STABILITY ESTIMATES FOR SCALAR CONSERVATION LAWS WITH MOVING FLUX CONSTRAINTS. Maria Laura Delle Monache. Paola Goatin

Transcription

1 Manuscript submitted to AIMS Journals Volume X, Number 0X, XX 200X doi: /xx.xx.xx.xx pp. X XX STABILITY ESTIMATES FO SCALA CONSEVATION LAWS WITH MOVING FLUX CONSTAINTS Maria Laura Delle Monache Department of mathematical sciences utgers University - Camden 311 N. 5th Street, Camden, NJ 08102, USA Paola Goatin Inria Sophia Antipolis - Méditerranée Université Côte d Azur, Inria, CNS, LJAD 2004, route des Lucioles - BP 93, Sophia Antipolis Cedex, FANCE Communicated by the associate editor name) Abstract. We study well-posedness of scalar conservation laws with moving flux constraints. In particular, we show the Lipschitz continuous dependence of BV solutions with respect to the initial data and the constraint trajectory. Applications to traffic flow theory are detailed. 1. Introduction. Motivated by the modeling of moving bottlenecks in traffic flow, we consider the Cauchy problem for a scalar conservation law with moving flux constraint t + x f) = 0, t, x) +, 1a) 0, x) = 0 x), x, 1b) ft, yt))) t)t, yt)) F α t)), t +, 1c) where t yt) is a given trajectory, starting from y0) = y 0. Systems of the form 1) arise in the modeling of moving bottlenecks in vehicular traffic [11, 15]: = t, x) [0, ] is the scalar conserved quantity and represents the traffic density, whose maximum attainable value is. The flux function f : [0, ] + is assumed to be strictly concave and such that f0) = f) = 0. The time-dependent variable y denotes the constraint position. In the present paper we consider a weakly coupled PDE-ODE system, in the sense that we assume that the constraint trajectory is given, and it does not depend on the solution of 1a). Let us detail the meaning of inequality 1c). A moving flux constraint located at x = yt) acts as an obstacle, thus hindering the flow as expressed by the unilateral constraint 1c). There, α ]0, 1[ is the dimensionless reduction rate of the road 2010 Mathematics Subject Classification. Primary: 35L65; Secondary: 35B35. Key words and phrases. Scalar conservation laws, moving flux constraints, PDE-ODE system. This research was supported by the INIA associated team Optimal Eroute Strategies for Traffic management OESTE). The first author s present affiliation is Inria Grenoble hône- Alpes, France. Corresponding author: Paola Goatin. 1

2 2 MAIA LAUA DELLE MONACHE AND PAOLA GOATIN capacity the maximal allowed density) at the bottleneck position. The inequality 1c) is derived by studying the problem in the constraint reference frame, i.e., setting t, x) := t, x + yt)) and rewriting the conservation law 1a) as t + x F ) = 0, F ) = f ). 2) In fact, let f α : [0, α] + be the rescaled flux function describing the reduced flow at x = yt), i.e. f α ) = αf/α), and α ]0, α[ such that F α α ) = f α α ) α = 0 f α α ) = with F α ) = f α ), see Figure 1. Then setting F α ) := F α α ) = f α α ) α and imposing that in the obstacle reference frame the flux F is less than the maximum value of the flux of the reduced flow, one gets 1c). Notice that the inequality 1c) is always satisfied if ft, yt)±)) t) =, t, yt)±) since the left-hand side is 0. Moreover, it is well defined even if has a jump at yt), because of the ankine-hugoniot conditions. f) f) F α 0 ˇ) α ˆ) α 0 α a) Fixed reference frame b) Obstacle reference frame Figure 1. Graphical representation of the constraint action in the fixed left) and moving right) reference frames. Problem 2), 1b), 1c), can therefore be recast in the framework of conservation laws with fix local constraint, first introduced in [9], then developed in [3, 1] for scalar equations and extended in [12, 2, 13] to systems. Following [11, Definition 4.1] and [5, Definition 1 and 2], solutions of 1) are defined as follows. Definition 1.1. Let y W 1,1 + ; ) and 0 L 1 L ; [0, ]) be given. A function C 0 + ; L 1 L ; [0, ]) ) is a solution to 1) if

