the Glimm type functional

Υ(t) = Υ(ρ n (t, •)) = TV(ρ n ) + γ = j ρ n j+1 -ρ n j + γ, (3.6)
where γ is given by

γ = γ(t) = 0 if ρ n (t, y n (t)-) = ρα , ρ n (t, y n (t)+) = ρα 2|ρ α -ρα | otherwise. (3.7)
The value of γ is chosen to have the following uniform bound on Υ.

Lemma 2 For any n ∈ N, the map t → Υ(t) = Υ(ρ n (t, •)) at any interaction either decreases by at least 2 -n , or remains constant and the number of waves does not increase.

Lemma 2 in particular implies that the wave-front tracking procedure can be prolonged to any time T > 0.

Proof. In order to obtain a uniform bound on the total variation, we will consider the different types of interactions separately. In particular, we will assume that at any interaction time t = t either two waves interact or a single wave hits the bus trajectory.

(I1) We consider a classical collision between two waves (see Figure 2). In this case, either two shocks collide (which means that the number of waves diminishes) or a shock and a rarefaction cancel. However, in both cases, TV(ρ n ) is not increasing and γ is constant. Thus, we get Υ( t+) ≤ Υ( t-).

Inria ρ l ρ r ρ m

Figure 2: Interaction between two waves away from the bus trajectory

A particular case is when the bus trajectory coincides with one of the interacting waves.

In this situation, it must be a classical shock between a left state belonging to [0, ρα ] and a right state in [ρ α , ρ * ], and moving with speed equal to V b . This interaction cannot generate a non-classical shock, it can therefore be treated as the general case above.

(I2) Assume that a wave between two states ρ l , ρ r ∈ [0, ρα ] ∪ [ρ α , 1] hits the bus trajectory (see Figure 3). In this case, the front crosses the bus trajectory and no new wave is created. Notice that this collision may eventually lead to a modification of the bus trajectory (for example, if ρ r > ρ * , after the collision the bus takes the velocity v(ρ r ) = ω(ρ l )). Nevertheless, TV(ρ n ), Υ and the number of waves remain constant. (I3.1) Assume that the non-classical shock is present at t < t, and a shock between ρ l ∈ [0, ρα ] and ρα hits the bus trajectory on the left (Figure 4, left). After the collision, the number of discontinuities in ρ n diminishes and the functional Υ remains constant:

ρ l V b ρ r
∆Υ( t) = Υ( t+) -Υ( t-) = |ρ l -ρα | + 2|ρ α -ρα | -(|ρ l -ρα | + |ρ α -ρα |) = 0.
Assume now a shock between ρα and ρ r ∈ [ρ α , 1] hits the bus trajectory on the right (Figure 4, right). Then, after the collision, the bus takes the velocity v(ρ r ) of the traffic mainstream, the number of discontinuities in ρ n diminishes and the functional Υ remains constant: In this case, a new wave is created at t and the total variation is given by:

∆Υ( t) = Υ( t+) -Υ( t-) = |ρ α -ρ r | + 2|ρ α -ρα | -(|ρ α -ρ r | + |ρ α -ρα |) = 0. ρα ρα ρ l V b ρα ρα ρ r ρ r v(ρ r )
• TV( t-) = |ρ α -ρ l | ≥ 2 -n-1 ; • TV( t+) = |ρ α -ρα | + |ρ α -ρ l | ≤ 2|ρ α -ρα |,
where the second estimate is obtained by simple algebraic manipulation of the total variation TV( t+). Then we are able to compute the changes in the functional as follows:

∆Υ( t) = Υ( t+) -Υ( t-) = (|ρ α -ρα | + |ρ α -ρ l |) -(|ρ α -ρ l | + 2|ρ α -ρα |) = 2(ρ α -ρ l ) ≤ -2 -n ,
hence the functional is strictly decreasing. Let us consider now the case of a rarefaction front hitting the bus trajectory from the right (Figure 5, right). In this case, we have ρα ≤ ρ r < ρ l = ρα . A new wave is created at t and the total variation is given by: Since ρ n converge strongly to ρ in L 1 loc (R + × R), it is straightforward to pass to the limit in the weak formulation of the conservation law, proving that the limit function ρ satisfies (3.3a). Kružhkov entropy condition (3.3b) can be recovered in the same way.

