The Cauchy problem at a node with buffer

We consider the Lighthill-Whitham-Richards traffic flow model 
 on a network composed by an arbitrary number of incoming and 
 outgoing arcs connected together by a node with a buffer. 
 Similar to [15], 
 we define the solution to the Riemann problem at the node 
 and we prove existence 
 and well posedness of solutions to the Cauchy problem, 
 by using the wave-front tracking technique and the generalized tangent 
 vectors.


Introduction
Fluid dynamic models were developed in the literature in order to describe the macroscopic evolution of vehicular traffic in roads and in networks. In the network setting, different kinds of solutions at the intersections were recently proposed; see [6,7,8,9,14,15,16,17,20] and the references therein. The interest in this field was also motivated by other applications: data networks [8], supply chains [13], air traffic management [22], gas pipelines [1].
In this paper we consider the scalar Lighthill-Whitham-Richards model (see [19,21]) on a network composed by a single junction with a buffer with finite size and capacity. Nodes with buffers have been introduced in the case of supply chains in [14] and also for car traffic in [12,15]. These kinds of intersections take into account some dynamics inside the junction, described by ordinary differential equations depending on the difference between incoming and outgoing fluxes.
In the following sections, we prove existence and well posedness of solutions at the node with buffer and with an arbitrary number of incoming and outgoing roads. The results are obtained by means of the wave-front tracking method [3,18] and on the generalized tangent vectors [2,5]. In our case, the wave-front tracking method consists in producing piecewise constants approximate solutions both for the density of cars and for the load of the buffer and in proving uniform estimates for the approximate solutions in order to obtain compactness and so existence of solutions. Instead, the Lipschitz continuous dependence of the solution with respect to the initial condition is proved by viewing the vector space L 1 as a Finsler manifold and by considering the evolution in time of generalized tangent vectors along wave-front tracking approximate solutions. We remark that the results contained in [14] do not apply in our situation, while the papers [12,15] describe only special cases of Riemann problems.
The paper is organized as follows. Section 2 contains some preliminary notations and definitions, while Section 3 describes in details the solution of Riemann problems at the node. Sections 4 and 5 deal respectively with the existence of solution and with the continuous dependence of the solution with respect to the initial condition. Finally, we recall in the appendix, for reader's convenience, some technical results of [11].

Basic Definitions and Notations
Consider a node J with n incoming arcs I 1 , . . . , I n and m outgoing arcs I n+1 , . . . , I n+m . We model each incoming arc I i (i ∈ {1, . . . , n}) of the node with the real interval I i =] − ∞, 0]. Similarly, we model each outgoing arc I j (j ∈ {n + 1, . . . , n + m}) of the node with the real interval I j = [0, +∞[. On each arc I l (l ∈ {1, . . . , n + m}), we consider the partial differential equation where ρ l = ρ l (t, x) ∈ [0, ρ max ], is the density of cars, v l = v l (ρ l ) is the mean velocity of cars and f (ρ l ) = v l (ρ l ) ρ l is the flux. Moreover the real valued function r(t) ∈ [0, r max ] denotes the total number of cars in the buffer inside the node J at time t.
We make the following assumptions on the flux f : The definitions of entropic solutions on arcs and weak solutions at the node are as follows.

The Riemann Problem with buffer
Consider a node J with a buffer, whose demand and supply are equal to a constant µ ∈ ]0, max{n, m}f (σ)[. Fix ρ 1,0 , . . . , ρ n+m,0 ∈ [0, 1], r 0 ∈ [0, r max ] and consider the Riemann problem at J f (ρ j (t, 0+)), A solution to the Riemann problem at J is defined in the following way. (7) is a weak solution at J, in the sense of Definition 2, such that ρ l (0, x) = ρ l,0 for every l ∈ {1, . . . , n + m} and for a.e. x ∈ I l and such that r(0) = r 0 .

Definition 3 A solution to the Riemann problem
We introduce the concept of Riemann solver at J.

