A macroscopic traffic flow model with phase transitions and local point constraints on the flow

The two phase model for vehicular traffic flows recently appeared in \cite{goatin2006aw} and \cite{BenyahiaRosini01} is endowed with a point constraint on the flow to allow for the modelling of phenomena such as the effects of toll booths along a road. We describe in this paper two Riemann solvers for this model upon which an ulterior study of the cauchy problem should rely.


Introduction
Macroscopic traffic flow models has been a growing field of research in the last decade as it is finding real life applications related to traffic control, management and prediction, see the surveys [5,11,12], the books [8,14] and the reference therein. Among these models, two of most noticeable importance are Lighthill, Whitham [10] and Richards [13] model (LWR) and the Aw, Rascle [4] and Zhang [15] model (ARZ) Theses two models aim to predict the evolution in time t of the density ρ and the velocity v of vehicles moving along a homogeneous highway parametrized by the coordinate x ∈ R and with no entries and exits. Both of the models have drawbacks however in their modelling of the evolution of traffic flows. Indeed, LWR assigns a priori an explicit relation between density and velocity, that means that the model assumes that the velocity of the vehicles is entirely determined by the density and dismisses the possibility of different vehicle populations that might exhibit a different velocity for a given density. Empirical studies show however that the density-flux diagram can be approximated by this kind of models only up to a certain density, across which the traffic changes phase from free to congested, the latter being better approximated by a second order models such as ARZ. On the other hand ARZ is not well-posed near the vacuum. In particular, when the density is close to zero, the solution does not depend in general continuously on the initial data. This motivated the introduction in [9] of a two phase model that describes free and congested phases by means of respectively LWR and ARZ. Recall that this model was recently generalized in [6]. A couple of mathematical difficulties have to be highlighted. First, the model consists of two systems of equations that prescribe the evolution in time of the traffic in two different phases, free and congested, and prescribes a specific set of admissible phase transitions. One difficulty is that it is not known a priori the curves in the (x, t) plane dividing two phases. The problem cannot be reduced therefore into solving two different systems in two distinct regions with prescribed boundary conditions. As a consequence, defining a notion of weak or entropy solution via an integral condition (as it is standard in the field of PDEs) turns out to be a delicate task. In [6] it has been possible to do so when the characteristic field of the free phase is linearly degenerate, because in this case the flux in the free phase reduces to a restriction of the flux diagram of ARZ, which allowed to make use of the definition of weak and entropy solution introduced in [1]. Another difficulty is the possibility that two phase transitions may interact with each other and cancel themselves. In fact, for instance, it is perfectly reasonable to consider a traffic characterized by a single congested region C, with vehicles emerging at the front end of C and moving into a free phase region with a velocity higher than the tail of the queue at the back end of C, so that after a certain time the congested region disappears and all the traffic is in a free phase. For this reason a global approach for the study of the corresponding Cauchy problem can not be applied, as it would require a priori knowledge of the phase transition curves; it is instead preferable to apply the wave-front tracking algorithm, as it allows to track the positions of the phase transitions.
The present article deals with the Riemann problem for the two phase model introduced in [6], equipped with a local point constraint on the flow, meaning that we add the further condition that at the interface x = 0 the flow of the solution must be lower than a given constant quantity Q. This models, for instance, the presence of a toll gate across which the flow of the vehicles cannot exceed the value Q. The additional difficulty that this add to the mathematical modelling of the problem is that this time one can start with a traffic that is initially completely in the free phase, but congested phases arise in a finite time in the upstream of x = 0, as it is perfectly reasonable in the case of a toll gate with a very limited capacity. The aim of this paper is to establish two Riemann solvers for this model and to study their properties. These Riemann solvers will be the basis upon which to rely for the ulterior study of the Cauchy problem.
The paper is organized as follows. In Section 2 we state carefully the problem, introduce the needed notations and define the two Riemann solvers, see Definition 2.2 and Definition 2.3. Then in Section 3 we study their basic properties, some of which require lenghty and technical proofs that are postponed in the last section.

The model and the main result
The aim of this section is to propose two Riemann solvers for the two phase transition model developed in [6], coupled with a point constraint on the flow (1).