3 STABILITY OF SCALA CONSEVATION LAWS WITH MOVING FLUX CONSTAINTS 3 1. satisfies Kružhkov entropy conditions [14] on + ) \ {t, yt)): t + }, i.e. for every k [0, 1] and for all ϕ Cc 1 2 ; + ) and ϕt, yt)) = 0, t > 0, k t ϕ + sgn k) f) fk)) x ϕ) dx dt + + o k ϕ0, x) dx 0 ; 3a) 2. for a. e. t > 0 the left and right traces of at x = yt) satisfy t, yt) ), t, yt)+)) G α t)). The set G α ) in 13b) is defined as follows, see [4, 3, 5]. 3b) Definition 1.2. The admissibility germ G α ) [0, ] 2 for 1a), 1c) is the union G α ) := G 1 ) G 2 ) G 3 ), where G 1 ) := { c L, c ) [0, ] 2 : c L > c, fc L ) c L = fc ) c = F α ) }, G 2 ) := { c, c) [0, ] 2 : fc) c F α ) }, G 3 ) := { c L, c ) [0, ] 2 : c L < c, fc L ) c L = fc ) c F α ) }. The equivalence between Definition 1.1 and [11, Definition 4.1] can be proved as in [3, Proposition 2.6]. Systems of the type 1) arise in the modeling of moving bottlenecks in traffic flows, see [11, 15] and Section 3 below, where they are coupled with an ODE depending on the downstream traffic velocity and describing the trajectory of a slow moving vehicle a bus or a truck) acting as a bottleneck. This paper is a first step towards establishing well-posedness for the strongly coupled models [11, 15]. Section 2 presents the main result, stating the L 1 Lipschitz continuous dependence of solutions of 1) from the initial data and the constraint trajectory. Section 3 describes in details the related traffic flow model with moving bottleneck. Technical proof details are deferred to Appendix A. 2. Lipschitz continuous dependence of BV solutions with respect to. Let us fix y W 1,1 + ) and 0 BV; [0, ]), and let L + ; [0, ]) be a solution of 1) in the sense of Definition 1.1. Applying the coordinate transformation t, x) := t, x + yt)), is a weak entropy solution of the problem t + x F ) = 0, t, x) +, 4a) 0, x) = 0 x + y 0 ), x, 4b) F t, 0)) F α t)), t +, 4c) in the sense of [3, Proposition 2.6A)] where we have set F ) := f ). Existence and uniqueness for 4) are proved in [3, 9]. In particular, we have that for every k [0, 1] and for all ϕ Cc 1 2 ; + ) such that ϕt, 0) = 0, t > 0, k t ϕ + Φ;, k) x ϕ) dx dt + o x + y 0 ) k ϕ0, x) dx 0, + where we have set and Φ; a, b) := sgna b) fa) fb)) a b, for a, b, t, 0 ), t, 0+)) G α t)) for a. e. t > 0,

4 4 MAIA LAUA DELLE MONACHE AND PAOLA GOATIN see [3, Proposition 2.6]. concave, we have Moreover, we note that, since α < 1 and f is strictly F α ) = f α α ) α < f α ) α = max F ). 5) [0,] emark 1. Solution to 4) are not in BV in general, since y W 1,1 + ) and thus BV + ). Yet, if BV[0, t]; ), t, ) BV; [0, ]) due to 5). In any case, the left and right traces at x = 0 do exist, see [3, Section 2]. We compare solutions of 4) corresponding to different constraint trajectories y and z. Theorem 2.1. Assume y, z W 1,1 + ) with 0, < f 0), 0, σ 0 L, [0, ]) and 0 σ 0 L 1 ). Let, σ L + ; [0, ]) be the solutions of 4) corresponding respectively to y, 0 and z, σ 0. Moreover we assume that TV t, )) C for all t > 0. Then we have t, ) σt, ) L 1 ) 0 σ 0 L 1 ) + C + 2) L 1 [0,t]). 6) Proof. The two solutions, σ satisfy t k + x Φ;, k) 0, 7) t σ k + x Φ; σ, k) 0, 8) in D + ) where we have noted = \ {0}). Following the proofs of [5, Lemma 15] and [6, Theorem 3.1] we observe that t k + x Φ;, k) = t k + x Φ;, k) + x Φ;, k) x Φ;, k) therefore t k + x Φ;, k) x Φ;, k) x Φ;, k) x. 9) Applying the classical Kružkov doubling of variables technique [14], with a test function ψ Cc 1 2 ; + ) such that ψt, 0) = 0, we deduce the following Kato inequality σ t ψ + Φ;, σ) x ψ) dx dt σ 0 ψ0, x) dx + C ψ dt 0, + We now choose the test function ψt, x) = θ ε x)ξt, x), where ξ C 1 c 2 ; + ) is an approximation of the characteristic function of the trapezoid {s, x): x M + Lt s), 0 s t} where L V )and θ ε a smooth approximation of x min{ x /ε, 1}. Following the proof of [9, Proposition 4.4] and letting ε 0 we get M M t, x) σt, x) dx M+Lt M Lt t x) σ 0 x) dx + C t 0 s) s) ds Φ; t, 0+), σt, 0+)) Φ; t, 0 ), σt, 0 ))) ds.