Definition 4 A Riemann solver RS is a function
satisfying the following 1. for every i ∈ {1, . . . , n}, the classical Riemann problem is solved with waves with negative speed; 2. for every j ∈ {n + 1, . . . , n + m}, the classical Riemann problem is solved with waves with positive speed.
Introduce the following sets 2. for every j ∈ {n + 1, . . . , n + m} define 4. for every s ∈ 0, n+m j=n+1 max O j define In [15], the authors proposed to solve the Riemann problem (7) in the following way.
r(t) = r max r(t) = 0 Figure 1: The solution to the Riemann problem (7): the case Γ inc > Γ out on the left, the case Γ inc < Γ out on the right.
For future use, we need some additional definitions.

Definition 6
We say that a datum ρ i ∈ [0, 1] in an incoming arc is a good datum if ρ i ∈ [σ, 1] and it is a bad datum otherwise. We say that a datum ρ j ∈ [0, 1] in an outgoing arc is a good datum if ρ i ∈ [0, σ] and it is a bad datum otherwise. 7

Wave-front tracking
Since solutions to Riemann problems are given, we are able to construct piecewise constant approximations via the wave-front tracking algorithm; see [3] for the general theory and [10] in the case of networks. Definition 7 Given ε > 0, we say that the maps ρ ε = (ρ 1,ε , . . . , ρ n+m,ε ) and r ε are an ε-approximate wave-front tracking solution to (18) if the following conditions hold.

For every
6. For every l ∈ {1, . . . , n + m}, For every l ∈ {1, . . . , n + m}, consider a sequence ρ 0,l,ν of piecewise constant functions defined on I l such that ρ 0,l,ν has a finite number of discontinuities and lim ν→+∞ ρ 0,l,ν = ρ 0,l in L 1 loc (I l ; [0, 1]). For every ν ∈ N \ {0}, we apply the following procedure. At time t = 0, we solve the Riemann problem at J (according to RS r 0 ) and all Riemann problems in each arc. We approximate every rarefaction wave with a rarefaction fan, formed by rarefaction shocks of strength less than 1 ν travelling with the Rankine-Hugoniot speed. Moreover, if σ is in the range of a rarefaction shock, then its speed is zero. We repeat the previous construction at every time at which interactions between waves or of waves with J happen and at the times when the buffer becomes empty or full.
Remark 3 By slightly modifying the speed of waves, we may assume that, at every positive time t, at most one interaction happens. Moreover, at every interaction timet, exactly one of the following possibilities is verified.
1. Two waves interact in an arc.

A wave reaches the node J and
3. Some waves exit the node J, i.e.
Remark 4 For interactions in arcs, we split rarefaction waves into rarefaction fans just at time t = 0. At the node J, instead, we allow the formation of rarefaction fans at every positive time.
Let us introduce the notions of generation order for waves, of big shocks and of waves with increasing or decreasing flux. We need these definitions in the proof of existence of a wave-front tracking approximate solution and of an uniform bound for the total variation of the flux.
Definition 8 A wave of ρ ε , generated at time t = 0, is said an original wave or a wave with generation order 1.
If a wave with generation order k ≥ 1 interacts with J, then the produced waves are said of generation k + 1.
If a wave with generation order k ≥ 1 interacts in an arc with a wave with generation order k ′ ≥ 1, then the produced wave is said of generation min{k, k ′ }.
If a wave exits the node J at timet > 0 and then it has generation order 2 if in the time interval [0,t[ no wave interacts with J, otherwise it has generation order k + 1, where k is the generation order of a wave, which interacts with J at timet <t, and in the time interval ]t,t[ no wave interacts with J.

Definition 9
We say that a wave (ρ l , ρ r ) in an arc is a big shock if ρ l < σ < ρ r .

Definition 10
We say that a wave (ρ l , ρ r ) interacting with J from an incoming arc has decreasing flux (resp. increasing flux We say that a wave (ρ l , ρ r ) interacting with J from an outgoing arc has

