A Riemann solver at a junction compatible with a homogenization limit
AA Riemann solver at a junction compatible witha homogenization limit
M. Garavello ∗ F. Marcellini † November 6, 2018
Abstract
We consider a junction regulated by a traffic lights, with n incom-ing roads and only one outgoing road. On each road the Phase Transi-tion traffic model, proposed in [6], describes the evolution of car traffic.Such model is an extension of the classic Lighthill-Whitham-Richards one,obtained by assuming that different drivers may have different maximalspeed.By sending to infinity the number of cycles of the traffic lights, weobtain a justification of the Riemann solver introduced in [9] and in par-ticular of the rule for determining the maximal speed in the outgoingroad. Key words and phrases:
Phase Transition Model, Hyperbolic Systems ofConservation Laws, Continuum Traffic Models, Homogenization Limit.
This paper deals with the Phase Transition traffic model, proposed by Colombo,Marcellini, and Rascle in [6], at a junction with n ≥ × ∗ Dipartimento di Matematica e Applicazioni, Universit`a di Milano Bicocca, Via R. Cozzi55, 20125 Milano, Italy. E-mail: [email protected] † Dipartimento di Matematica e Applicazioni, Universit`a di Milano Bicocca, Via R. Cozzi55, 20125 Milano, Italy. E-mail: [email protected] a r X i v : . [ m a t h . A P ] D ec n providing a concept of solution at nodes. A possible way to do this is toconstruct a Riemann solver at nodes, i.e. a function which associates to eachRiemann problem at the node a solution. A reasonable Riemann solver hasto satisfy the mass conservation, a consistency condition; it should producewaves with negative speed in the incoming edges and with positive speed in theoutgoing ones. A Riemann solver satisfying such properties is proposed in [9]. Inparticular, it prescribes that the maximal speed in the outgoing road is a convexcombination of the maximal speed in the incoming arcs. Similar conditions arealso present in [11, 12, 13].In this paper we are going to investigate the delicate issue of how the maximalspeed changes through the junction. To this aim, we consider a single junctionregulated by a time-periodic traffic lights. At each time the green light appliesonly at one incoming road. Vehicles, in the remaining incoming roads, arethen stopped, waiting for their green light. With a limit-average procedure,we are able to find the relation between the incoming maximal speeds and theoutgoing one. In this way, the maximal outgoing speed turns out to be a convexcombination of the n incoming ones and it satisfies the corresponding conditionprescribed by the Riemann solver in [9].The paper is organized as follow. In the next section we recall the 2-PhasesTraffic Model introduced in [6] and the solution to the classical Riemann problemalong a single road of infinite length. In Section 3 we consider a time periodictraffic lights regulating the intersection and we study the solution in the outgoingroad as the time period of the traffic lights tends to 0. More precisely, inSubsection 3.1 we describe in details the solution in the simple situation with n = 2 incoming roads and, finally, in Subsection 3.2 we generalize the previousstudy to the case of n ≥ The Phase Transition model, introduced in [6], is given by: (cid:26) ∂ t ρ + ∂ x ( ρ v ( ρ, η )) = 0 ∂ t η + ∂ x ( η v ( ρ, η )) = 0 with v ( ρ, η ) = min (cid:26) V max , ηρ ψ ( ρ ) (cid:27) , (2.1)where t denotes the time, x the space, ρ ∈ [0 , R ] is the traffic density, η is ageneralized momentum, v ∈ [0 , V max ] is the speed of cars, and V max is a uniformbound of the cars’ speed.It is obtained as an extension of the Lighthill-Whitham-Richards model [15,18], by assuming that different drivers have different maximal speed, denotedby the quantity w = η/ρ ∈ [ ˇ w, ˆ w ]. It is characterized by two phases, the freeone and congested one, which are described by the sets F = { ( ρ, η ) ∈ [0 , R ] × [0 , ˆ wR ] : ˇ wρ ≤ η ≤ ˆ wρ, v ( ρ, η ) = V max } , (2.2) C = (cid:26) ( ρ, η ) ∈ [0 , R ] × [0 , ˆ wR ] : ˇ wρ ≤ η ≤ ˆ wρ, v ( ρ, η ) = ηρ ψ ( ρ ) (cid:27) , (2.3)see Figure 1. As in [6, 9], we assume the following hypotheses.2 F ρRC ρF C Rρv