• TV( t-) = |ρ α -ρ r | ≥ 2 -n-1 ; • TV( t+) = |ρ α -ρα | + |ρ α -ρ r | ≤ 2|ρ α -ρα |,
ρ n (t, y n (t)+) -ρ + (t) ≤ |ρ n (t, y n (t)+) -ρ n (t, z n )| + ρ n (t, z n ) -ρ + (t) ≤ 3ǫ *
for n large enough.

If ρ(t, •) has a discontinuity of strength greater than ǫ * at y(t), then it holds also for the approximate solutions |ρ n (t, y n (t)+) -ρ n (t, y n (t)-)| ≥ ǫ * /2 for n sufficiently large, and we proceed as in [START_REF] Bressan | Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws[END_REF]Section 4]. That is, we set ρ n,+ = ρ n (t, y n (t)+) and we show that for each ε > 0 there exists δ > 0 such that for all n large enough there holds ρ n (s, x) -ρ n,+ < ε for |s -t| ≤ δ, |x -y(t)| ≤ δ, x > y n (s).

(3.10)

In fact, if (3.10) does not hold, we could find ε > 0 and sequences t n → t, δ n → 0 such that TV (ρ n (t n , •); ]y n (t n ), y n (t n ) + δ n [) ≥ ε. By strict concavity of the flux function f , there should be a uniformly positive amount of interactions in an arbitrarily small neighborhood of (t, y(t)), giving a contradiction. Therefore (3.10) holds and we get

ρ n (t, y n (t)+) -ρ + (t) ≤ |ρ n (t, y n (t)+) -ρ n (t, z n )| + ρ n (t, z n ) -ρ + (t) ≤ 2ε
for n large enough, thus proving (3.9). Combining (3.8c) and (3.9) we get ẏ(t) = ω(ρ(t, y(t)+)) for a.e. t > 0.

Proof of (3.3d)

In order to verify that the limit solutions satisfy the constraint (3.3d), we can use directly the convergence result proved in the previous section or proceed as follows. Let us introduce the sets

Ω ± = {(t, x) ∈ R + × R : x ≶ y(t)}
and Ω ± n = {(t, x) ∈ R + × R : x ≶ y n (t)}.

Inria

Let us now consider a test function ϕ ∈ C 1 c (R + × R), ϕ ≥ 0, such that supp(ϕ) ∩ {(t, y(t)) : t > 0} = 0 and supp(ϕ) ∩ {(t, y n (t)) : t > 0} = 0. Then by conservation on Ω + n we have (f (ρ) -ω(ρ)ρ) (t, x) ≤ F α for a. e. t > 0. The left-hand limit can be handled in the same way using the sets Ω + n and Ω + .

χ Ω + n ρ n ∂ t ϕ + f n (ρ n )∂ x ϕ

( 3 . 4 )

 34 where y n satisfies ẏn (t) = ω(ρ n (t, y n (t)+)),y n (0) = y 0 . (3.5)Note that the solutions to the constrained Riemann problem in (3.4), left, coupled with (3.5), is constructed by means of R α , see Definition 1.

Figure 3 :

 3 Figure 3: Interaction between a wave and the bus trajectory

Figure 4 :Figure 5 :

 45 Figure 4: Interaction between a shock and the bus trajectory

Inria

3. 3 . 1

 31 Proof of (3.3a) and(3.3b)

  Inria3. the constraint is satisfied, in the sense that for a.e. t ∈ R +

	lim x→y(t)±
	2. y is a Carathéodory solution of the ODE, i.e., for a.e. t ∈ R +
	t
	y(t) = y 0 +	ω(ρ(s, y(s)+)) ds ;	(3.3c)
	0

  By pointwise convergence a.e. of ρ n to ρ, there exists a sequence z n ≥ y n (t) such that z n → y(t) and ρ n (t, z n ) → ρ + (t).For a.e. t > 0, the point (t, y(t)) is for ρ(t, •) either a continuity point, or it belongs to a discontinuity curve (represented by y(•)) that can be either a classical shock or a non-classical discontinuity between ρ(t, y(t)-) = ρα and ρ(t, y(t)+) = ρα .Fix ǫ Then by weak convergence of measures (see[START_REF] Bressan | Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws[END_REF] Lemma 15]) we have TV (ρ n (t, •); ]y(t) -δ, y(t) + δ[) ≤ 2ǫ * for n large enough, and we can estimate