Bound on the total flux variation
Fix a wave-front tracking approximate solution for the Cauchy problem (18). We prove in Corollary 1, that the total variation of the flux is uniformly bounded by a constant which depends on the initial data. We need some preliminary results.
Then exactly one of the following possibilities holds.
1. If r ε (t−) = 0, then some waves are generated at J at timet only in the outgoing arcs.
2. If r ε (t−) = r max , then some waves are generated at J at timet only in the incoming arcs.
Proof. Define by Γ 1± inc , Γ 1± out , Γ ± inc and Γ ± out the values, att− andt+, of the quantities introduced in Section 3. Finally with Γ 1 inc , Γ 1 out , Γ inc and Γ out we denote the values at timet of the quantities introduced in Section 3. By Remark 3, at timet no wave interacts with J. Hypothesis (19) implies that either r ε (t) = 0 or r ε (t) = r max . If We have also Γ + inc = Γ inc = Γ − inc = Γ + out and so Q(t+) = 0. Moreover, no waves exit from the incoming arcs, while by (13) and the fact that r ε (t) = 0 for t in a left neighborhood oft, Γ − out > Γ out and so some waves exit from the outgoing arcs. By Lemma 4 in [11], we deduce that all the waves generated at timet have decreasing flux. This implies that We have also Γ + inc = Γ − out = Γ out = Γ + out and so Q(t+) = 0. Moreover, no waves exit from the outgoing arcs, while by (12) and the fact that r ε (t) = 0 for t in a left neighborhood of t, Γ − inc > Γ inc and so some waves are generated in the incoming arcs. By Lemma 4 in [11], we deduce that all the waves generated at timet have decreasing flux. This implies that This concludes the proof. 2 Lemma 3 Assume that a wave (ρ l , ρ r ) interacts with J at timet and suppose that r ε (t) = 0. Then Υ(t+) = Υ(t−) and TV f (t+) ≤ TV f (t−).
In this case Γ 1+ inc < µ and Γ + inc = Γ 1+ inc . Therefore no wave is generated in I 1 and the waves generated in the other incoming arcs have increasing flux. Moreover Γ + inc = Γ + out < Γ − out . By [11,Lemma 4], the waves generated in the outgoing arcs have decreasing flux and so and the conclusion follows, since Q(t+) = 0.

Γ
inc and Γ − out = Γ + out . Therefore no waves are generated in the outgoing arcs and in I 1 . By [11,Lemma 5], the waves produced in the other incoming arcs have increasing fluxes if f (ρ l ) < f (ρ r ), and decreasing fluxes if f (ρ l ) > f (ρ r ). Thus and we conclude, since Q(t+) = 0.
In this case we have that Γ − inc < µ and so ρ − i ≤ σ for every i ∈ {1, . . . , n}. Since the wave (ρ l , ρ r ) has positive speed, then ρ − 1 = ρ r < ρ l ≤ σ. Therefore, in the incoming arcs, either no waves are produced (in the case Γ 1+ inc ≤ µ) or waves with decreasing flux are generated (in the case Γ 1+ inc > µ). In the outgoing arcs, by (13) we easily deduce that Γ + out ≥ Γ − out , and so either no waves are created or waves with increasing flux are generated, see [11,Lemma 4].
Finally, suppose that the wave (ρ l , ρ r ) interacts with J from an outgoing arc, say I n+1 , and so ρ r ≥ σ and ρ l = ρ − n+1 . In this case Γ − inc = Γ + inc and so no waves are produced in the incoming arcs. There are three different possibilities.
1. Γ + out < Γ − out . In this case Γ + out = Γ 1+ out < Γ + inc and so no wave is generated in I n+1 , while in the other outgoing arcs at most m − 1 waves are generated and they have increasing flux. Therefore and so Υ(t+) = Υ(t−).
2. Γ + out = Γ − out . In this case Γ + out = Γ − out = Γ + inc and no wave is generated in I n+1 . In the other outgoing arcs, at most m − 1 waves are generated.
The proof is similar to that of Lemma 3 and so we omit it.
Proof. We denote by (ρ − 1 , . . . , ρ − n+m ) and by (ρ + 1 , . . . , ρ + n+m ) the states at J respectively before and after the interaction. Define also by Γ 1± inc , Γ 1± out , Γ ± inc and Γ ± out the values, att− andt+, of the quantities introduced in Section 3. Since 0 < r ε (t) < r max in a left neighborhood oft, we have that Assume that the wave (ρ l , ρ r ) interacts with J from an incoming arc; say I 1 . Thus ρ l ≤ σ and ρ r = ρ − 1 . Moreover Γ + out = Γ − out and so no wave is produced in the outgoing arcs. We have three possibilities.
In this case at most n waves are generated in the incoming arcs. If f (ρ l ) < f (ρ r ), then ρ + 1 = ρ l and so no wave is produced in I 1 , while the waves generated in the other incoming arcs have increasing flux, by [11,Lemma 5]. If f (ρ l ) > f (ρ r ), then f (ρ r ) ≤ f (ρ + 1 ) ≤ f (ρ l ) and the waves generated in I 2 , . . . , I n have decreasing flux, by [11,Lemma 5]. Thus by the previous considerations. Moreover Q(t−) = Q(t+) and so we conclude that Υ(t−) ≥ Υ(t+).
The case of the wave (ρ l , ρ r ) interacting with J from an outgoing arc is similar to the previous one. 2 Lemmas 2-5 imply that the functional Υ is decreasing, as stated in the following Proposition.
Proposition 1 For a.e. t > 0, we have that Proof. The functional Υ is piecewise constant in time and it can vary only when two waves interact inside an arc or when a wave hits or exits from the node. If two waves interact in an arc, then TV f is non-increasing and Q remains constant; hence Υ is non-increasing. Consider therefore the case of a wave interacting or exiting from the node at timet. For simplicity, we denote by Γ 1± inc , Γ 1± out , Γ ± inc and Γ ± out the values, at t− andt+, of the quantities introduced in Section 3. At the node, we have the following two cases.
• A wave (ρ l , ρ r ) hits the node at a certain timet. We have three different possibilities.
• A wave exits the node at a certain timet. In this case Lemma 2 states that Υ(t+) = Υ(t−).
The proof is so finished. Corollary 1 For every t > 0, we have that Proof. By Proposition 1, we deduce that for every t > 0. The conclusion follows by the fact that 0 ≤ Q(t) ≤ (n + m)f (σ) for every t ≥ 0.