Figure 1: The free phase F and the congested phase C resulting from (2.1) inthe coordinates, from left to right, ( ρ, η ) and ( ρ, ρv ). Note that F and C areclosed sets and F ∩ C (cid:54) = ∅ . Note also that F is 1–dimensional in the ( ρ, ρv )plane, while it is 2–dimensional in the ( ρ, η ) coordinates. In the ( ρ, η ) plane,the curve η = V max ψ ( ρ ) ρ divides the two phases. (H-1) R, ˇ w, ˆ w, V max are positive constants, with V max < ˇ w < ˆ w . (H-2) ψ ∈ C ([0 , R ]; [0 , ψ (0) = 1, ψ ( R ) = 0, and, for every ρ ∈ (0 , R ), ψ (cid:48) ( ρ ) ≤ d dρ ( ρ ψ ( ρ )) ≤ (H-3) Waves of the first family in the congested phase C have negative speed.By (H-1) , (H-2) , and (H-3) , system (2.1) is strictly hyperbolic in C , see [6],and λ ( ρ, η ) = η ψ (cid:48) ( ρ ) + v ( ρ, η ) , λ ( ρ, η ) = v ( ρ, η ) ,r ( ρ, η ) = (cid:20) − ρ − η (cid:21) , r ( ρ, η ) = (cid:34) η (cid:16) ρ − ψ (cid:48) ( ρ ) ψ ( ρ ) (cid:17) (cid:35) , ∇ λ · r = − d dρ [ ρ ψ ( ρ )] , ∇ λ · r = 0 , L ( ρ ; ρ o , η o ) = η o ρρ o , L ( ρ ; ρ o , η o ) = ρ v ( ρ o , η o ) ψ ( ρ ) , ρ o < R, where λ i and r i are respectively the eigenvalues and the right eigenvectors ofthe Jacobian matrix of the flux, and L i are the Lax curves. When ρ o = R , the2-Lax curve through ( ρ o , η o ) is given by the segment ρ = R , η ∈ [ R ˇ w, R ˆ w ].In view of the results of the next section, we recall the description of thesolutions of the Riemann problem for the model (2.1). First, we enumerate allthe possible waves for (2.1). • A Linear wave is a wave connecting two states in the free phase. It alwaystravels with speed V max . • A Phase Transition Wave is a wave connecting a left state ( ρ l , η l ) ∈ F with a right state ( ρ r , η r ) ∈ C satisfying η l ρ l = η r ρ r . It always travels withspeed given by the Rankine-Hugoniot condition. • A Wave of the First Family is a wave connecting a left state ( ρ l , η l ) ∈ C with a right state ( ρ r , η r ) ∈ C such that η l ρ l = η r ρ r . It is either a rarefactionwave or a shock wave. 3 A Wave of the Second Family is a wave connecting a left state ( ρ l , η l ) ∈ C with a right state ( ρ r , η r ) ∈ C such that v ( ρ l , η l ) = v ( ρ r , η r ). It alwaystravels with speed v ( ρ l , η l ). Under the assumptions (H-1) , (H-2) and (H-3) , for all states ( ρ l , η l ) and( ρ r , η r ) ∈ F ∪ C , the Riemann problem consisting of (2.1) with initial data ρ (0 , x ) = (cid:26) ρ l if x < ρ r if x > η (0 , x ) = (cid:26) η l if x < η r if x > ρ, η ) = ( ρ, η )( t, x ) constructed asfollows:(1) If ( ρ l , η l ) , ( ρ r , η r ) ∈ F , then the solution attains values in F and consists ofa linear wave separating ( ρ l , η l ) from ( ρ r , η r ).(2) If ( ρ l , η l ) , ( ρ r , η r ) ∈ C , then the solution attains values in C and consists of awave of the first family (shock or rarefaction) between ( ρ l , η l ) and a middlestate ( ρ m , η m ), followed by a wave of the second family between ( ρ m , η m )and ( ρ r , η r ). The middle state ( ρ m , η m ) belongs to C and is uniquely char-acterized by the two conditions η m ρ m = η l ρ l and v ( ρ m , η m ) = v ( ρ r , η r ).(3) If ( ρ l , η l ) ∈ C and ( ρ r , η r ) ∈ F , then the solution attains values in F ∪ C andconsists of a wave of the first family separating ( ρ l , η l ) from a middle state( ρ m , η m ) and by a linear wave separating ( ρ m , η m ) from ( ρ r , η r ). The middlestate ( ρ m , η m ) belongs to the intersection between F and C and is uniquelycharacterized by the two conditions η m ρ m = η r ρ r and v ( ρ m , η m ) = V max .(4) If ( ρ l , η l ) ∈ F and ( ρ r , η r ) ∈ C , then the solution attains values in F ∪ C and consists of a phase transition wave between ( ρ l , η l ) and a middle state( ρ m , η m ), followed by a wave of the second family between ( ρ m , η m ) and( ρ r , η r ). The middle state ( ρ m , η m ) is in C and is uniquely characterized bythe two conditions η m ρ m = η l ρ l and v ( ρ m , η m ) = v ( ρ r , η r ). Fix a junction with n incoming roads and a single outgoing road. In [9], it isintroduced a Riemann solver at the junction, which conserves the mass n (cid:88) i =1 ρ i v ( ρ i , η i ) = ρ n +1 v ( ρ n +1 , η n +1 ) , (3.5)and prescribes that the maximal speed in the outgoing road is given by w n +1 = n (cid:88) i =1 σ i ρ i v ( ρ i , η i ) w in (cid:88) i =1 σ i ρ i v ( ρ i , η i ) , (3.6)4or suitable coefficients σ > , · · · , σ n >
0, satisfying σ + · · · + σ n = 1.In this section we provide a justification for the rule (3.6). To this aim, fix apositive time T > (cid:96) ∈ N \ { } cycles in the timeinterval [0 , T ] and that each cycle of length T(cid:96) is divided into n subintervals oflength τ (cid:96) , · · · , τ (cid:96)n , which represent respectively the duration of the green lightfor the corresponding incoming road. The first cycle (cid:2) , T(cid:96) (cid:2) is thus composed by (cid:20) , T(cid:96) (cid:20) = (cid:2) , τ (cid:96) (cid:1) (cid:91) (cid:2) τ (cid:96) , τ (cid:96) + τ (cid:96) (cid:1) (cid:91) · · · (cid:91) (cid:2) τ (cid:96) + · · · + τ (cid:96)n − , τ (cid:96) + · · · + τ (cid:96)n (cid:1) , where (cid:2) , τ (cid:96) (cid:1) is the time interval of the green for the road I and so on.Denote, for every i ∈ { , . . . , n − } , σ i = τ (cid:96)i τ (cid:96)n , (3.7)which we suppose that it does not depend on (cid:96) . Thus the constant σ i is theratio between the green time interval for roads I i and I n . For simplicity we put σ n = 1. In this subsection we treat only