5 STABILITY OF SCALA CONSEVATION LAWS WITH MOVING FLUX CONSTAINTS 5 By Lemma A.1, the last integrand can be bounded by t 0 Φ; t, 0+), σt, 0+)) Φ; t, 0 ), σt, 0 ))) ds 2 Letting M, we recover 6). 3. Application to traffic modeling. Setting f) := v), t 0 s) s) ds. where v) = V 1 /) is the mean traffic speed, V being the maximal velocity allowed on the road, problem 1) can be used to describe the situation of a moving bottleneck along a road, see [11]. In this case, we get f α ) = V 1 α = α 2 1 V ), so that 10) α) and F α ) = α 4V V )2. Let us suppose that a slow and large vehicle, like for example a bus or a truck moves on the road. The slow vehicle, that in the following we will refer as bus, reduces the road capacity and moves with a trajectory given by the following ODE: { t) = ωt, yt)+)), t +, 11) y0) = y 0, where the velocity of the bus is given by the following traffic density dependent function see Figure 2) { Vb if ω) =. = 1 Vb /V ), 12) v) otherwise. This means that if the traffic is not too congested, the bus moves at its own maximal speed V b < V. When the surrounding traffic density becomes too high, the bus adapts its velocity accordingly. In particular, it is not possible for the bus to overtake the cars. ω) Bus speed Cars speed v) V b Figure 2. Bus and cars speed. Solutions of the coupled system 1), 11) for general initial data are defined as follows. Definition 3.1. A couple, y) C 0 + ; L 1 BV; [0, 1]) ) W 1,1 + ; ) is a solution to 1) if

6 6 MAIA LAUA DELLE MONACHE AND PAOLA GOATIN 1. satisfies Kružhkov entropy conditions [14] on + ) \ {t, yt)): t + }, i.e. for every k [0, 1] and for all ϕ Cc 1 2 ; + ) and ϕt, yt)) = 0, t > 0, k t ϕ + sgn k) f) fk)) x ϕ) dx dt + + o k ϕ0, x) dx 0 ; 13a) 2. for a. e. t > 0 the one-sided traces of at x = yt) satisfy t, yt) ), t, yt)+)) G α t)) ; 3. y is a Carathéodory solution of 11), i.e. for a.e. t + yt) = y o + t 0 ωs, ys)+)) ds. 13b) 13c) The proof of existence of solutions for the general Cauchy problem 1) strongly coupled with the bus trajectory 11) with BV initial data can be found in [11]. For completeness, we recall the definition the solution of a iemann problem, as given in [11]. Let us consider a iemann type initial datum { L if x < 0, 0 x) = y if x > 0, 0 = 0. 14) Denote by the standard i.e., without the constraint 1c)) iemann solver for 1a)-1b)-14), i.e., the right continuous) map t, x) L, )x/t) given by the standard weak entropy solution, see for instance [7, Chapter 5]. Moreover, assume that is constant and let ˇ = ˇ) and ˆ = ˆ), with ˇ ˆ, be the intersection points of the flux function f) with the line f α α ) + α ) see Figure 1a)): ˇ) = ) 1 1 α V ), 2V ˆ) = 15) ) α V ). 2V Definition 3.2. The constrained iemann solver α : [0, ] 2 L 1 loc ; [0, ]) corresponding to 1), 11), 14) is defined as follows. 1. If f L, )V b )) > F α + V b L, )V b ), then { α L, ˆV L, )x/t) = b ))x/t) if x < V b t, ˇV b ), )x/t) if x V b t, and yt) = V b t. 2. If V b L, )V b ) f L, )V b )) F α + V b L, )V b ), then α L, ) = L, ) and yt) = V b t. 3. If f L, )V b )) < V b L, )V b ), i.e., v L, )V b )) < V b then α L, ) = L, ) and yt) = v )t. Note that, when the constraint is enforced point 1. in the above definition), a non-classical shock arises between ˆV b ) and ˇV b ), which satisfies the ankine- Hugoniot condition but violates the Lax entropy condition, see Figure 3 for an example.