	3.3.2 Proof of (3.3c)
	We will prove that
	lim n→∞	ρ n (t, y n (t)+) = ρ + (t) = ρ(t, y(t)+) for a.e. t ∈ R + .	(3.9)

* > 0 and assume TV (ρ(t, •); ]y(t) -δ, y(t) + δ[) ≤ ǫ * , for some δ > 0.

  n ∂ t ϕ + f n (ρ n )∂ x ϕ dx dt (ρ n (t, y n (t)+)) -ẏn (t)ρ n (t, y n (t)+) ϕ(t, y n (t)) dt (ρ n (t, y n (t)+)) -ω(ρ n (t, y n (t)+))ρ n (t, y n (t)+) ϕ(t, y n (t)) dt F α ϕ(t, y n (t)) dt,(3.11)where we have used the fact that wave-front tracking approximations satisfy the constraint (2.6) by construction. The same can be done for the limit solutions ρ and y(t) that, by conservation on Ω, satisfyχ Ω + ρ∂ t ϕ + f (ρ)∂ x ϕ dx dt

			dx dt
	+∞	+∞
	=
	0	yn(t)
	+∞
	= f n = 0 +∞ 0 f n ≤ T
	0
	+∞	+∞
	=
	0	y(t)

ρ ρ∂ t ϕ + f (ρ)∂ x ϕ dx dt = +∞ 0 f (ρ(t, y(t)+)) -ẏ(t)ρ(t, y(t)+) ϕ(t, y(t)) dt = +∞ 0 f (ρ(t, y(t)+)) -ω(ρ(t, y(t)+))ρ(t

, y(t)+) ϕ(t, y(t)) dt (3.12) By (3.8a) and (3.8b) we can pass to the limit in (3.11) and (3.12), which gives +∞ 0 f (ρ(t, y(t)+)) -ω(ρ(t, y(t)+))ρ(t, y(t)+) -F α ϕ(t, y(t)) dt ≤ 0. Since the above inequality holds for every test function ϕ ≥ 0, we have proved that lim x→y(t)±

RR n°8119

route des Lucioles -BP 93 06902 Sophia Antipolis Cedex Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

Acknowledgments

This research was supported by the ERC Starting Grant 2010 under the project "TRAffic Management by Macroscopic Models".

The functional changes as follows:

making the functional strictly decreasing.

Convergence of approximate solutions

In this section we prove that the limit of wave-front tracking approximations provides a solution (ρ, y) of the PDE-ODE model (1.1) in the sense of Definition 2.

We start showing the convergence of the wave-front tracking approximations.

Lemma 3 Let ρ n and y n , n ∈ N, be the wave-front tracking approximations to (1.1) constructed as detailed in Section 3.1, and assume TV(ρ 0 ) ≤ C be bounded, 0 ≤ ρ 0 ≤ 1. Then, up to a subsequence, we have the following convergences

Proof. Lemma 2 gives a uniform bound on the total variation of approximate solutions. Thus, we have TV(ρ n (t, •)) ≤ Υ(t) ≤ Υ(0). A standard procedure based on Helly's Theorem (see [START_REF] Bressan | Hyperbolic systems of conservation laws[END_REF]Theorem 2.4]) ensures the existence of a subsequence converging to some function ρ ∈ C 0 R + ; L 1 ∩ BV(R) , proving (3.8a). Since | ẏn (t)| ≤ V b , the sequence {y n } is uniformly bounded and equicontinuous on any compact interval [0, T ]. By Ascoli-Arzelà Theorem, there exists a subsequence converging uniformly, giving (3.8b).

In order to prove (3.8c), we have to show that TV ( ẏn ; [0, T ]) is uniformly bounded. In fact, the analysis performed in Section 3.2 shows that ẏn can change only at interactions with waves coming from its right. We can estimate the speed variation at interactions times t by the size of the interacting front:

In particular, ẏn is non-increasing at interactions with shock fronts and non-decreasing at interactions with rarefaction fronts, which must be originated at t = 0. In fact, the analysis performed in Section 3.2 shows that no new rarefaction front can arise at interactions. Therefore,

is uniformly bounded. Above, PV ( ẏn ; [0, T ]) denotes the positive variation of ẏn , i.e., the total amount of positive jumps in the interval [0, T ].