the special junction with n = 2 incoming roads(namely I and I ) and a single outgoing road I . We assume that assump-tions (H-1) , (H-2) , and (H-3) hold. As introduced in (3.7), the positiveconstants σ = τ (cid:96) τ (cid:96) , σ = 1do not depend on the number of cycles (cid:96) . This means that the ratio betweenthe green and red times is constant in each incoming road.Given, for every i ∈ { , , } , initial conditions (¯ ρ i , ¯ η i ) ∈ F ∪ C , we denotewith ( ρ (cid:96),i ( t, x ) , η (cid:96),i ( t, x )) ( i ∈ { , , } ) the solution to the Riemann problem,the junction being governed by the traffic lights with (cid:96) cycles.Introduce the following notation. With (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) we call thepoints in the congested region C satisfying η (cid:93) ρ (cid:93) = ¯ w , η (cid:93) ρ (cid:93) = ¯ w , v (cid:16) ρ (cid:93) , η (cid:93) (cid:17) = v (cid:16) ρ (cid:93) , η (cid:93) (cid:17) = v (¯ ρ , ¯ η ) . Moreover with (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and (cid:0) ρ (cid:91) , η (cid:91) (cid:1) we call the points in the intersection betweenthe free and congested region F ∩ C satisfying η (cid:91) ρ (cid:91) = ¯ w , η (cid:91) ρ (cid:91) = ¯ w , v (cid:16) ρ (cid:91) , η (cid:91) (cid:17) = v (cid:16) ρ (cid:91) , η (cid:91) (cid:17) = V max . Note that the points (cid:16) ρ (cid:93)i , η (cid:93)i (cid:17) and (cid:0) ρ (cid:91)i , η (cid:91)i (cid:1) , i = 1 ,
2, are uniquely defined; seeFigure 2.The next lemmas describe the solution at the junction with the traffic lightsfor the possible different situations that may happen.5 F ρRC (¯ ρ , ¯ η ρ , ¯ η
2) (¯ ρ , ¯ η (cid:16) ρ(cid:93) , η(cid:93) (cid:17)(cid:16) ρ(cid:93) , η(cid:93) (cid:17) η F ρRC (¯ ρ , ¯ η ρ , ¯ η (cid:16) ρ(cid:91) , η(cid:91) (cid:17)(cid:16) ρ(cid:91) , η(cid:91) (cid:17) Figure 2: Left, an example of representation for the points (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) ,right an example of representation for the points (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and (cid:0) ρ (cid:91) , η (cid:91) (cid:1) . Lemma 3.1
Assume that the initial conditions belong to the congested phase,i.e., for every i ∈ { , , } , (¯ ρ i , ¯ η i ) ∈ C . Then the solution in the outgoing road I is ( ρ (cid:96), , η (cid:96), ) ( t, x ) = (¯ ρ , ¯ η ) , if < t < T, x > v (¯ ρ , ¯ η ) t, (cid:16) ρ (cid:93) , η (cid:93) (cid:17) , if ( t, x ) ∈ A (cid:96) , (cid:16) ρ (cid:93) , η (cid:93) (cid:17) , if ( t, x ) ∈ A (cid:96) , (3.8) where A (cid:96) = (cid:96) − (cid:91) i =0 (cid:26) ( t, x ) : 0 < t < T, x > t − τ (cid:96) − i T(cid:96) < xv (¯ ρ , ¯ η ) < t − i T(cid:96) (cid:27) (3.9) and A (cid:96) = (cid:96) (cid:91) i =1 (cid:26) ( t, x ) : 0 < t < T, x > t − i T(cid:96) < xv (¯ ρ , ¯ η ) < t − i T(cid:96) + τ (cid:96) (cid:27) . (3.10) Proof.
In the time interval [0 , τ (cid:96) [ the traffic lights is green in the first incomingroad; this permits to study the Riemann problem as a classical one consideringa unique road given by the union of the I and I : the classic Riemann problembetween (¯ ρ , ¯ η ) and (¯ ρ , ¯ η ) produces a first family wave between (¯ ρ , ¯ η ) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and a second family wave between (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and (¯ ρ , ¯ η ), see Figure 3.In I , instead, the flow at the junction is equal to zero and so the trace of thesolution is ( R, R ¯ w ). The solution in the road I is given by a shock wave ofthe first family connecting (¯ ρ , ¯ η ) to ( R, R ¯ w ); see Figure 4.At time t = τ (cid:96) , the traffic lights becomes red for the road I and green for I . This situation remains constant in the whole time interval [ τ (cid:96) , τ (cid:96) + τ (cid:96) [; thispermits to study the Riemann problem as a classical one considering a uniqueroad given by the union of the I and I ; see Figure 5. We have to solve theRiemann problem between ( R, R ¯ w ) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) . The solution is given bya rarefaction curve of the first family between ( R, R ¯ w ) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) followedby a second family wave between (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) , see Figure 5. In I ,instead, the flow at the junction is equal to zero and so the trace of the solution6 F ρRC (¯ ρ , ¯ η ρ , ¯ η (cid:16) ρ(cid:93) , η(cid:93) (cid:17) (¯ ρ , ¯ η t x (¯ ρ , ¯ η
1) 0 (¯ ρ , ¯ η ρ , ¯ η I I (cid:16) ρ(cid:93) , η(cid:93) (cid:17) τ(cid:96) Figure 3: The situation of Lemma 3.1 in the time interval [0 , τ (cid:96) [ for the firstincoming road and the outgoing road in the coordinates, from left to right, ( ρ, η )and ( x, t ). η F ρRC ρ , ¯ η
2) (
R, R ¯ w tx ρ , ¯ η ρ , ¯ η
2) (