Existence of a wave-front tracking solution
In this subsection, we prove the existence of a wave-front tracking approximate solution. We have the following proposition, whose proof is very similar to that of [11,Proposition 10]. Here we give the proof for completeness.
Proposition 2 For every ν ∈ N \ {0} the construction in Subsection 4.1 can be done for every positive time, producing an 1 ν -approximate wave-front tracking solution to (18) with respect to the Riemann solver described in Definition 4.
where Kν = 2(n + m)ν. This bound is due to the fact that each wave with generation order k can interact with J and produce at mostν waves with generation order k + 1 in each arc (in the case of rarefactions) and the same can happen at a second time, when the function r ε reaches 0 or r max . Now, there exists 0 < η < T such that no wave with generation order 1 interacts with J in the time interval (T − η, T ). Equation (22) implies also that in (T − η, T ) there is an infinite number of interactions of waves with J. Since waves of generation order 1 do not interact in (T − η, T ), the only possibility is that a wave with generation order k ≥ 2 comes back to J producing waves of order k + 1, some of which come back to J producing waves of order k + 2 and so on. Moreover, by Lemma 4.3.7 of [10] (see the Appendix), if a wave of generation order k ≥ 2 interacts with J from an arc in (T − η, T ), then, after the interaction, the datum in that arc is bad, since the wave can not interact with waves of generation order 1 and come back to J. In an arc a bad datum at J can change only in the following cases: 1. an original wave interacts with J from the arc; 2. a wave, which is a big shock, is originated at J on the arc and the new datum at J is good.
Obviously, in the time interval (T −η, T ) the first possibility can not happen; so only the second possibility may happen. Assume that there exist t 1 , t 2 ∈ (T − η, T ) with t 1 < t 2 such that a big shock is originated at J at time t 1 in an arc and comes back to J at time t 2 . In this arc, the datum before t 1 is bad, since a big shock is originated at time t 1 . Moreover the big shock comes back to J at time t 2 , and so an original wave cannot interact with the big shock; hence the bad datum of the big shock does not change. Therefore, in that arc after the time t 2 , the datum is bad and is the same as the datum before t 1 . Thus every arc I l may take only a precise bad valueρ l , otherwise good values. The key point is that, at every time t ∈ (T − η, T ), there are finitely many possible combinations of bad data at the node J (obtained choosing the arcs which present a bad datum at J, the precise value being fixed). Since the Riemann solvers RS rε(t) are indeed at most three (RS 0 , RS rmax and RSr withr ∈]0, r max [ arbitrary) and since each of them satisfies the property (P1) of [11] (i.e. the image of a Riemann solver depends only on bad data, for a proof see [11,Section 4.2]) we deduce that, for t ∈ (T − η, T ), ρν(t) at J may take only a finite number of values, thus waves produced by J have a finite set of possible velocities. Denote with G the set of all l ∈ {1, . . . , n+m} such that ρν ,l (t, 0) is a good datum for every time t in a left neighborhood of T . Considerl ∈ G. We claim that there exists a constant Cl > 0 such that Nl ,ν (t) ≤ Cl for every time t in a left neighborhood of T . Indeed, the number of different states, which can be produced at J, is finite by the previous considerations. Since all states are good, there is a minimal size of a flux jump along a discontinuity. Then the total number of discontinuities is necessary bounded by Corollary 1. Consider nowl ∈ {1, . . . , n + m} \ G. If ρν ,l (t, 0) is a bad datum for every time t in a left neighborhood of T , then clearly Nl ,ν (t) is uniformly bounded in the same time interval. The other case is that a big shock is originated in the arc Il and comes back to J infinitely many times. We claim that there exists a constant Cl > 0 such that Nl ,ν (t) ≤ Cl for every time t ∈ [t 1 ,t 2 ], wheret 1 andt 2 are the times, at which a big shock respectively is originated at J in Il and comes back to J. In fact, in the time interval ]t 1 ,t 2 [, the datum ρν ,l (t, 0) is good and the number of possible different states between J and the big shock is finite. Therefore, as before, if the number of discontinuity can not be bounded by a constant, then also the total variation of the flux can not be bounded and this is not true, by Corollary 1.
This concludes the proof by contradiction.