Gilles Debizet
email: gilles.debizet@ujf-grenoble.fr

PUBLIC AUTHORITIES CHALLENGE TO IMPLEMENT ENERGY RENOVATION OF CONDOMINIUM BUILDINGS IN FRANCE

Keywords: boundary object, housing improvement, mitigation, OPATB, thermal refurbishment

In France in 2008, the "Grenelle de l'Environnement" Forum agreed and planned on energy renovation on private buildings to reduce greenhouse gases; the cost was estimated as hundreds of billion euros. Its implementation is actually more difficult than expected: in the absence of the obligation to renovate, public initiatives are required. Government strengthened incentive tax breaks, but they benefits mainly to the wealthiest individual home owners-occupiers. Public accompaniment of owners had always been necessary to improve residential co-owned buildings. Numerous local authorities helped them with the support of a national program first developed thirty years ago for social issues. Reducing carbon emissions recently became a new goal of the housing policies. This paper exposes a new device that French government has developed especially for energy renovation and that a few local authorities have experimented. It demonstrates that fight against climate change needs not only public grants but also local convergences with other aims and strong devices to convinced housing building co-owners.

INTRODUCTION

The residential and tertiary buildings represent 43% of final energy consumption and 23% of greenhouse gazes (GHG) emissions in France. Although this sector emits less as transport, it appears to be the best way to reduce French GHG emissions and, so, to mitigate climate change. Since 1975, the construction sector has adapted to successive reinforcements of energy regulation. Thank to a high thermal insulation, new buildings consume nearly ten times less energy than the average of existing buildings. This makes plausible a factor 4 (a 75% decreased in GHG) by 2050 if a substantial part of the current buildings stock will be renovated. Housing represents the majority of the built surfaces. Private owners own 4/5th of all housings. The disjunction [START_REF] Berdoulay | Lécologie urbaine et lurbanisme. Aux fondements des enjeux actuels[END_REF] between the scale of the problem (global warming) and the decision scale (each housing) cannot be greater. The possible solidarity towards the planet from the owners will not be sufficient to reach the factor 4 in France; energy renovation requires governmental actions. The French government strengthened incentive market measures in 2005: the tax break is the main one. In 2008, the State spent therefore 2.8 billion for 1.6 million households who spent themselves 8.2 billion of work. This tax break benefited almost entirely (95%) to occupiers owners (57% of housing belong to occupiers 13/09/12 10:25 owners and 25% to renter owners). Among the French households, the fifth wealthiest benefited six time more of tax breaks than the fifth poorest. Owners of single-family dwelling are overrepresented (56% of French housings, 81% of tax break beneficiaries) (Clerc & Mauroux, 2010). The limits of tax break for housing improvement are known for several decades. During 70s, the Housing Ministry had implemented a grant device in partnership with local authorities: this device targeted non-occupiers owners and modest occupiers owners. It has often been essential for condominium improvement for a main reason: work decision requires a large majority of co-owners. Unlike tax breaks, partnership devices involve local governments during several years. At least, three levels of "government" are involved: national, local and, of course, condominium. How these different scales of government integrate the issue of climate change mitigation? How mitigation policies should be implemented for the condominium buildings?

We assume that the content of mitigation policies is determined by involved organizations: not by one of them but by their interactions. Therefore, we consider partnership projects and multi-scale devices as boundary objects (Star & Griesemer 1989[START_REF] Aibar | Constructing a city: The Cerdà Plan for the Extension of Barcelona[END_REF][START_REF] Trompette | Revisiting the notion of boundary object[END_REF][START_REF] Debizet | Expertise and Methodology in Building Design for Sustainable Development : A FrancoBritish Comparison[END_REF]. This paper focuses on the experimental partnership device developed by the state and applied by several local authorities in order to mitigation: OPATB device. The first part considers the OPATB device as a boundary object involving different ministries with their specific organization: knowledge, procedures and aims. The second part shows the high degree of investment of local governments and the interactions with territorial issues: it also illustrates the necessity to implement mitigation on a local scale.