R, R ¯ w I τ(cid:96) Figure 4: The situation of Lemma 3.1 in the time interval [0 , τ (cid:96) [ for the secondincoming road in the coordinates, from left to right, ( ρ, η ) and ( x, t ). η F ρRC R, R ¯ w (cid:16) ρ(cid:93) , η(cid:93) (cid:17) (cid:16) ρ(cid:93) , η(cid:93) (cid:17) (cid:16) ρ(cid:93) , η(cid:93) (cid:17) t xτ(cid:96) ρ , ¯ η τ(cid:96) τ(cid:96) (cid:16) ρ(cid:93) , η(cid:93) (cid:17) ( R, R ¯ w R, R ¯ w I I ρ , ¯ η (cid:16) ρ(cid:93) , η(cid:93) (cid:17) Figure 5: The situation of Lemma 3.1 in the time interval [ τ (cid:96) , τ (cid:96) + τ (cid:96) [ for thesecond incoming road and the outgoing road in the coordinates, from left toright, ( ρ, η ) and ( x, t ).is ( R, R ¯ w ). More precisely a shock wave of the first family starts from thepoint (cid:0) τ (cid:96) , (cid:1) connecting the states (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and ( R, R ¯ w ); see Figure 6.Similarly, in the time interval [ τ (cid:96) + τ (cid:96) , τ (cid:96) + τ (cid:96) [, the traffic light is green7 F ρRC
R, R ¯ w (cid:16) ρ(cid:93) , η(cid:93) (cid:17) tx (¯ ρ , ¯ η ρ , ¯ η
1) (
R, R ¯ w I (cid:16) ρ(cid:93) , η(cid:93) (cid:17)(cid:16) ρ(cid:93) , η(cid:93) (cid:17) τ(cid:96) τ(cid:96) τ(cid:96) Figure 6: The situation of Lemma 3.1 in the time interval [ τ (cid:96) , τ (cid:96) + τ (cid:96) [ for thefirst incoming road in the coordinates, from left to right, ( ρ, η ) and ( x, t ). η F ρRC R, R ¯ w (cid:16) ρ(cid:93) , η(cid:93) (cid:17)(cid:16) ρ(cid:93) , η(cid:93) (cid:17) t x ( R, R ¯ w I I ρ , ¯ η (cid:16) ρ(cid:93) , η(cid:93) (cid:17) (cid:16) ρ(cid:93) , η(cid:93) (cid:17) τ(cid:96) τ(cid:96) τ(cid:96) τ(cid:96) (cid:16) ρ(cid:93) , η(cid:93) (cid:17)(cid:16) ρ(cid:93) , η(cid:93) (cid:17) Figure 7: The situation of Lemma 3.1 in the time interval [ τ (cid:96) + τ (cid:96) , τ (cid:96) + τ (cid:96) [ forthe first incoming road and the outgoing road in the coordinates, from left toright, ( ρ, η ) and ( x, t ).for road I and red for I ; so we need to consider a Riemann problem between( R, R ¯ w ) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) , see Figure 7. The solution consists in a rarefaction curveof the first family between ( R, R ¯ w ) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) followed by a second familywave between (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) . The situation of I is analogous to thatrepresented in Figure 4. More precisely at the point (cid:0) τ (cid:96) + τ (cid:96) , (cid:1) a shock wavewith negative speed is generated and it connects (cid:16) ρ (cid:93) , η (cid:93) (cid:17) with ( R, R ¯ w ).We proceed in the same way until we arrive at time t = T . In this way wededuce that the solution in I is given by (3.8); see Figure 8. (cid:3) Lemma 3.2
Assume that the initial conditions satisfy (¯ ρ , ¯ η ) ∈ F, (¯ ρ , ¯ η ) ∈ C, (¯ ρ , ¯ η ) ∈ C. Then either the solution in the outgoing road I is given by (3.8) or has astructure similar to (3.8) except for a set, whose Lebesgue measure is boundedby a constant times (cid:96) . t I τ(cid:96) τ(cid:96) τ(cid:96) τ(cid:96) τ(cid:96) τ(cid:96) τ(cid:96) (cid:16) ρ(cid:93) , η(cid:93) (cid:17)(cid:16) ρ(cid:93) , η(cid:93) (cid:17)(cid:16) ρ(cid:93) , η(cid:93) (cid:17)(cid:16) ρ(cid:93) , η(cid:93) (cid:17)(cid:16) ρ(cid:93) , η(cid:93) (cid:17) (¯ ρ , ¯ η A(cid:96) A(cid:96) A(cid:96) A(cid:96) Figure 8: The situation of Lemma 3.1 in the time interval [0 , τ (cid:96) + 2 τ (cid:96) [ for theoutgoing road in the coordinates ( x, t ). The two states (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) ,separated by second family waves, alternate periodically. Proof.
We proceed in the same way as the previous Lemma 3.1. In the timeinterval [0 , τ (cid:96) [ the traffic lights is green in the first incoming road and we studythe Riemann problem as a classical one considering a unique road given bythe union of the I and I . We have two possible cases: the Riemann problembetween (¯ ρ , ¯ η ) and (¯ ρ , ¯ η ) produces a shock wave with negative speed between(¯ ρ , ¯ η ) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and a second family wave between (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and (¯ ρ , ¯ η ),or produces a shock wave with positive speed between (¯ ρ , ¯ η ) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and a second family wave between (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and (¯ ρ , ¯ η ), see Figure 9. In I the situation is the same as that of the previous Lemma 3.1: the flow at thejunction is equal to zero and the solution is given by a shock wave of the firstfamily connecting (¯ ρ , ¯ η ) to ( R, R ¯ w ), see Figure 4. • First case: a shock with negative speed.
In this case, from time τ (cid:96) to time t = T , the solution becomes as that described in the previous Lemma 3.1and it is given by (3.8); see Figure 8. • Second case: a shock with positive speed.