Assumptions and notations
Let us first introduce the notation to the reader and explain its usage.
For later convenience, see Figure 1, we introduce the following notation: To avoid technicalities in the exposition, we will throughout assume that p is defined in (0, +∞) and that there it satisfies (H2).
Some simple choices for V and p, see [3,4,9], are where v max , R, γ, v ref and ρ max are strictly positive parameters, that can be chosen so that (H1)-(H3) are satisfied, see [6] for the details.
Fix V c ∈ ]0, V min ] and let u . and are the domains of respectively free phases and congested phases. Observe that Ω f and Ω c are invariant domains for respectively LWR and ARZ. Introduce also the domains , Figure 1: Geometrical meaning of notations used through the paper.

The unconstrained Riemann problem
The Riemann problem for the two phase model introduced in [6] has the form where u ℓ , u r ∈ Ω and the maps Q : Above ρ and v denote respectively the density and the speed of the vehicles, while V and p give respectively the speed of the vehicles in a free flow and the "pressure" of the vehicles in a congested flow. In the free phase the characteristic speed is In the following table we collect the informations on the system modelling the congested phase: Above r i is the i-th right eigenvector, λ i is the corresponding eigenvalue and L i is the i-th Lax curve. By the assumptions (H1) and (H2) the characteristic speeds are bounded by the velocity, λ f (u) ≤ v, λ 1 (u) ≤ λ 2 (u) = v, and λ 1 is genuinely non-linear, ∇λ 1 · r 1 (u) 0. Beside the Riemann solver introduced in [6], here denoted by R 1 , we introduce in the following definition also a second Riemann solver R 2 . Roughly speaking, the motivation is that the solution corresponding to R 2 has flow through the constraint higher than that corresponding to R 1 . We denote by R LWR and R ARZ the Riemann solvers for respectively LWR and ARZ. To simplify our notation, we let q * . = Q(u * ) and w * . = W(u * ). Recall that for any u ℓ , u r ∈ Ω with ρ ℓ ρ r , the speed of propagation of a discontinuity between u ℓ and u r is σ(u ℓ , u r ) .
Definition 2.1. The Riemann solver R 1 : Ω 2 → L ∞ (R; Ω) is defined as follows: The Riemann solver R 2 : Ω 2 → L 1 loc (R; Ω) is defined as follows: (R 2 ) If (u ℓ , u r ) ∈ Ω f × Ω c , ρ ℓ 0 and w ℓ < w r , then We remark that, differently from the phase transitions from Ω ′′ f to Ω c introduced by R 1 , those introduced by R 2 may not satisfy the second Rankine-Hugonot condition, namely they may not conserve the generalized momentum.
In [6,Proposition 4.2] we proved that R 1 is L 1 loc -continuous and consistent in Ω. Let us recall that a Riemann solver S : It is easy to prove that R 2 is L 1 loc -continuous but is not consistent in Ω. Indeed, for instance, R 2 does not satisfy (II) with u ℓ ∈ Ω ′′ f and u m , u r ∈ Ω c such that w ℓ = w m < w r and v m = v r .