OPATB AS A NATIONAL DEVICE FOR MITIGATION, COMBINING SOCIAL AND ECONOMIC ISSUES

Housing and Economic Ministries have overseen national policies relative to housing improvement. Climate change leaded the Ministry of Environment to promote building energy renovation. In absence of obligation to renovate, they developed together the OPATB device.

An obligation to renovate legally or politically difficult to set up

Since its inception in 1975, building regulation has established minimum energy efficiency; it applies only to new buildings, i.e. those submitted to an authorization procedure. Concerning existing buildings, an energy performance assessment must be associated since 2009 with the sale (or rent) of housing; however, it only informs the buyer (or tenant). In addition, minimum performances of products are imposed on professionals: i.e. when they replace windows. In 2007, the mitigation of climate change was one of the main issues of the French presidential race. Incumbent President Sarkozy avoided specific commitments, but vowed a broad consultation on the environment and climate change. During the "Grenelle Environnement" of the Forum, all participants agreed to mandatory energy renovation of buildings. However, a few months later, the "operational building committee" was reluctantly argue about the solvency of owners and the difficulty in defining the verifiable obligations. Laws that apply the "Grenelle Environnement" does not actually impose a general obligation. In the absence of requirement to undertake energy renovation, incentive provisions have been promoted for several years and shall be developed for may years. Preexisting economic incentives for owners With 1.5 million jobs, the construction sector is one of the country's first economic sectors. As such, tax cuts for individual investors and zero-interest loans have sustained construction. Progressively, French government strengthened these incentives in case of energy efficiency work. Tax cut and loan subsidies have always been insufficient to stimulate the landlords and modest homeowners. The former do not benefit directly of energy savings. The latter have little capacity to finance work. These two categories can receive grants from the national agency for housing improvement (Anah). Created by the State in 1971, Anah "encourages and facilitates the execution of improvement work and housing adaptation." In 2008, Anah distributed two thirds of the 526 million euro of grants within local scheduled operations of housing improvement (OPAH: Opérations programmées d'Amélioration de l'Habitat). Specifically, local authorities increase the fund and define the perimeter, the goals and some grants level in consultation with Anah. Finally, the owners receive more grants than Anah usually gives outside of an OPAH perimeter [START_REF] Anah | Les chiffres clés[END_REF]). The first scheduled operations were signed with local governments in 1977, as part of urban renewal of neglected and unhealthy historic districts [START_REF] Gravejat | Sur les opérations programmées d'amélioration de l'habitat à Lyon : 1975-1985[END_REF]. Gradually, Anah has expanded the device to economic and social aims. During 90s, energy savings appeared progressively as a lever of housing improvement. Before partnership convention, pre-operational studies clarify the scope, work types and calibrate both subsidy levels and partners financial budget. The subsidy rate should be high enough to trigger reinvestment from private actors and low enough to respect the budget of each partner. The result is uncertain because owners (or condominium corporation) actually choose the work and, therefore, decide whether or not to use the subsidies.

OPATB national device genesis

Ministry of Housing (supervisory authority of Anah) and the Ministry of Environment (supervisory of Ademe) introduced OPATB device in February 2002 as a call for local authority projects. "The OPATB are intended to act locally on residential and commercial buildings to reduce their energy consumption and CO2 emissions" [START_REF] Ademe | Secrétariat d'Etat au logement[END_REF] The OPATB device actually fitted into the mold of Scheduled Operations of Housing Improvement (OPAH) that have proven efficient since their creation in 1977 by Anah. However, the principles of call for project and the criteria to select communities correspond to Ademe procedures.

Agency for the Environment and Energy Management2 (Ademe) implements environmental policies carried by Ministry of environment. It also offers its expertise to local authorities. Ademe calibrates its devices to bring out innovative experiments and facilitate its diffusion. Then, national or local governments often contribute, by grants or tax cuts, to the diffusion of a sustainable innovation initially funded by Ademe. Ademe considers OPATB (operations rather than the device itself) as innovative experiments: "The multiplicity of targets must create genuine competition among professionals and increase synergies [...]