In the time interval [ τ (cid:96) , τ (cid:96) + τ (cid:96) [the traffic light is red for the road I and green for I and we study theRiemann problem as a classical one considering a unique road given by theunion of the I and I . We have to solve the Riemann problem between( R, R ¯ w ) and (¯ ρ , ¯ η ). The solution is given by a rarefaction curve ofthe first family between ( R, R ¯ w ) and (cid:0) ρ (cid:91) , η (cid:91) (cid:1) ∈ F ∩ C followed by alinear wave wave between (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and (¯ ρ , ¯ η ). At time t = ¯ t , the linearwave generated at the time t = τ (cid:96) between (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and (¯ ρ , ¯ η ) interactswith the shock with positive speed genearated at time t = 0 between(¯ ρ , ¯ η ) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) , see Figure 10. The intersection point is determined9 (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) ρF C Rρv (¯ ρ , ¯ η
1) (¯ ρ , ¯ η (cid:16) ρ(cid:93) , η(cid:93) (cid:17) (¯ ρ , ¯ η t x (¯ ρ , ¯ η
1) 0 (¯ ρ , ¯ η ρ , ¯ η I I (cid:16) ρ(cid:93) , η(cid:93) (cid:17) τ(cid:96) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) ρF C Rρv (¯ ρ , ¯ η
1) (¯ ρ , ¯ η (cid:16) ρ(cid:93) , η(cid:93) (cid:17) (¯ ρ , ¯ η t x (¯ ρ , ¯ η ρ , ¯ η ρ , ¯ η I I (cid:16) ρ(cid:93) , η(cid:93) (cid:17) τ(cid:96) Figure 9: The situation of Lemma 3.2 in the time interval [0 , τ (cid:96) [ for the firstincoming road and the outgoing road in the coordinates, from left to right,( ρ, ρv ) and ( x, t ). Above, the first case, a shock with negative speed; below, thesecond case, a shock with positive speed.by solving the system: (cid:26) x ( t ) = v s tx ( t ) = V max ( t − τ (cid:96) ) , (3.11)where v s = ¯ ρ V max − ρ (cid:93) v ( ρ (cid:93) ,η (cid:93) )¯ ρ − ρ (cid:93) . We denote the intersection point by(¯ t , ¯ x ) = (cid:18) V max τ (cid:96) V max − v s , v s V max τ (cid:96) V max − v s (cid:19) . (3.12)At this point we have to solve a Riemann problem between (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) that generates a first family wave between (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) ∈ C and a second family wave between (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) , see Figure 10.The first family wave between (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) could interact againat a time t = t with the linear wave generated at time t = τ (cid:96) when thetraffic lights is red for the road I and green for I , producing again a firstfamily wave and a second family wave. A first family wave with negativespeed could be produced at each interaction up to a time t = t ∗ , whenit is absorbed and the solution becomes as that described in the previousLemma 3.1 and it is given by (3.8) except for a set, whose Lebesguemeasure is bounded by a constant times (cid:96) . Indeed, if for example wesuppose that the first famiy wave is absorbed at time t ∗ = ¯ t and, denoting10 (cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) t x ρ , ¯ η I (cid:16) ρ(cid:93) , η(cid:93) (cid:17) τ(cid:96) ρ , ¯ η t = ¯ t t = ¯ t t = ¯ x (cid:16) ρ(cid:91) , η(cid:91) (cid:17)(cid:16) ρ(cid:93) , η(cid:93) (cid:17) Figure 10: The situation of Lemma 3.2 in the time interval [0 , ¯ t [ for the outgoingroad in the coordinates ( x, t ): the interaction between the shock with positivespeed genearated at time t = 0 between (¯ ρ , ¯ η ) and (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and the linearwave generated at the time t = τ (cid:96) between ( ρ m , η m ) and (¯ ρ , ¯ η ).with L the Lebesgue measure of a set, we estimate the area of the triangle A (cid:96) generated up to time t = ¯ t , see Figure 10. By posing ¯ t = K (cid:96) and¯ t = K ¯ x , we have: L ( A (cid:96) ) = 12 v s K (cid:96) (cid:18) K (cid:96) + K v s K (cid:96) (cid:19) = K (cid:96) , (3.13)for K , K , K positive costants. (cid:3) Lemma 3.3
Assume that the initial conditions satisfy (¯ ρ , ¯ η ) ∈ F, (¯ ρ , ¯ η ) ∈ C, (¯ ρ , ¯ η ) ∈ F. Then the solution in the outgoing road I is ( ρ (cid:96), , η (cid:96), ) ( t, x ) = (¯ ρ , ¯ η ) , if < t < T, x > V max t, (¯ ρ , ¯ η ) , if (cid:26) < t < T, x < V max t,x > max (cid:8) , V max (cid:0) t − τ (cid:96) (cid:1)(cid:9) , (cid:0) ρ (cid:91) , η (cid:91) (cid:1) , if ( t, x ) ∈ A (cid:96) , (cid:0) ρ (cid:91) , η (cid:91) (cid:1) , if ( t, x ) ∈ A (cid:96) , (3.14) where A (cid:96) = (cid:96) − (cid:91) i =0 (cid:26) ( t, x ) : 0 < t < T, x > t − τ (cid:96) − i T(cid:96) < xV max < t − i T(cid:96) (cid:27) (3.15) and A (cid:96) = (cid:96) (cid:91) i =1 (cid:26) ( t, x ) : 0 < t < T, x > t − i T(cid:96) < xV max < t − i T(cid:96) + τ (cid:96) (cid:27) ; (3.16)11 F ρRC (¯ ρ , ¯ η ρ , ¯ η
3) (¯ ρ , ¯ η t x (¯ ρ , ¯ η ρ , ¯ η ρ , ¯ η I I τ(cid:96) Figure 11: The situation of Lemma 3.3 in the time interval [0 , τ (cid:96) [ for the firstincoming road and the outgoing road in the coordinates, from left to right, ( ρ, η )and ( x, t ). η F ρRC ρ , ¯ η
2) (
R, R ¯ w tx ρ , ¯ η ρ , ¯ η
2) (
R, R ¯ w I τ(cid:96) Figure 12: The situation of Lemma 3.3 in the time interval [0 , τ (cid:96) [ for the secondincoming road in the coordinates, from left to right, ( ρ, η ) and ( x, t ). Proof.
