BOUNDARY VALUE PROBLEM FOR A PHASE TRANSITION MODEL

. We consider the boundary value problem for the phase transition (PT) model, introduced in [4] and in [7]. By using the wave-front tracking tech- nique, we prove existence of solutions when the initial and boundary conditions have ﬁnite total variation.

1. Introduction. The paper deals with the initial-boundary value problem for the phase transition model (PT for short), introduced in [4] and in [7] for modeling car traffic in an unidirectional road. Traffic models based on differential equations can be divided mainly in two classes: microscopic and macroscopic. The PT model belongs to the class of macroscopic traffic models, and it is composed by a system of two differential equations, which impose the conservation of the number of vehicles and of a momentum.
The fluid dynamic and macroscopic approach for car traffic was, in turn, initiated by the seminal work of Lighthill-Whitham and Richards [12,13], known as the LWR model. It consists in a differential equation stating the conservation of the number of vehicles. It well describes the evolution of free traffic, but it is not accurate for congested traffic. Hence second order models, i.e. system with two equations, were introduced in the literature. Among these we mention the model proposed by Aw and Rascle in [3] and independently by Zhang in [14]. In 2002, Colombo proposed a second order model with phase transitions [6], in order to describe the different behaviors of traffic in free and congested regimes. In free flow the model is a classical LWR model, whereas in the congested flow it is a system of two equations: the first one is the conservation of the number of vehicles, while the second one describes the evolution of a linearized momentum. The system considered in this paper is a modification of the Colombo phase-transition model.
Here we consider the following initial-boundary value problem for the PT model if (ρ, q) ∈ Ω c , (ρ, q) (t, a) = (ρ a (t), q a (t)) (ρ, q) (t, b) = (ρ b (t), q b (t)) (ρ, q) (0, x) = (ρ 0 (x), q 0 (x)) , where (ρ 0 , q 0 ) : (a, b) → Ω f ∪ Ω c is the initial datum, (ρ a , q a ) : (0, +∞) → Ω f ∪ Ω c and (ρ b , q b ) : (0, +∞) → Ω f ∪ Ω c are the boundary data and Ω f , Ω c denote respectively the free and congested phase (see Section 2 for a rigorous definition). We prove existence of solutions by means of the wave-front tracking technique, provided the initial and boundary conditions are BV functions. The wave-front tracking technique consists in constructing a piecewise constant approximate solution and in proving that every limit point is indeed a solution of the problem. The key estimate to obtain compactness for the sequence of approximated solutions is a uniform bound of a functional measuring the strength of waves. Moreover bounds on the number of waves and interactions for wave-front tracking approximate solutions are given. We remark that imposing boundary conditions for systems of balance laws is a quite delicate issue, especially for characteristic boundary conditions; see [1,2,9] and the references therein. In particular, the PT system with boundary is characteristic, since phase-transition waves can travel with zero speed. The paper is organized as follows. In Section 2 the phase transition model is presented, and, in Section 3, a theorem about the existence of solutions for the Cauchy problems with boundary data is stated and proved.
2. Description of the phase transition model. We describe the phase transition model, introduced in [4], with the Newell-Daganzo velocity function. The PT model describes the evolution of traffic trough the macroscopic variables ρ and q, representing respectively the density and the linearized momentum; see also [6]. In this model, it is assumed that cars behave differently, depending on the fact that the traffic is low or heavy. Therefore there are two phases: the free phase, denoted by Ω f , and the congested one, denoted by Ω c . These two phases are described by the sets where V > 0 is the velocity in the free phase, R > 0 is the maximal density, is the velocity; see Figure 1. The PT model assumes that the velocity of cars in the free phase Ω f is constantly equal to V , while in the congested phase Ω c is a perturbation of the equilibrium velocity v eq (ρ). The equilibrium velocity v eq (ρ) represents the desired speed of cars when the density is ρ. Moreover, the constant σ is called critical density and corresponds to the density for which the flux is maximal in scalar models. The constants q − and q + are respectively the minimum and the maximum value of the momentum. When q = 0, the velocity of cars v coincides with the equilibrium velocity. Finally, the constants σ − > 0 and σ + > 0 are defined respectively by the ρ-component of the solutions in Ω c to the systems Figure 1. The fundamental diagram for the phase transition model in (ρ, q) and (ρ, ρv) coordinates. The free phase Ω f is the one-dimensional region in blue. The congested phase Ω c is the two-dimensional region in red. Note that Ω f ∩ Ω c = ∅.