GRENOBLE AND ECHIROLLES OPATB

Six of the thirteen OPATB operations implemented in 2006 were located in the Rhône-Alpes, one of the most dynamic regions in terms of sustainable building [START_REF] Roudil | Translations" and Boundary Objects : Amateurs and Professionals in Berkely's Museum of Vertebrate Zoology, 1907-39[END_REF]. We choose the cases of Grenoble and Echirolles 3 because these OPATB mainly concern condominiums. Anah and Ademe selected Grenoble in 2002 and Echirolles in 2004. After describing the OPATB process, we explain how local authorities approach OPATB. Then, we summarize the results of Grenoble OPATB . A lengthy and complex implementation: i.e. Grenoble OPATB Three years were required to define the Grenoble OPATB (two years for Echirolles). The implementation convention, signed by Ademe, Anah, the city of Grenoble, the State and Grenoble Alpes Métropole (the metropolitan authority) in October 2005, described the quantitative targets, the financial commitments, the subsidy rules and the terms of animation, monitoring, evaluation and control. Once signed the partnership convention, the operational stage began with a long and tedious work of information to storekeepers, condominium boards and managers in order to encourage them to initiate an energy audit. After its energy audit, the condominium board was accompanied for one or two years until a majority of coowners votes (or does not vote) for thermal work during the annual condominium meeting. 13/09/12 10:25 When many boards of condominium planned to propose energy renovation at their annual meeting, the city left proactive communication. It avoided to involve new condominiums and, therefore, to exceed the OPATB budget.

The figure below illustrates the process. The city ("Ville de Grenoble") steered the OPATB process and the condominium board ("copropriété") steered the building renovation project. During work, city subsidized 20% of work cost to the condominium. The housing owners received subsidies based on their individual situation. Funded by the signatories of the OPATB agreement (blue or dark gray), the animation device (middle) aimed condominiums (green or light gray) to drive the energy renovation project: each condominium recruited a contractor. Each category had a technical expertise (orange or gray): this facilitated the knowledge transfer to architects ("conception") and contractors ("travaux") and the control of thermal performances by Ademe. OPATB animation combined the contribution of technical, organizational, financial and communication experts. A non-profit association (PACT), specialized in assisting housing improvement, coordinated the animation team.

City of Grenoble: an urban project became an illustration of the energy-climate local policy

At the stage of pre-operational studies, perimeter covered a band of four hundreds meters around the new tramway lane through the city. Three kinds of building were targeted: tertiary public buildings, small shops and condominium housing. The scope of these last two had been restricted. The 'Small shops' only concerns facades along the new tram lane (orange line below). The perimeter "Habitat" (green line below) applies to all blocs along the tram lane (green line below). 13/09/12 10:25 Realized where fortifications protected Grenoble city center until 1938 [START_REF] Parent | Grenoble deux siècles d'urbanisation: projets d'urbanisme et réalisations architecturales 1815-1965[END_REF], the Grands boulevards are currently lined by tall buildings built between 1945 and 1967. Their energy performance was poor. Several major urban projects concerned this area: highway bridge demolition, new tram lane, traffic calming measures, building facelifts and also building a green district nearby (De Bonne). The city council expected to reduce urban break between the wealthy historic center and the suburbs built during the post-war boom. "The intervention on the Grands Boulevards is part of a global project of territory conquest and transformation." explained the mayor in the 2006 brochure of the OPATB addressed to the inhabitants. It also supported the tram lane success, which some storekeepers opposed. The brochure advertised three aims: revitalizing the shops, improving buildings thermal performance, and enhancing architecture. OPATB's slogan "new air on the Grands Boulevards" exalted the aesthetic as much as climate. Three years later, in 2009, the tramway lane was so popular with inhabitants and most of the storekeepers, that the ideal of urban continuity became useless. Now, apart from technical communication to owners, the city exposes OPATB as a flagship project of its "Grenoble Factor 4" program. "Grenoble Factor 4" is de facto the Municipal Climate Action Plan and a new ideal for the city's future [START_REF] Henry | L'expérience de Grenoble depuis 2006, Introduction à l'atelier 3[END_REF]. The renewal of the City Council in 2008 confirmed this trend by moving the OPATB political leadership: the deputy mayor in charge of Sustainable Development replaced the deputy mayor in charge of Urban Development.

