A LWR model with constraints at moving interfaces

We propose a mathematical framework to the study of scalar conservation laws with moving interfaces. This framework is developed on a LWR model with constraint on the flux along these moving interfaces. Existence is proved by means of a finite volume scheme. The originality lies in the local modification of the mesh and in the treatment of the crossing points of the trajectories.


Introduction
-The authors of [14,17] considered a model very similar to (1.2).In their framework, (  )  represented the trajectories of autonomous vehicles, and the authors aimed at modeling the regulation impact on a few autonomous vehicles on the traffic flow.In the same framework but with different applications in mind, the model of [22] accounts for the boundedness of traffic acceleration.Note that in each of these models, the trajectories of the moving interfaces (  )  were not given a priori, but rather obtained as solutions to an ODE involving the density of traffic, a mechanism reminiscent of [4,11,24] for instance.Let us also mention the work of [18] where the authors studied a different model for the situation of several moving bottlenecks.
A. SYLLA -The numerical aspect of (1.2) was treated in [8] (for one trajectory) and [12] (for multiple trajectories), where the authors modeled the moving bottlenecks created by buses on a road.-In a class of problems close to (1.2), i.e. without constraint on the flux, but still with coupling interfaces/density, the authors of [16] described the interaction between a platoon of vehicles and the surrounding traffic flow on a highway.-Problem (1.2) can be seen as a conservation law with discontinuous flux and special treatments at the interfaces.In that directions, the authors of [1,2,6,20,26] studied such problems but with the classical vanishing viscosity coupling at the interfaces.
In several of these works [17,22], the existence issue is tackled using the wave-front tracking procedure which is very sensitive to the details of the model.On the other hand, when numerical schemes are considered, see [8,12], the numerical analysis is usually left out.
The contribution of this paper is to provide a robust mathematical setting both in the theoretical and numerical aspects of (1.2).The proof of uniqueness is based upon a combination of Kruzhkov classical method of doubling variables and the theory of dissipative germs in the framework of discontinuous flux [5] and it is analogous to the one of [2].To prove existence, we build a finite volume scheme with a grid that adapts locally to the trajectories (  )  and to their crossing points, but remains a simple cartesian grid away from the interfaces.Our work can serve as a basis for constructing solutions to more involved models, e.g.via the splitting approach.As an example of application, we can point out the variant of our recent work [24] with multiple slow vehicles involved; this is a mildly non-local analogue of the problem considered numerically in [12].
As the fundamental ingredient of the well-posedness proof and numerical approximation of (1.2), we will first tackle the one trajectory/one constraint problem: ( () − ẏ())| =() ≤ ()  > 0, (1.3) with  ∈ W 1,∞ loc ((0, +∞)) and  ∈ L ∞ loc ((0, +∞)).Models in the class of (1.3) have been greatly investigated in the past few decades.Motivated by the modeling of tollgates and traffic lights for instance, the authors of [9] considered (1.3) with the trivial trajectory  ≡ 0 and proved a well-posedness result in the BV framework (i.e. with both  and  0 with bounded variation, locally).The authors of [4] then extended the well-posedness in the L ∞ framework and also constructed a convergent numerical scheme.More recently, in [11,13,24], the authors studied a variant of (1.3) in which  and ẏ were coupled via an ODE.The coupling was thought to model the influence of a slow vehicle, traveling at speed ẏ, on road traffic.
The reduction of (1.2) to localized problem (1.3) requires the construction of a finite volume scheme in the original coordinates (, ), while the treatment of (1.3) in the literature is most often based upon the rectification of the interface via a variable change, see [11,13,24].For (1.2), this approach leads to a cumbersome and singular construction, see [2].In our well-posedness analysis and approximation of (1.3), having in mind (1.2), we will not change the coordinate system.
Let us detail how the paper is organized.Sections 2 and 3 are devoted to Problem (1.3).We start by giving two definitions of solutions.One, most frequently used in traffic dynamics (see [3,9]), is composed of classical Kruzhkov entropy inequalities with reminder term taking into account the constraint and of a weak formulation for the constraint, see Definition 2.1.The second definition emanates from the theory of conservation laws with dissipative interface coupling (see [1,5]).It consists of Kruzhkov entropy inequalities with test functions that vanish along the interface { = ()} and of an explicit treatment of the traces of the solution along the interface, see Definition 2.6.Before tackling the well-posedness issue, we prove that these two definitions are equivalent, see Propositions 2.8 and 2.9, similarly to what the authors of [4] did.Uniqueness follows from the stability obtained in Section 2, see Theorem 2.11.In Section 3, we construct a finite volume scheme for (1.3) and prove of its convergence.In the construction, we do not rectify the trajectory but instead we locally modify the mesh to mold the trajectory.Moreover, we fully make use of techniques and results put forward by the author of [25] to derive localized BV estimates away from the interface, essential to obtain strong compactness for the approximate solutions created by the scheme, see Corollary 3.9.This is a way to highlight the generality of the compactness technique of [25].
In Section 4, we get back to the original problem (1.2).Our strategy is to assemble the study of (1.2) from several local studies of (1.3) with the help of a partition of unity argument.This concerns, in particular, the convergence of finite volume approximation of (1.2) which is addressed via a localization argument.However, the scheme needs to be defined globally, which makes it impossible to use the rectification strategy as soon as the interfaces have crossing points, cf.[2] for a singular rectification strategy.