Existence of a solution
This part deals with the proof of Theorem 1.
Concerning r ε , Ascoli-Arzelà Theorem guarantees the uniform convergence of a subsequence r ε k → r. Moreover, Dunford-Pettis Theorem implies the weak compactness of {r ′ ε } ε in L 1 ([0, T ]), thus, up to a subsequence, r ′ ε k ⇀ s weakly in L 1 ([0, T ]) and r ′ = s in the weak sense. Thus, passing to the limit in the wave-front tracking approximations, we obtain that (ρ, r) satisfies points 3. and 4. of Theorem 1. 2

Dependence of solutions on initial data
In this section we prove that, for every type of nodes, the solution to (18) depends in a Lipschitz continuous way with respect to the initial condition.
We use the technique of generalized tangent vectors, introduced in [4,5] for hyperbolic systems of conservation laws. A complete description, in the case of scalar conservation laws on networks, is in [11]. Here we just analyze the estimates on the shifts of waves along wave-front tracking approximate solutions at the node. We recall the definition of shift of wave.
Definition 11 Fix ξ ∈ R and a wave (ρ l , ρ r ) of an ε-approximate wave-front tracking solution to (18). We say that ξ forms a shift for the wave (ρ l , ρ r ) if we consider the same ε-approximate wave-front tracking solution, except for the position of the wave (ρ l , ρ r ), which is translated by the quantity ξ in the x-direction.
The proof of the continuous dependence is based on the following lemmas.
If ξ − k is a shift in the wave defined by the statesρ k , ρ − k , then the function r ε becomes .
Lett > t 1 + h be the first time at which r h ε (t) = 0 or r h ε (t) = r max . Thus the waves (ρ + l ,ρ + l ) are shifted in time byt − t 2 = v 2 −v 1 v 2 h. This permits to conclude. Proof. First consider variations in the ρ component of the initial condition. As in the proof of Theorem 17 of [11], we can restrict the study to the evolution of shifts. Fix a timet > 0; we have the following possibilities.
a) No interaction of waves takes place in any arc att and no wave interacts with J. In this case the shifts are constant. b) Two waves interact att on an arc and no other interaction takes place. In this case the norms of the tangent vectors are decreasing by Lemma 2.7.2 of [10].
c) A wave interacts with J at a timet from the arc I k and no other interaction takes place. Denote byρ k = ρ − k the other side of the interacting wave. Using Lemma 6 and its notations, we deduce by Lemmas 3, 4, 5.
d) Waves exit J at a timet > 0 and no other interaction takes place. Definet ∈ [0,t[ in the following way:t = 0 if no interaction at J happens in the time interval (0,t), otherwiset is the time at which a wave reaches J and no other interaction at J happens in the time interval (t,t). Ift = 0 and since no variation in r 0 occurs, then no shift appears.