Stability estimates for scalar conservation laws with moving flux constraints

. 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 f (ρ(t, y(t))) −ẏ(t)ρ(t, y(t)) ≤ F α (ẏ(t)), where t → y(t) is a given trajectory, starting from y(0) = y 0 . Systems of the form (1) arise in the modeling of moving bottlenecks in vehicular traffic [11,15]: ρ = ρ(t, x) ∈ [0, R] is the scalar conserved quantity and represents the traffic density, whose maximum attainable value is R. The flux function f : [0, R] → R + is assumed to be strictly concave and such that f (0) = f (R) = 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 = y(t) 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 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., settingρ(t, x) := ρ(t, x + y(t)) and rewriting the conservation law (1a) as In fact, let f α : [0, αR] → R + be the rescaled flux function describing the reduced flow at x = y(t), i.e. f α (ρ) = αf (ρ/α), and ρ α ∈ ]0, αR[ such that Figure 1. Then setting 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 ifẏ since the left-hand side is 0. Moreover, it is well defined even if ρ has a jump at y(t), because of the Rankine-Hugoniot conditions. 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.
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.
be the solutions of (4) corresponding respectively to y,ρ 0 and z,σ 0 . Moreover we assume that TV(ρ(t, ·)) ≤ C for all t > 0. Then we have Proof. The two solutionsρ,σ satisfy Applying the classical Kružkov doubling of variables technique [14], with a test function ψ ∈ C 1 c (R 2 ; R + ) such that ψ(t, 0) = 0, we deduce the following Kato inequality is an approximation of the characteristic function of the trapezoid where v(ρ) = V (1 − ρ/R) 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 (ρ) = V ρ 1 − ρ αR and ρ α = αR 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: where the velocity of the bus is given by the following traffic density dependent function (see Figure 2) 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.  Solutions of the coupled system (1), (11) for general initial data are defined as follows.
Note that, when the constraint is enforced (point 1. in the above definition), a non-classical shock arises betweenρ(V b ) andρ(V b ), which satisfies the Rankine-Hugoniot condition but violates the Lax entropy condition, see Figure 3 for an example.

Remark 2.
Unfortunately, no result about the Lipschitz continuous dependence of the solution y = y(t) of (11) from the solution ρ = ρ(t, x) of (1) is known at present. Related 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).