We proceed in the same way of the previous lemmas. In the timeinterval [0 , τ (cid:96) [ the traffic lights is green in the first incoming road and we studythe Riemann problem as a classical one considering a unique road given bythe union of the I and I : the Riemann problem between (¯ ρ , ¯ η ) and (¯ ρ , ¯ η )produces a linear wave between (¯ ρ , ¯ η ) and (¯ ρ , ¯ η ), see Figure 11. In I , theflow at the junction is equal to zero and the solution in the road I is given bya shock wave of the first family connecting (¯ ρ , ¯ η ) to ( R, R ¯ w ), see Figure 12.In the time interval [ τ (cid:96) , τ (cid:96) + τ (cid:96) [, the traffic lights becomes red for the road I and green for I and the solution of the Riemann problem in the unique roadgiven by the union of the I and I between ( R, R ¯ w ) and (¯ ρ , ¯ η ) is given by ararefaction curve of the first family between ( R, R ¯ w ) and (cid:0) ρ (cid:91) , η (cid:91) (cid:1) followed bya linear wave between (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and (¯ ρ , ¯ η ), see Figure 13. In I , instead, theflow at the junction is equal to zero and so the trace of the solution is ( R, R ¯ w ),see Figure 14. More precisely a shock wave with negative speed starts from thepoint (cid:0) τ (cid:96) , (cid:1) connecting the states (¯ ρ , ¯ η ) and ( R, R ¯ w ).Similarly, in the time interval [ τ (cid:96) + τ (cid:96) , τ (cid:96) + τ (cid:96) [, the traffic light is greenfor road I and red for I ; so we need to consider a Riemann problem between( R, R ¯ w ) and (cid:0) ρ (cid:91) , η (cid:91) (cid:1) , see Figure 15. The solution consists in a rarefactioncurve of the first family between ( R, R ¯ w ) and (cid:0) ρ (cid:91) , η (cid:91) (cid:1) followed by a linearwave between (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and (cid:0) ρ (cid:91) , η (cid:91) (cid:1) . The situation of I is analogous to that12 F ρRC
R, R ¯ w (cid:16) ρ(cid:91) , η(cid:91) (cid:17) (¯ ρ , ¯ η
1) (¯ ρ , ¯ η t x (¯ ρ , ¯ η
2) (¯ ρ , ¯ η R, R ¯ w R, R ¯ w I I ρ , ¯ η (cid:16) ρ(cid:91) , η(cid:91) (cid:17) τ(cid:96) τ(cid:96) τ(cid:96) Figure 13: The situation of Lemma 3.3 in the time interval [ τ (cid:96) , τ (cid:96) + τ (cid:96) [ for thesecond incoming road and the outgoing road in the coordinates, from left toright, ( ρ, η ) and ( x, t ). η F ρRC R, R ¯ w ρ , ¯ η tx (¯ ρ , ¯ η ρ , ¯ η
1) (
R, R ¯ w I τ(cid:96) τ(cid:96) τ(cid:96) Figure 14: The situation of Lemma 3.3 in the time interval [ τ (cid:96) , τ (cid:96) + τ (cid:96) [ for thefirst incoming road in the coordinates, from left to right, ( ρ, η ) and ( x, t ). η F ρRC R, R ¯ w (cid:16) ρ(cid:91) , η(cid:91) (cid:17)(cid:16) ρ(cid:91) , η(cid:91) (cid:17) t x ( R, R ¯ w I I ρ , ¯ η τ(cid:96) τ(cid:96) τ(cid:96) τ(cid:96) (cid:16) ρ(cid:91) , η(cid:91) (cid:17)(cid:16) ρ(cid:91) , η(cid:91) (cid:17) (¯ ρ , ¯ η
1) (¯ ρ , ¯ η Figure 15: The situation of Lemma 3.3 in the time interval [ τ (cid:96) + τ (cid:96) , τ (cid:96) + τ (cid:96) [for the first incoming road and the outgoing road in the coordinates, from leftto right, ( ρ, η ) and ( x, t ).represented in Figure 12. More precisely at the point (cid:0) τ (cid:96) + τ (cid:96) , (cid:1) a shock wavewith negative speed is generated and it connects (cid:0) ρ (cid:91) , η (cid:91) (cid:1) with ( R, R ¯ w ).We proceed in the same way until we arrive at time t = T . The solution in13 t I τ(cid:96) τ(cid:96) τ(cid:96) τ(cid:96) τ(cid:96) τ(cid:96) τ(cid:96) ρ , ¯ η ρ , ¯ η (cid:16) ρ(cid:91) , η(cid:91) (cid:17)(cid:16) ρ(cid:93) , η(cid:93) (cid:17)(cid:16) ρ(cid:91) , η(cid:91) (cid:17)(cid:16) ρ(cid:91) , η(cid:91) (cid:17) A(cid:96) A(cid:96) A(cid:96) Figure 16: The situation of Lemma 3.3 in the time interval [0 , τ (cid:96) + 2 τ (cid:96) [ for theoutgoing road in the coordinates ( x, t ). The two states (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and (cid:0) ρ (cid:91) , η (cid:91) (cid:1) ,separated by linear waves, alternate periodically. I is given by (3.14); see Figure 16. (cid:3) Lemma 3.4
Assume that the initial conditions satisfy (¯ ρ , ¯ η ) ∈ C, (¯ ρ , ¯ η ) ∈ C, (¯ ρ , ¯ η ) ∈ F. Then the solution in the outgoing road I is ( ρ (cid:96), , η (cid:96), ) ( t, x ) = (¯ ρ , ¯ η ) , if < t < T, x > V max t, (cid:0) ρ (cid:91) , η (cid:91) (cid:1) , if ( t, x ) ∈ A (cid:96) , (cid:0) ρ (cid:91) , η (cid:91) (cid:1) , if ( t, x ) ∈ A (cid:96) , (3.17) where A (cid:96) = (cid:96) − (cid:91) i =0 (cid:26) ( t, x ) : 0 < t < T, x > t − τ (cid:96) − i T(cid:96) < xV max < t − i T(cid:96) (cid:27) (3.18) and A (cid:96) = (cid:96) (cid:91) i =1 (cid:26) ( t, x ) : 0 < t < T, x > t − i T(cid:96) < xV max < t − i T(cid:96) + τ (cid:96) (cid:27) . (3.19) Proof.