Uniqueness and stability for the single trajectory problem
The content of this section is not original in the sense that it is a rigorous adaptation and assembling of existing techniques reminiscent of [4,5,9,21,27].
Remark 2.3.As it happens, the time-continuity regularity is actually a consequence of inequalities (2.2).Indeed, Theorem 1.2 of [7] (or [10,21]) states that if  is an open subset of R and if for all test functions satisfies the following entropy inequalities: then  ∈ C 0 (R + ; L 1 ( )).Moreover, since  is bounded and  ∖ has a Lebesgue measure 0,  ∈ C 0 (R + ; L 1 loc ( )).Taking  = R * ensures that  ∈ C 0 (R + ; L 1 loc (R)).A simple translation ensures that any bounded functions satisfying (2.2) is in C 0 (R + ; L 1 loc (R)).The BV regularity is there to ensure the existence of traces, see also Definition 2.6.Definition 2.1 is well suited for passage to the limit of a.e.convergent sequences of exact or approximate solutions.However, we cannot derive uniqueness by the standard arguments like in the classical case of Kruzhkov.Using an equivalent notion of solution, which we adapt from [5], based on explicit treatment of traces of  on Γ, we rather combine the arguments of [21,27].In this definition a couple plays a major role, the one which realizes the equality in the flux constraint in (1.3).More precisely, fix first  ≥ 0. By (1.1) and concavity of  , for all  ∈ [0, max   ), the equation   () =  admits exactly two solutions in [0, 1], see Figure 1, left.The same way, if  ≤ 0, then for all  ∈ [− ṡ, max   ), the equation still admits two solutions in [0, 1].The couple formed by these two solutions, denoted by (̂︀   (), q   ()) in Definition 2.4 below, will serve both in the prove of uniqueness and existence.
Lemma 2.5.For all  ∈ R + and  ∈ [0, max   ), and for all  ∈ R − and  ∈ [−, max   ), the admissibility germ   () is L 1 -dissipative in the sense that:  for any nonempty interval (, ) ⊂ (0, 1),  |(,) is not constant. (2.9) In the context of traffic flow, however, we sometimes consider fluxes which do not verify (2.9).Such fluxes, which have linear parts, usually model constant traffic velocity for small densities.In those situations, and when  ≡ 0, one can prove that under a mild assumption on the constraint, if the initial data has bounded variation, then solutions to (1.3) are in L ∞ ((0,  ); BV(R)), and once again, traces are to be understood in the sense of BV functions, see Theorem 3.2 of [24].Also note that the germ formalism can be adapted to the situations where the flux is degenerate and no variation bound is assumed, see Remarks 2.2 and 2.3 of [5].
. Using a suitable approximation of the characteristic function of the trapezoid in Kato inequality, we obtain: )︂ d.
The computations leading to this inequality are standard and can be adapted from the one of the proofs of Proposition 4.4 from [9] or Proposition 2.10 from [4].What is left to do is to take the limit when  → +∞ and to estimate the last two terms of the right-hand side of the previous inequality.The following table, in which we dropped the -indexing, summarizes which values can take the difference ∆() according to which parts of their respective germs the couples ((()−, ), (()+, )) and ((()−, ), (()+, )), respectively denoted by (  ,   ) and (  ,   ) belong to.

Existence for the single trajectory problem
We build a simple finite volume scheme and prove its convergence to an admissible entropy solution to (1.3).From now on, we denote by  ∨  = max{, } and  ∧  = min{, }.

Adapted mesh and definition of the scheme
We start by defining the sequence of approximate slopes: and the sequence of approximate trajectories: The same way, we define ( Δ ) Δ , the sequence of approximate constraints: Remark 3.1.Note that with our choices, from (2.5), we deduce that This fact will come in handy in the proof of stability for the scheme.
Fix now  > 0 and a spatial mesh size ∆ > 0 with  = ∆/∆ fixed, verifying the CFL condition For all  ∈ N, there exists a unique index   ∈ Z such that   ∈ [  ,  +1 ), see Figure 2. Introduce the sequence (   ) ∈Z defined by We define the cell grids: where for all  ∈ N and  ∈ Z,   +1/2 is the rectangle We start by discretizing the initial data  0 with )︁
Fix  ∈ N. To simplify the reading, we introduce the notations: We now proceed to the definition of the scheme.It comes from a discretization of the conservation law written in each volume control   +1/2 ( ∈ N,  ∈ Z).Away from the trajectory/constraint, it is the standard 3-point marching formula and when  ∈ {  − 1,   }, we have to deal with both the constraint and the interface which is not vertical.Three cases have to be considered when describing the marching formula of the scheme, but we really give the details for only one of them.
Finally, the approximate solution  Δ is defined almost everywhere on Ω: The other cases ( +1 =   or  +1 =   − 1) follow from similar geometric considerations.Note that in the context of traffic dynamics,  would be the trajectory of a stationary or a forward moving obstacle and therefore, we should have ẏ ≥ 0. This implies that for all  ∈ N, either  +1 =   or  +1 =   + 1.This is why we will focus on the case presented in Figure 2.