The constrained Riemann problem
In this section we consider the Riemann problem (1) coupled with a pointwise constraint on the flux where Q 0 ∈ (0, q max ) is a fixed constant. In general, [(t, x) → R i [u ℓ , u r ](x/t)] does not satisfy (2). For this reason we introduce the sets and for any (u ℓ , u r ) ∈ N i , we replace [(t, x) → R i [u ℓ , u r ](x/t)] by another self-similar weak solution [(t, x) → R c i [u ℓ , u r ](x/t)] to (1), satisfying (2) and obtained by juxtaposing weak solutions constructed by means of R i . It is easy to see that To simplify our notation, we let * q . = Q( * u) and * w .
In the following proposition we show that R c 2 is well defined. Proposition 2.2. For any (u ℓ , u r ) ∈ N 2 , (û,ǔ) ∈ Ω c × Ω is uniquely selected by (6), (7) as follows: , then we distinguish the following cases: 2 ) If (u ℓ , u r ) ∈ N 2 ∪ N 4 2 , then we distinguish the following cases: It is easy to prove that in general both R c 1 and R c 2 fail to be consistent. Proposition 2.4. In general, both R c 1 and R c 2 satisfy neither (I) nor (II) in Ω.
Proof. For any u ℓ , u m , u r ∈ Ω such that u m = u r and q r > Q 0 , by the finite speed of propagation of the waves, there exists x > 0 such that R c 1 [u ℓ , u r ](x) = R c 2 [u ℓ , u r ](x) = u r . Then the property R c 1 [u r , u r ](x) = R c 2 [u r , u r ](x) = u r for any x <x required in (I) cannot be satisfied because otherwise Q(R c 1 [u r , u r ](0 ± )) = Q(R c 2 [u r , u r ](0 ± )) = q r > Q 0 and this gives a contradiction because R c 1 [u r , u r ] and R c 2 [u r , u r ] satisfy (2), see Proposition 2.3. Moreover, if Q 0 ∈ [Q(u c (W c )), Q(u c (W max ))], then we can take u ℓ , u r ∈ Ω f , u m ∈ Ω c with w ℓ = w m and Q(u m ) = Q 0 = Q(u r ), and see that (II) is not satisfied by both R c 1 and R c 2 . We conclude this section by considering the total variation of the two constrained Riemann solvers in the Riemann invariant coordinates. We provide two examples showing that in general the comparison of their total variation can go in both ways. This suggests that the total variation is not a relevant selection criteria for choosing a wave-front tracking algorithm based on one or the other Riemann solver.

Basic properties of
The proof is rather technical and is therefore deferred to Section 4.1.
and is not L 1 loc -continuous in any point of C ∩ N 1 . Proof. Assume that Q 0 > Q(u c (W c )) and let u 0 ∈ Ω f be such that Q(u 0 ) = Q 0 . Then it suffices to take u ℓ = u 0 , u r = (R ′′ f , V min ) and u n ℓ ∈ Ω f with ρ n ℓ .
= ρ 0 + 1/n. Indeed in this case (u n ℓ ) n converges to u ℓ but R c 1 [u n ℓ , u r ] does not converge to R c 1 [u ℓ , u r ] in L 1 loc (R; Ω). More precisely, R c 1 [u ℓ , u r ] ≡ u ℓ in R − and by (T 1 1 a) the restriction of R c 1 [u n ℓ , u r ] to R − converges to Proving that R c 1 is L 1 loc -continuous in Ω 2 in any other situation is now a matter of showing that R 1 [u n ℓ ,û n ] → R c 1 [u ℓ , u r ] pointwise in {x < 0}, R 1 [ǔ n , u n r ] → R c 1 [u ℓ , u r ] pointwise in {x > 0}, and applying the dominated convergence theorem of Lebesgue. For this, it suffices to observe that eitherû n → R 1 [u ℓ , u r ](0 − ) and the result follows then by the continuity of R 1 , or σ(u n ℓ ,û n ) → 0 and R 1 [u ℓ , u r ] is constant equal to u ℓ in {x < 0} and we obtain therefore again that R 1 [u n ℓ ,û n ] → R 1 [u ℓ , u r ] pointwise in {x < 0}. A similar analysis proves that R 1 [ǔ n , u n r ] → R 1 [u ℓ , u r ] pointwise in {x > 0}.
In the next proposition we study the invariant domains of R c 1 . However, since in [6] we did not consider the invariant domains of R 1 , let us first point out that for any 0 ≤ ρ min < ρ max ≤ R ′′ f , 0 ≤ v min < v max ≤ V c and W c ≤ w min < w max ≤ W max the following sets are invariant domains for R 1 .
containing Ω c . The proof is postponed to Section 4.2.

Proof of Proposition 3.1
To simplify the exposition of the proof, we divide it into the following two lemmas.
Lemma 4.2. R c 1 satisfies (II) in D 1 . Proof. Fix u ℓ , u m , u r ∈ D 1 ,x ∈ R and assume that R c 1 [u ℓ , u m ](x) = u m = R c 1 [u m , u r ](x). Then we consider the following cases: • If u ℓ , u r ∈ Ω f , then also u m ∈ Ω f . Hence (II) follows from the consistency of R 1 , because in this case (u ℓ , u r ), (u ℓ , u m ), (u m , u r ) ∈ C 1 ⊆ C 1 .