We proceed in the same way of the previous lemmas. In the timeinterval [0 , τ (cid:96) [ the traffic lights is green in the first incoming road and we studythe Riemann problem as a classical one considering a unique road given bythe union of the I and I : the Riemann problem between (¯ ρ , ¯ η ) and (¯ ρ , ¯ η )produces a rarefaction curve of the first family between (¯ ρ , ¯ η ) and (cid:0) ρ (cid:91) , η (cid:91) (cid:1) followed by a linear wave between (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and (¯ ρ , ¯ η ), see Figure 17. In I , theflow at the junction is equal to zero and the solution in the road I is given bya shock wave of the first family connecting (¯ ρ , ¯ η ) to ( R, R ¯ w ), see Figure 18.In the time interval [ τ (cid:96) , τ (cid:96) + τ (cid:96) [, when the traffic lights becomes red for the14 F ρRC (¯ ρ , ¯ η ρ , ¯ η (cid:16) ρ(cid:91) , η(cid:91) (cid:17) (¯ ρ , ¯ η t x (¯ ρ , ¯ η
1) 0 (¯ ρ , ¯ η ρ , ¯ η I I τ(cid:96) (cid:16) ρ(cid:91) , η(cid:91) (cid:17) Figure 17: The situation of Lemma 3.4 in the time interval [0 , τ (cid:96) [ for the firstincoming road and the outgoing road in the coordinates, from left to right, ( ρ, η )and ( x, t ). η F ρRC ρ , ¯ η
2) (
R, R ¯ w tx ρ , ¯ η ρ , ¯ η
2) (
R, R ¯ w I τ(cid:96) Figure 18: The situation of Lemma 3.4 in the time interval [0 , τ (cid:96) [ for the secondincoming road in the coordinates, from left to right, ( ρ, η ) and ( x, t ).road I and green for I , the solution of the Riemann problem in the uniqueroad given by the union of the I and I between ( R, R ¯ w ) and (cid:0) ρ (cid:91) , η (cid:91) (cid:1) is givenby a rarefaction curve of the first family between ( R, R ¯ w ) and (cid:0) ρ (cid:91) , η (cid:91) (cid:1) followedby a linear wave between (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and (cid:0) ρ (cid:91) , η (cid:91) (cid:1) , see Figure 19. In I , instead,a shock wave with negative speed starts from the point (cid:0) τ (cid:96) , (cid:1) connecting thestates (cid:0) ρ (cid:91) , η (cid:91) (cid:1) and ( R, R ¯ w ), see Figure 20.From time τ (cid:96) + τ (cid:96) to time t = T , the solution becomes as that describedin the previous Lemma 3.3 and it is given by (3.17) with a structure similar toFigure 16. (cid:3) Assume that the values σ > , · · · , σ n − > , σ n = 1, defined in (3.7), are allpositive constant, not depending on the number of cycles (cid:96) .We can now state and prove the main result of the paper. Theorem 3.5
Assume (H-1) , (H-2) , and (H-3) hold. Fix, for every i ∈{ , . . . , n + 1 } , (¯ ρ i , ¯ η i ) ∈ F ∪ C . Consider the Riemann problem at the junction F ρRC
R, R ¯ w (cid:16) ρ(cid:91) , η(cid:91) (cid:17)(cid:16) ρ(cid:91) , η(cid:91) (cid:17) t x (¯ ρ , ¯ η R, R ¯ w R, R ¯ w I I ρ , ¯ η (cid:16) ρ(cid:91) , η(cid:91) (cid:17) τ(cid:96) τ(cid:96) τ(cid:96) (cid:16) ρ(cid:91) , η(cid:91) (cid:17)(cid:16) ρ(cid:91) , η(cid:91) (cid:17) Figure 19: The situation of Lemma 3.4 in the time interval [ τ (cid:96) , τ (cid:96) + τ (cid:96) [ for thesecond incoming road and the outgoing road in the coordinates, from left toright, ( ρ, η ) and ( x, t ). η F ρRC R, R ¯ w (cid:16) ρ(cid:91) , η(cid:91) (cid:17) tx (¯ ρ , ¯ η ρ , ¯ η
1) (
R, R ¯ w I τ(cid:96) τ(cid:96) τ(cid:96) (cid:16) ρ(cid:91) , η(cid:91) (cid:17) Figure 20: The situation of Lemma 3.4 in the time interval [ τ (cid:96) , τ (cid:96) + τ (cid:96) [ for thefirst incoming road in the coordinates, from left to right, ( ρ, η ) and ( x, t ). for the phase transition model (2.1) where the initial conditions are given by (¯ ρ , ¯ η ) , · · · , (¯ ρ n +1 , ¯ η n +1 ) .Denote with ( ρ (cid:96),i ( t, x ) , η (cid:96),i ( t, x )) ( i ∈ { , · · · , n + 1 } ) the solution to the Rie-mann problem, where the junction is governed by the traffic lights with (cid:96) cycles.If (cid:96) → + ∞ , then there exists a function ( (cid:101) ρ n +1 ( t, x ) , (cid:101) η n +1 ( t, x )) , defined inthe outgoing road I n +1 , such that ( ρ (cid:96),n +1 , η (cid:96),n +1 ) (cid:42) ∗ ( (cid:101) ρ n +1 , (cid:101) η n +1 ) (3.20) converges in the weak ∗ topology of L ∞ ([0 , T ] × I n +1 ) and the limit function ( (cid:101) ρ n +1 ( t, x ) , (cid:101) η n +1 ( t, x )) is a weak solution to (2.1) on I n +1 . oreover, if (¯ ρ n +1 , ¯ η n +1 ) ∈ F , then (cid:101) ρ n +1 ( t, x ) = ¯ ρ n +1 , if x > V max t, < t < T, (cid:34) n (cid:88) i =1 σ i (cid:35) n (cid:88) i =1 σ i ρ (cid:91)i , if < x < V max t, < t < T, (cid:101) η n +1 ( t, x ) = ¯ η n +1 , if x > V max t, < t < T, (cid:34) n (cid:88) i =1 σ i (cid:35) n (cid:88) i =1 σ i η (cid:91)i , if < x < V max t, < t < T. (3.21) Instead, if (¯ ρ n +1 , ¯ η n +1 ) ∈ C , then (cid:101) ρ n +1 ( t, x ) = ¯ ρ n +1 , if x > λt, < t < T, (cid:34) n (cid:88) i =1 σ i (cid:35) n (cid:88) i =1 σ i ρ (cid:93)i , if < x < λt, < t < T, (cid:101) η n +1 ( t, x ) = ¯ η n +1 , if x > λt, < t < T, (cid:34) n (cid:88) i =1 σ i (cid:35) n (cid:88) i =1 σ i η (cid:93)i , if < x < λt, < t < T, (3.22) where λ = v (¯ ρ n +1 , ¯ η n +1 ) .Finally, for a.e. t ∈ [0 , T ] , the trace at x = 0 + of the maximal speed (cid:101) w n +1 = (cid:101) η n +1 (cid:101) ρ n +1 satisfies (cid:101) w n +1 ( t, + ) = 1 σ + · · · + σ n [ σ ¯ w + · · · + σ n ¯ w n ] . (3.23) Proof.