City of Echirolles: OPATB as an action of the sustainable development program complementary to major urban projects

Second town of the Grenoble urban area, Echirolles has been urbanized since 30s. In the 60s, social housing and condominiums buildings had started to grow alongside the working-class individual house area around main industrial sites. For the last twenty years, the city has built a true urban center around a tramway lane, which is the communal territory epicenter. Ten years ago, the city council launched an Agenda 21. It has mobilized residents and city staff around shared objectives by focusing on social and environmental synergies. It took care to cover all areas of the city and, both, the 13/09/12 10:25 diversity of sectorial thematic and the different categories of people. Since then, it has led to systematic follow-up actions with the residents. The OPATB "Echirolles West" conveniently filled the hollows of Agenda 21 and of a vigorous urban policy. First, the western districts did not benefit from the public facilities of the Echirolles city center. Second, the various actions on buildings conducted by the city did not relate to existing private buildings. The western districts focused on the higher proportion of private housing (mainly condo) and weaker political relays of the city leftist majority. Compared to Grenoble, OPATB communication emphasized household interest and, implicitly, global solidarity but it ignored the urban dimensions. Grenoble Alpes Métropole, the metropolitan authority: embrace a new skill Partner of Grenoble and Echirolles OPATB, the metropolitan authority (commonly called "The Metro") abounded them on its own budget. This institution is in charge of metropolitan transport, economic development, social housing and some environmental issues (atmospheric pollution, road noises, waste...) but not urban planning. The Metro created in 1999 the Local Energy Agency, which provides expertise and advice to local partners, businesses and residents. The Metro was one of the first French Métropole to develop a Local Climate Action Plan (presented in May 2005). It introduced early the climate change mitigation goal in its housing and transport policies, and mobilized the cities of the area to develop climate policies [START_REF] Debizet | Expertise and Methodology in Building Design for Sustainable Development : A FrancoBritish Comparison[END_REF]. It steers the "Mur Mur" device that succeed to OPATB from 2010. Restricted to an animation and incitement in urban planning role, the Metro will initiate and facilitate innovation 4 in the building industry that remains one of the main economic sectors and provides the majority of the metropolitan taxes. Minor contributors to both OPATB operations 13/09/12 10:25

The Rhône-Alpes Region funded energy audits and some solar water heaters or photovoltaic panels. In addition to GHG reducing, the action of the Region has been part of an economic development of solar and green building sectors [START_REF] Roudil | Translations" and Boundary Objects : Amateurs and Professionals in Berkely's Museum of Vertebrate Zoology, 1907-39[END_REF]. Non-signatory to the OPATB, the Region did not participate in their targeting. Under the European Concerto program, European Community funded a tenth of the OPATB animation. This funding was acquired several months before signing the partnership convention; City of Grenoble might make choices more vigorous.

Results of Grenoble OPATB

When Grenoble OPATB period (2007[START_REF] Debizet | La rénovation énergétique des bâtiments en France entre marché, actions territoriales et dispositions nationales[END_REF] ended, a quarter of eligible condominiums (22 condos grouping 637 apartments) had decided to realize thermal improvement work. Unlike previous OPATB partnership convention, their work consisted almost entirely in adding external insulation. OPATB granted 1.6 million of the 5.5 million work and spent 0.6 million for animation and expertise [START_REF] Debizet | La rénovation énergétique des bâtiments en France entre marché, actions territoriales et dispositions nationales[END_REF]. Grants represented 30% of work cost for an average owner. However, this rate depended on owner status and income. It varied from 35 to 80% for non-occupiers owners (landlords) and from 20 to 80% for occupiers owners. These rates did not only result of social issues. They were required to find a majority in favor of energy work during the annual condominium meeting: -Because national law restricts the rent increase, landlords do not benefit of energy renovation when they keep the housing. Therefore, they need grants to maintain profitability of investment. Among them, the less wealthy and the older have to be helped because they cannot get a bank loan.