The solution of each system exists and is unique. The first system describes the point (ρ, q) ∈ Ω c such that v(ρ, q) = V , i.e. (ρ, q) belongs also to Ω f , and q ρ = q − R , i.e. (ρ, q) belongs to the lower side of Ω c ; see Figure 1. The second system, instead, describes the point (ρ, q) ∈ Ω c such that v(ρ, q) = V , i.e. (ρ, q) belongs also to Ω f , and q ρ = q + R , i.e. (ρ, q) belongs to the upper side of Ω c ; see Figure 1. The model in the free phase Ω f reads while in the congested phase Ω c reads The eigenvalues for (4) are while the respectively eigenvectors are Moreover the first characteristic speed is genuinely nonlinear if q = 0 and it is linearly degenerate if q = 0. Instead the second characteristic speed is linearly degenerate. A deeper analysis of (4) is contained in [4]. We denote with L 1 (ρ; ρ 0 , q 0 ) and L 2 (ρ; ρ 0 , q 0 ) respectively the Lax curves of the first and second family for system (4); see Figure 2. We recall that (ρ, L 1 (ρ; ρ 0 , q 0 )) are lines in the (ρ, q) plane passing through the origin and (ρ, L 2 (ρ; ρ 0 , q 0 )) are lines in the (ρ, ρv) plane passing through the origin. Let us introduce the functions such that, for every (ρ 0 , q 0 ) ∈ Ω c , ψ 1 (ρ 0 , q 0 ), ψ − 2 (ρ 0 , q 0 ) and ψ + 2 (ρ 0 , q 0 ) are defined respectively by the solutions in Ω c to the systems Remark 1. As in the case of the definitions of σ − and σ + , each system in (8) admits a unique solution, due to the fact that the Lax curve of the first and second families are transverse. The first system in (8) selects the point (ρ,q) ∈ Ω c on the Lax curve of the first family through (ρ 0 , q 0 ) such that v(ρ,q) = V , i.e. (ρ,q) belongs also to the free phase Ω f ; see Figure 2.
Similarly the second and third system in (8) select, the point (ρ,q) ∈ Ω c on the Lax curve of the second family through (ρ 0 , q 0 ), i.e. v(ρ,q) = v (ρ 0 , q 0 ), such that it belongs respectively on the lower and upper side of Ω c ; see Figure 2.
The following lemma implies that waves of the first family have negative speed, while waves of second family have positive speed.
Lemma 2.1. The first eigenvalue λ 1 is strictly negative in Ω c . The second eigenvalue λ 2 is positive in Ω c .
For a proof, see [10,Proposition 3.7]. For the PT model, there is a third kind of waves, namely phase-transition waves. They are waves connecting a state ρ l , q l ∈ Ω f \ Ω c (on the left) with a state (ρ r , q r ) ∈ Ω c (on the right). More precisely, if one considers the following Riemann problem on the whole line R with ρ l , q l ∈ Ω f \ Ω c and (ρ r , q r ) ∈ Ω c \ Ω f , then its solution is given by the following phase-transition wave where (ρ m , q m ) ∈ Ω c is the unique solution to i.e. (ρ m , q m ) is the point belonging to the lower part of the congested phase Ω c and to the Lax curve of second family through (ρ r , q r ), and Note that λ 1 is given by the Rankine-Hugoniot condition; for complete details see [4]. Introduce the following definition. Figure 2. Shape of Lax curves in (ρ, q) and (ρ, ρv(ρ, q)) planes.
Definition 2.2. Let I be a real interval. We say that a function (ρ, q) : Remark 2. Note that, since 0 <v < V , a point (ρ,q) satisfying v(ρ,q) ≥v either belongs to the free phase Ω f or belongs to a subset of the congested phase Ω c .
in invariant for the solution to the Riemann problem. Moreover, assumption (H-1) of Definition 2.2 guarantees that the total variation of a solution can be controlled by the total variation of the Riemann invariants of the solution itself.
We introduce also a distance in the set Ω f ∪ Ω c in the following way. First define the function ω : Secondly, given two states ρ l , q l , (ρ r , q r ) in the set Ω f ∪ Ω c : 2. if ρ l , q l ∈ Ω c and (ρ r , q r ) ∈ Ω c , then, denoting by (ρ m , q m ) ∈ Ω c the point satisfying q m = L 1 ρ m ; ρ l , q l and q r = L 2 (ρ r ; ρ m , q m ), Remark 4. Note that equations (12) and (13) can be both rewritten as since in case 1. v ρ l , q l = v (ρ r , q r ), and in case 2. ω ρ l , q l = ω (ρ m , q m ) and v (ρ m , q m ) = v (ρ r , q r ).