7 STABILITY OF SCALA CONSEVATION LAWS WITH MOVING FLUX CONSTAINTS 7 emark 2. Unfortunately, no result about the Lipschitz continuous dependence of the solution y = yt) of 11) from the solution = t, x) of 1) is known at present. elated results [8, 10] concerning uniqueness and continuous dependence for ODEs of the form 11) hold under hypothesis on the speed function ω that are not satisfied by 12). Appendix A. Proof of 10). Lemma A.1. For any, + ) G α ) and σ, σ + ) G α ), it holds Φ; +, σ + ) Φ;, σ ) 2. Proof. For simplicity, we consider the case f) = V 1 /). First of all, let us remark that Φ; +, σ + ) Φ;, σ ) 16) = sgn + σ + ) f + ) + fσ + ) + σ + ) sgn σ ) f ) fσ ) + σ ) =λ +, σ + ) ) + σ + λ, σ ) ) σ, 17) where f) fσ) λ, σ) = = V 1 + σ ). σ Without loss of generality, we can assume <. Therefore we get F α ) F α ) = α 2V ) ) > 0. 18) 4V We distinguish the following cases: 1., + ) G 1 ): we observe that = ˆ, + = ˇ and f ) = f + ) + = F α ). Depending on the values of σ, σ +, we different situations can occur as shown in Figure 4: 1.1 σ, σ + ) G 1 ): in this case σ = ˆσ, σ + = ˇσ, as shown in Figure 4a and fσ ) σ = fσ + ) σ + = F α ), therefore 16) = F α ) F α )) F α ) + F α )) = + ) ). 1.2 σ, σ + ) G 2 ): we set σ := σ = σ + and fσ) σ =: F σ) F α ). The following cases can occur: 0 σ ˇσ: 16) = F α ) F σ)) F α ) + F σ)) = + ) ).

8 8 MAIA LAUA DELLE MONACHE AND PAOLA GOATIN f) t ˆ α V b V b ˇ α 0 ˇ α α ˆ α α 1 a) Example of solution corresponding to case 1. of Definition 3.2 for L = =, ˇ α < < ˆ α. x f) t V b L V b L 0 ˇ α α ˆ α α 1 x b) Example of solution corresponding to case 2. of Definition 3.2 for 0 < L < ˇ α and ˆ α < <. f) t v ) V b L L 0 ˇ α α ˆ α α 1 c) Example of solution corresponding to case 3. of Definition 3.2 for L, >. x Figure 3. Different solutions of the iemann problem 14). Each subfigure illustrates a point of the Definition 3.2: fundamental diagram representation left) and space-time diagram right).

9 STABILITY OF SCALA CONSEVATION LAWS WITH MOVING FLUX CONSTAINTS 9 f) f) σ + σ + ˇσ + ˆσ σ a) Case 1.1 b) Case 1.2 f) σ ˇσ ˇ ˆσ ˆ σ + c) Case 1.3 Figure 4. Case 1 ˆσ σ ˆ, see Figure 4b: 16) = F α ) F σ)) F α ) + F σ)) = 2F σ) 2F α ) ) ) 2F α ) 2F α ) + 2 ) = α 2V ) ) + 2 ) 2V 2. ˆ σ : 16) = F α ) F σ)) + F α ) + F σ)) = + ) ) σ, σ + ) G 3 ): we set fσ ) σ = fσ + ) σ + =: F σ) F α ).

10 10 MAIA LAUA DELLE MONACHE AND PAOLA GOATIN We observe that sgn + σ + ) < 0 and sgn σ ) > 0, see Figure 4c. Therefore 16) = F α ) F σ)) F α ) + F σ)) = 2F σ) 2F α ) ) ) 2F α ) 2F α ) + 2 ) = α 2V ) ) + 2 ) 2V 2. 2., + ) G 2 ): we set := = + and f) =: F ) F α ). as illustrated in Figure 5. f) f) 0 ˇσ ˇ ˆσ ˆ 0 ˇσ ˇ ˆσ σ ˆ a) Case 2.1 b) Case 2.2 f) σ ˇσ ˇ ˆσ σ + ˆ c) Case 2.3 Figure 5. Case σ, σ + ) G 1 ): 0 ˇσ: 16) = F ) + F α )) + F ) + F α )) = 0.