-

The poorer and the older occupiers owners have also to be more helped. First, they cannot benefit of tax breaks and bank loan. Second, the profitability of their energy renovation investment is lower than that of the wealthiest occupiers owners because their initial energy bill is low (they already try to save energy). Owners can also benefit from national tax breaks on the balance after OPATB grants. Failing of specific statistics, we assess that tax breaks represent roughly 10% of the global work cost. According professionals involved in the OPATB, tax breaks are not decisive for modest owners. The Grenoble OPATB helped reduce annual carbon emission of 575 tons. Therefore, energy renovation cost 10,000 euros to reduce annually one ton of carbon: OPATB partners (including state) funded directly 4,000 euros and state funded 1,000 euros through tax breaks.

*** Local authorities consider OPATB as an opportunity for a wider project (or policy): sustainability program (Echirolles), urban project or climate plan (Grenoble). The specific priorities of local governments influence perimeter and targeted population. Anyway, the initial convention and the partnership steering committee ensure that the implementation respects the aims of mitigation and energy saving. We missed place to describe the cognitive and organizational aspects. The concept of boundary object could also be useful to describe the interactions of local governments and national agencies especially in order to analyze the outstanding issues of energy renovation policies.

CONCLUSION 13/09/12 10:25

The energy renovation appears a consensual mean to reduce GHG emission. However, it requires a high level of investment. The wealthier occupiers owners of individual dwelling can expect a financial gain thank to energy saving; they benefit easily from tax breaks and are able to quickly decide insulation work. The collective housing owners do not have the same abilities. A local accompaniment is necessary; it combines variable grants and a socio-technical mediation. Unlike one ministry, local authorities consider the scheduled operations of energy improvement as a transversal element to their different policies. They will have to face the antagonism between mitigation and social goals [START_REF] Debizet | La rénovation énergétique des bâtiments est-elle possible ?[END_REF]. Maximizing GHG emission reduction of a given budget leads to orientate public funds toward the wealthier condominiums. On another scale, French government should adjust the rate of tax break according household income and dwelling type. It should also increase the part of scheduled operations within its overall budget available for energy renovation. Anyway, it cannot avoid the debate between mitigation goal and social priorities. Further analyzes of the local implementations of mitigation could feed this debate.

  between building partners". 13/09/12 10:25 Ministries validated the OPATB device. They ensured compliance guidance to key moments and the communication fallout. Then, Anah and Ademe have implemented the device in partnership with local governments: -Anah fulfills its mission: aid landlords and modest tenants. It masters the operational methods: targeting buildings, defining scope and ways of grants... -Ademe funded energy studies, technical animation, final evaluation and some high performance work. It manages also the experiences exchange network. Ademe and Anah envisaged twenty-six local OPATB: "The future multiplication of OPATB depends on the success of these first OPATB." Only 16 OPATB were underway in 2007. However, a growing number of OPAH incorporates "energy" as one of their goals: 273 of 715 ongoing OPAH in 2009. *** During the OPATB device gestation period (2000-2002), Ministry of Housing (Besson 2000) appropriated the goal of climate change mitigation (targeting global solidarity) in addition to initial goal of saving energy (targeting the purse of owners and occupiers). OPATB device hybridizes OPAH (Anah) and innovation funding (Ademe), it finally encourages Anah to develop energy goal in the OPAHs. Ministry of Environment and Ademe increased their legitimacy and their visibility on the housing sector. OPATB device appears as a transient boundary object between the sectorial Ministries and between their agencies. Let us observe how the OPATB device implements the mitigation on a local scale.

Figure 1

 1 Figure 1 OPATB Grands Boulevards process. (Source: Debizet based on figure of Carré, PACT de l'Isère 2009)

Figure 2

 2 Figure 2 Perimeter and and results of Grands Boulevards OPATB (Source City of Grenoble : OPATB assessment)

Figure 3

 3 Figure 3 Perimeter and posters of West Echirolles OPATB. (Source: City of Echirolles)

13/09/12 10:25

ADEME is inFrance what the EPA is in the U.S.A.

Grenoble (160,000 inhabitants) and Echirolles (37,000) are the main cities of Grenoble urban area (500,000 inhabitants).

COBRA2012_DEBIZET_energy-renovation-local-policiy_grenoble-France_final.doc

Grenoble is often considered as an urban laboratory. First world low-platform tramway was intented in 1987 à Grenoble in conjunction with associations for the disabled and the authorities. The Caserne de Bonne urban project was awarded as best French green district.