Remark 5. The function ω, defined in (11), is indeed a continuous function. It is clear that q We have: This proves that ω is continuous. (12), (13) and (14), is a metric on Ω f ∪ Ω c .
Proof. First we prove that d ρ l , q l , (ρ r , q r ) = 0 if and only if ρ l , q l = (ρ r , q r ). It is clear that if ρ l , q l = (ρ r , q r ), then d ρ l , q l , (ρ r , q r ) = 0. Assume therefore that ρ l , q l = (ρ r , q r ). We have different possibilities.
1. ρ l , q l ∈ Ω f and (ρ r , q r ) ∈ Ω f . We have that the function ω restricted to the set Ω f can be viewed as a strictly increasing function of the variable ρ. Since ρ l , q l = (ρ r , q r ), we may suppose, without loss of generalities, that ρ l < ρ r and so d ρ l , q l , (ρ r , q r ) = ω ρ l , q l − ω (ρ r , q r ) = ω (ρ r , q r ) − ω ρ l , q l > 0.
2. ρ l , q l ∈ Ω c and (ρ r , q r ) ∈ Ω c . Call (ρ m , q m ) ∈ Ω c the point defined in the point 2. of the construction of the function d. Since ρ l , q l = (ρ r , q r ), either ρ l , q l = (ρ m , q m ) and so 4. ρ l , q l ∈ Ω c and (ρ r , q r ) ∈ Ω f . This case is completely analogous to the previous one.
We prove now that d is symmetric. Fix two states ρ l , q l and (ρ r , q r ) in Ω f ∪ Ω c . If both states belong to Ω f , then the conclusion easily follows by (12).
If both states belong to Ω c , then define the states (ρ m , q m ) ∈ Ω c and (ρ s , q s ) ∈ Ω c by One can easily deduce that and so In the remaining case, the conclusion easily follows from the fact that the point We prove now that the triangular inequality holds for d. Fix three states (ρ 1 , q 1 ), (ρ 2 , q 2 ), and (ρ 3 , q 3 ) in Ω f ∪ Ω c . If they both belong either to Ω f or to Ω c , then, by equation (15), the triangular inequality easily follows. The triangular inequality for the general case follows from the previous case.
3. The initial-boundary value problem. Fix a, b ∈ R with a < b. Consider the following initial boundary value problem First introduce the definition of solution to (16).

Remark 6.
As usual in conservation laws, boundary conditions are a delicate issue. In this paper the boundaries are characteristic, since there are phase-transition waves with zero speed. Conditions 3 and 4 of Definition 3.1 are exactly the same as the boundary condition in the characteristic case in [1].
We state the main result of the paper whose proof is contained in the next subsections.

It holds that
5. It holds that, for a.e. t > 0, the Riemann problem with initial condition is solved with waves with non positive speed. 6. It holds that, for a.e. t > 0, the Riemann problem with initial condition is solved with waves with non negative speed.
Consider three sequences (ρ 0,ν , q 0,ν ), (ρ a,ν , q a,ν ), and (ρ b,ν , q b,ν ) of piecewise constant functions with a finite number of discontinuities such that For every ν ∈ N \ {0}, we apply the following procedure. At time t = 0, we solve the boundary Riemann problems at x = a and x = b and all Riemann problems for x ∈ (a, b). We approximate every rarefaction wave with a rarefaction fan, formed by rarefaction shocks of strength less than 1 ν traveling with the Rankine-Hugoniot speed. At every discontinuity time for (ρ a,ν , q a,ν ) or for (ρ b,ν , q b,ν ), we solve the corresponding Riemann problem at x = a or x = b. At every interaction between two waves, we solve the corresponding Riemann problem. Finally, when a wave interacts with the boundary x = a or x = b, we solve the corresponding boundary Riemann problem.
Remark 7. As usual, by slightly modifying the speed of waves or the position of the discontinuities for the boundary values, we may assume that, at every positive time t, at most one of the following possibilities happens: 1. two waves interact together at a point x ∈ (a, b); 2. a wave interacts with the boundary x = a or with the boundary x = b; 3. t is a point of discontinuity either for (ρ a,ν , q a,ν ) or for (ρ b,ν , q b,ν ). ∈ (a, b), we split rarefaction waves into rarefaction fans just at time t = 0. At the boundary x = a or x = b, instead, we allow the formation of rarefaction fans at every positive time, but only for changes of the boundary data.

Remark 8. For interactions at a point x
Remark 9. By assumption, the initial condition and the boundary data satisfy hypothesis (H-1). Therefore there existsv > 0 such that the velocity v of every states in Ω c , generated by the previous construction, is greater than or equal tov.