First consider the case n = 2, i.e. the junction with n = 2 incomingroads and one outgoing road. In the time interval [0 , τ (cid:96) [ the traffic lights forthe incoming road I is red. Hence the trace of the solution in I has maximaldensity R . This means that we may assume that(¯ ρ , ¯ η ) = ( R, ¯ η ) ∈ C. (3.24)If the initial condition for the road I belongs to the free phase F , thenLemma 3.3 and Lemma 3.4 imply that the sequence of solutions in I is ofRademacker type, see for instance [3, Exercise 4.18]. Hence we deduce that thelimit of such sequence is given by (3.21).If the initial condition for the road I belongs to the congested phase C ,then Lemma 3.1 and Lemma 3.2 imply that the sequence of solutions in I isagain of Rademacker type, and so the limit of such sequence is given by (3.22).The functions (3.21) and (3.22) are piecewise constant and the discontinuitytravels with speed satisfying the Rankine-Hugoniot condition. Hence they areweak solutions to (2.1).Finally, consider the maximal speed (cid:101) w ( t, + ) = (cid:101) η ( t, + ) (cid:101) ρ ( t, + )17f the solution ( (cid:101) ρ , (cid:101) η ) at the junction. If (¯ ρ , ¯ η ) ∈ F , then (cid:101) w ( t, + ) = σ η (cid:91) + σ η (cid:91) σ ρ (cid:91) + σ ρ (cid:91) = σ ¯ w ρ (cid:91) + σ ¯ w ρ (cid:91) σ ρ (cid:91) + σ ρ (cid:91) = σ ρ (cid:91) V max σ ρ (cid:91) V max + σ ρ (cid:91) V max ¯ w + σ ρ (cid:91) V max σ ρ (cid:91) V max + σ ρ (cid:91) V max ¯ w = γ γ + γ ¯ w + γ γ + γ ¯ w , with γ = σ ρ (cid:91) V max and γ = σ ρ (cid:91) V max . Therefore the maximal speed (cid:101) w ( t, + )in the outgoing road is a convex combination of the maximal speeds in the twoincoming roads ¯ w and ¯ w and it coincides with the condition on the maximalspeed proposed in [9].If (¯ ρ , ¯ η ) ∈ C , then (cid:101) w ( t, + ) = σ η (cid:93) + σ η (cid:93) σ ρ (cid:93) + σ ρ (cid:93) = σ ¯ w ρ (cid:93) + σ ¯ w ρ (cid:93) σ ρ (cid:93) + σ ρ (cid:93) = σ ρ (cid:93) v (¯ ρ , ¯ η ) σ ρ (cid:93) v (¯ ρ , ¯ η ) + σ ρ (cid:93) v (¯ ρ , ¯ η ) ¯ w + σ ρ (cid:93) v (¯ ρ , ¯ η ) σ ρ (cid:93) v (¯ ρ , ¯ η ) + σ ρ (cid:93) v (¯ ρ , ¯ η ) ¯ w = γ γ + γ ¯ w + γ γ + γ ¯ w , with γ = σ ρ (cid:93) v (cid:16) ρ (cid:93) , η (cid:93) (cid:17) and γ = σ ρ (cid:93) v (cid:16) ρ (cid:93) , ¯ η (cid:93) (cid:17) , since v (cid:16) ρ (cid:93) , η (cid:93) (cid:17) = v (cid:16) ρ (cid:93) , η (cid:93) (cid:17) = v (¯ ρ , ¯ η ) . We conclude as in the previous case.The proof for the general case n ≥ R . Instead, in the incoming road where the traffic lights is green, then the initialcondition for such road is propagated in I n +1 with speed v (¯ ρ n +1 , ¯ η n +1 ). Hencethe solution (˜ ρ (cid:96),n +1 , ˜ η (cid:96),n +1 ) is similar to that of Lemmas 3.1-3.4 in the sense thatthere exist subsets A (cid:96) , · · · A (cid:96)n of [0 , T ] × I n +1 with a “periodic” structure in which(˜ ρ (cid:96),n +1 , ˜ η (cid:96),n +1 ) is given respectively by (¯ ρ , ¯ η ) , · · · , (¯ ρ n , ¯ η n ). This permits toconclude. (cid:3) Acknowledgments
The authors were partially supported by the INdAM-GNAMPA 2017 project“Conservation Laws: from Theory to Technology”.
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