11 STABILITY OF SCALA CONSEVATION LAWS WITH MOVING FLUX CONSTAINTS 11 ˇσ ˇ, this case is displayed in Figure 5a: 16) = F ) + F α )) + F ) + F α )) = 2F ) 2F α ) + 2 ) 2 F α ) F α )) α. ˆ : 16) = F ) + F α )) F ) + F α )) = σ, σ + ) G 2 ), illustrated in Figure 5b: We observe that sgn + σ + ) = sgn σ ) = sgn σ), therefore 16) = sgn σ) [F ) + F σ)) F ) + F σ))] = σ, σ + ) G 3 ) shown in Figure 5c: If sgn σ + ) = sgn σ ), we get 16) = sgn σ + ) [F ) + F σ)) F ) + F σ))] = 0. Otherwise, we have that σ ˇ or ˆ σ +. In this case λ +, σ + ) = V 1 + σ ) + V 1 σ ) + + σ = V σ ) 0, λ, σ ) = V 1 + σ ) V 1 σ ) + + σ = V σ + ) 0. Therefore, 16) 0 by 17). 3., + ) G 3 ): we set f ) = f + ) + =: F ) F α ). See Figure 6, for a graphical representation. 3.1 σ, σ + ) G 1 ), see Figure 6a. 16) = F ) F α )) + F ) + F α )) = 2F ) 2F α ) ) ) 2 F α ) F α )) α. 3.2 σ, σ + ) G 2 ): If sgn + σ) = sgn σ), we get 16) = sgn + σ) [F ) F σ)) F ) + F σ))] + ) ). Otherwise as shown in Figure 6b, we have that σ ˇσ or ˆσ σ +. In this case λ +, σ + ) λ +, σ + ) = V 1 ) + + σ V 1 ) + + = V σ) 0. Moreover, we observe that λ, σ ) >. Indeed λ, σ ) = V 1 ) + σ V 1 ) + + = V + σ) 0.

12 12 MAIA LAUA DELLE MONACHE AND PAOLA GOATIN f) f) σ + σ + σ + a) Case 3.1 b) Case 3.2 f) σ σ + + c) Case 3.3 Figure 6. Case 3 Therefore, by 17) 16). 3.3 σ, σ + ) G 3 ): We observe that one of the following relations must hold σ < σ + +, σ < σ + +, σ < + σ +. For an example see Figure 6c. Therefore λ +, σ + ) = V 1 ) + + σ + V 1 σ ) + + σ = V σ + ) 0, λ, σ ) = V 1 ) + σ V 1 σ ) + + σ = V σ + ) 0, hence 16) 0 by 17). Acknowledgments. The authors are grateful to Boris Andreianov for suggesting links with the results in [5].

13 STABILITY OF SCALA CONSEVATION LAWS WITH MOVING FLUX CONSTAINTS 13 EFEENCES [1] B. Andreianov, C. Donadello, U. azafison, and M. D. osini. iemann problems with nonlocal point constraints and capacity drop. Math. Biosci. Eng., 122): , [2] B. Andreianov, C. Donadello, and M. D. osini. A second-order model for vehicular traffics with local point constraints on the flow. Math. Models Methods Appl. Sci., 264): , [3] B. Andreianov, P. Goatin, and N. Seguin. Finite volume schemes for locally constrained conservation laws. Numer. Math., 1154): , With supplementary material available online. [4] B. Andreianov, K. H. Karlsen, and N. H. isebro. A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux. Arch. ation. Mech. Anal., 2011):27 86, [5] B. Andreianov, F. Lagoutière, N. Seguin, and T. Takahashi. Well-posedness for a onedimensional fluid-particle interaction model. SIAM J. Math. Anal., 462): , [6] F. Bouchut and B. Perthame. Kružkov s estimates for scalar conservation laws revisited. Trans. Amer. Math. Soc., 3507): , [7] A. Bressan. Hyperbolic systems of conservation laws, volume 20 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, The onedimensional Cauchy problem. [8] A. Bressan and W. Shen. Uniqueness for discontinuous ODE and conservation laws. Nonlinear Anal., 345): , [9]. M. Colombo and P. Goatin. A well posed conservation law with a variable unilateral constraint. J. Differential Equations, 2342): , [10]. M. Colombo and A. Marson. A Hölder continuous ODE related to traffic flow. Proc. oy. Soc. Edinburgh Sect. A, 1334): , [11] M. L. Delle Monache and P. Goatin. Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result. J. Differential Equations, 25711): , [12] M. Garavello and P. Goatin. The Aw-ascle traffic model with locally constrained flow. J. Math. Anal. Appl., 3782): , [13] M. Garavello and S. Villa. The Cauchy problem for the Aw- ascle- Zhang traffic model with locally constrained flow. Preprint, [14] S. N. Kružkov. First order quasilinear equations with several independent variables. Mat. Sb. N.S.), ): , [15] S. Villa, P. Goatin, and C. Chalons. Moving bottlenecks for the Aw-ascle-Zhang traffic flow model. Preprint, July address: maria-laura.delle monache@inria.fr address: paola.goatin@inria.fr