Given an ε-approximate wave-front tracking solutionū ε = (ρ ε ,q ε ) with boundary dataū a,ε = (ρ a,ε ,q a,ε ) andū b,ε = (ρ b,ε ,q b,ε ), define, for a.e. t > 0, the following functionals (a detailed explanation is contained in Remark 10) Note that the previous functionals may vary only at times at which the boundary datum changes or at timest when two waves interact or a wave reaches the boundary.
Remark 10. The functional W is composed by 5 terms. The first and second term measure the strength of waves of first and second family. The third term measures the phase transition waves. Finally, the last two terms measure the distance of the boundary term from the trace at the boundary of the approximate solution.
3.0.2. Interaction estimates. We consider here estimates for wave interactions. In the following, as in [10], we describe wave interactions by the nature of the involved waves. For example, if a wave of the second family hits a wave of the first family producing a phase-transition wave, we write 2-1/PT. Here the symbol "/" divides the waves before and after the interaction. The functional W does not increment for wave interaction inside the phase transition system; hence it can only increases for interaction of waves with the interface x = 0. Lemma 3.4. Assume that the waves ((ρ l , q l ), (ρ m , q m )) and ((ρ m , q m ), (ρ r , q r )) interact at the point (t,x) witht > 0 andx ∈ (a, b). Then W (t+) ≤ W (t−).
The proof is completely identical to that of Lemma 4.29 of [10]; hence we omit it.
Lemma 3.5. Assume that the wave ((ρ l , q l ), (ρ r , q r )) interacts with the boundary at the point (t, a). Then W (t+) ≤ W (t−). Moreover at timet either no wave or one wave is generated. In the latter case, the possible interactions are 1/PT (if (ρ r , q r ) ∈ Ω f ) or 1/2 (if (ρ r , q r ) ∈ Ω f ).
Proof. First note that ∆W b (t) = 0, since the interaction happens at x = a. Moreover, since the boundary Riemann problem does not generate waves, then the states (ρ a,ε ,q a,ε )(t) and (ρ l , q l ) are connected through waves with non positive speed. We have therefore several possibilities.
2. The states (ρ a,ε ,q a,ε )(t) and (ρ l , q l ) are connected by a wave of the first family. In this case both the states (ρ a,ε ,q a,ε )(t) and (ρ l , q l ) belong to Ω c and so the interacting wave also is of the first family; at timet, it is absorbed and no other wave is generated.
3. The states (ρ a,ε ,q a,ε )(t) and (ρ l , q l ) are connected by a phase-transition wave with non positive speed.
The proof is so finished.
Proof. First note that ∆W a (t) = 0, since the interaction happens at x = b. Moreover, since the boundary Riemann problem does not generate waves, then the states (ρ r , q r ) and (ρ b,ε ,q b,ε )(t) are connected through waves with non negative speed. We have therefore several possibilities.
2. The states (ρ r , q r ) and (ρ b,ε ,q b,ε )(t) are connected by a wave of the second family.
Lemma 3.8. Assume thatt is a discontinuity point for the boundary datum at Proof. In general, at timet, a wave with negative speed emerges from the boundary x = b. In all the cases, denoting with (ρ,q) the trace of the approximate solution before timet at x = b−, we have by the triangular inequality. The proof is so concluded.
Proposition 2. For a.e. t > 0, we have where M > 0 is a constant.
3.0.3. Existence of a wave-front tracking solution. We now want to bound the number of waves and of interactions.
Definition 3.9. A wave ofū ε , generated at time t = 0, is said an original wave.
Definition 3.10. A wave of the second family (ρ l , q l ), (ρ r , q r ) is said special if Definition 3.11. A wave of the second family (ρ l , q l ), (ρ r , q r ) is said semi-special either if Lemma 3.12. Special waves can not emerge by interactions of waves inside the phase transition system. They can be generated only at time t = 0 or at the boundary x = a.
For a proof see [10].
Lemma 3.13. A wave of the second family (ρ l , q l ), (ρ r , q r ) with (ρ l , q l ), (ρ r , q r ) ∈ Ω f \ Ω c either is an original wave or can be generated by a variation of the boundary data at x = a. This holds in particular for semi-special waves.
Proof. Note first that the wave has positive speed; hence can not be generated at the boundary x = b. Assume first that it is generated at a point (t,x) witht > 0 andx ∈ (a, b). In this case, it should emerge also a wave of the first family or a phase-transition wave connecting a state (ρ,q) to (ρ l , q l ). This is clearly not possible.
Assume now that a wave with negative speed ((ρ,q), (ρ r , q r )) interacts with the boundary x = a at a timet > 0. Since (ρ r , q r ) ∈ Ω f this interactive wave is not a phase-transition wave. The remaining possibility is a wave of the first family, which is not possible, since (ρ r , q r ) ∈ Ω c . This completes the proof.
The following proposition holds.  Table 1. List of interaction types, which can happen, in principle, an infinite number of times.
The number of waves can increase in the cases 1., 3., 4., and 5. By construction, the case 4., and 5. happen at most a finite number of times. In case 1., the only interaction, which produces an increment of the number of waves, is 2-1/PT-1-2.
In this situation, a special wave interacts with a wave of the first family, producing three waves; see [10]. The wave of the second family, generated by the interaction is not special. By Lemma 3.12, the number of special waves is bounded by N ν (0+) (number of original waves) plus the number of discontinuities of the boundary datum (ρ a,ν , q a,ν ).
In case 3., the interaction, producing an increment of the number of waves, is 2/PT-1. In this case the interacting wave of the second family ρ l , q l , (ρ r , q r ) has the property that both ρ l , q l and (ρ r , q r ) belong to Ω f \ Ω c . By Lemma 3.13, such a wave is either an original wave or a wave produced at the boundary x = a and so, its number is bounded by N ν (0+) (number of original waves) plus the number of discontinuities of the boundary datum (ρ a,ν , q a,ν ).
The previous analysis shows that the interactions, producing an increment of the number of waves N ν , can happen at most a finite number of times. Therefore the interactions (inside the domain (a, b)) 1-1/1, 2-PT/PT, PT-1/2, PT-1/PT, 2-1/PT can happen at most a finite number of times, since we have a uniform bound on the number of waves. So it remains to consider and to bound the number of interactions of the types included in Table 1. Consider first the interactions of Table 1 generating a wave of the second family and happening inside the domain (a, b). In all such interactions, a wave of the second family is present also before the interaction itself. Moreover this interacting wave of the second family is not a semi-special wave; see Definition 3.11. Instead the interaction 1/2 at the boundary x = a produces a semi-special wave. Therefore the interactions 2-1/1-2, 2-PT/1-2, and 2-1/PT-2 can happen a finite number of times, since the interacting waves of the second family can be generated either at time 0 or at the boundary x = a when the boundary datum changes or by an interaction happening at most a finite number of times. For the same reason (the interacting wave of the second family is not semi-special), also the interaction 2-1/PT-1 inside the domain and 2/PT at the right boundary can happen at most a finite number of times. We can therefore limit the study to the following interactions: Interaction's position Interaction's types Left boundary 1/PT, 1/2 Right boundary 2/1 Inside the domain PT-1/PT-1 Let Λ be the set containing the speed of all possible waves in the phase-transition model and define K = sup {|λ| : λ ∈ Λ}. It is sufficient to prove that the number of interactions is finite in each time interval [t 1 , t 2 ] such that t 2 − t 1 < b−a 2K . Indeed this choice implies that the boundary at x = a does not influence the boundary at x = b and viceversa. Under this assumption, the interactions 1/2 at x = a and 2/1 at x = b can happen at most a finite number of times.
Therefore only PT-1/PT-1 (inside (a, b)) and 1/PT (at x = a) can happen an infinite number of times.
If 1/PT happens a finite number of times, then also PT-1/PT-1 does; see [10, Proof of Proposition 4.37]. Thus the only possibility for having an infinite number of interactions is that the wave of the first family produced by PT-1/PT-1 goes back to the boundary x = a in order to produce 1/PT. If this happens the number of waves strictly decreases, since the generated phase-transition wave of PT-1/PT-1 is at left of the wave of the first family. Hence the combination of interactions can not happen an infinite number of times. The proof is so concluded.
3.0.4. Existence of a solution. In this part we conclude the proof of Theorem 3.2.
Proof of Theorem 3.2. Fix an ε-approximate wave-front tracking solutionū ε to (16), in the sense of Definition 3.3. By Proposition 2, we deduce that there exists a constant M > 0, depending on the total variation of the flux of the initial datum, such that W 1 (t) + W 2 (t) + W P T (t) ≤ M, (24) for a.e. t > 0. Since (H-1) holds, then inequality (24) implies that the functional Tot.Var. ((ρ ε ,q ε )(t, ·)) is uniformly bounded for a.e. t > 0. Hence, at least by a subsequence, there is a function (ρ,q), which is a solution to (16) in the sense of Definition 3.1. This permits to conclude.