Stability and discrete entropy inequalities
proving the monotonicity of H   .Similar computations show that H  −1 is nondecreasing with respect to its arguments as well.Stability.We now turn to the proof of (3.8), which is done by induction on . )︁

𝑗
. Suppose now that (3.8) holds for some integer  ≥ 0 and let us show that it still holds for  + 1.
Note that 0 and 1 are stationary solutions to the scheme.It is obviously true in the case (3.4).The definitions of H  −1 and H   do not change this fact.For instance, H  −1 (0, 0, 0) = 0 since   ≥ 0 and because of (3.1), we also have: Similar computations would ensure that it holds also for H   .Using now the monotonicity of H  −1 for instance, we deduce that where Φ   and Φ   denote the numerical entropy fluxes: Proof.This result is mostly a consequence of the scheme monotonicity.When the interface/constraint does not enter the calculations i.e. when  / ∈ { +1 − 1,  +1 }, the proof follows Lemma 5.4 of [15].The key point is not only the monotonicity, but also the fact that in the classical case, all the constants states  ∈ [0, 1] are stationary solutions of the scheme.This observation does not hold when the constraint enters the calculations.Suppose for example that  =  +1 (which corresponds to the function H   ).Here, we have and it implies: We deduce: which is exactly (3.9) in the case  =  +1 .The obtaining of (3.9) in the case  =  +1 − 1 is similar so we omit the details of the proof for this case.

Continuous inequalities for the approximate solution
The next step of the reasoning is to derive continuous inequalities, analogous to (2.2) and (2.3), verified by the approximate solution  Δ , starting from the discrete entropy inequalities (3.9) and the marching formula We start by deriving continuous entropy inequalities verified by  Δ .Let us define the approximate entropy flux: Proposition 3.4 (Approximate entropy inequalities).Fix  ∈ N and  ∈ [0, 1].Then we have (3.10) Proof.For all  ∈ Z∖{ +1 − 2}, we multiply the discrete entropy inequalities (3.9) by  +1 +1/2 and take the sum to obtain: This inequality can be rewritten as We now proceed to the Abel's transformation and reorganize the terms of the inequality.This leads us to: with We recognize inequality (3.11) as the discrete analogous to inequality (3.10).The remaining of the proof consists in estimating the difference between the terms appearing in (3.11) and their continuous counterparts.For instance, The computations for the other terms can be found in the proof of Proposition 4.2.3 from [23].

Compactness and convergence
The remaining part of the reasoning consists in obtaining sufficient compactness for the sequence ( Δ ) Δ in order to pass to the limit in (3.12)-(3.14).To doing so, we adapt techniques and results put forward by Towers in [25].With this in mind, we suppose in this section that the flux function, still bell-shaped, is also strictly concave.By continuity, We denote for all  ∈ N and  ∈ Z, }︁ .
We will also use the notation In [25], the author dealt with a discontinuous in both time and space flux and the specific "vanishing viscosity" coupling at the interface.The discontinuity in space was localized along the curve { = 0}.Here, we deal with a smooth flux but we have a flux constraint along the curve { = ()}.The applicability of the technique of [25] for our case with moving interface and flux-constrained interface coupling relies on the fact that one can derive a bound on D +1  as long as the interface does not enter the calculations for D +1  i.e. as long as  ∈ ̂︀ Z +1 in the case  +1 =   + 1.

.16)
Proof.For the sake of completeness, the proof, largely inspired by [25], can be found in Appendix A.

3𝑛
. Consider now the open subset Using the BV bounds (3.19) and (3.20) and the uniform L ∞ bound (3.8), Appendix A of [19] provides a subsequence of ( Δ ) Δ which converges almost everywhere in any rectangular bounded domains of   .Using a covering argument, we proved that a subsequence of ( Δ ) Δ converges a.e. on   to some bounded function   ∈ L ∞ (  ).Now, a diagonal procedure provides the a.e.convergence of a subsequence of ( Δ ) Δ on any compact subsets of the set A further extraction yields the a.e.convergence on Ω to some  ∈ L ∞ (Ω).
Proof.Existence comes from Theorem 3.10 while uniqueness was established by Theorem 2.11.

Well-posedness for the multiple trajectory problem
We now get back to the original problem (1.2).Let us detail the organization of this section.First, we construct a partition of the unity to reduce the study of (1.2) to an assembling of several local studies of (1.3), see Section 4.1.Using the definition based on germs, analogous to Definition 2.6, we will prove a stability estimate, leading to uniqueness, see Theorem 4.3.Then in Section 4.3, we construct a finite volume scheme in which we fully use the precise study of Section 3. A special treatment of the crossing points is described, see Section 4.3.1.
Let us recall that we are given a finite (or more generally locally finite) family of trajectories and constraints (  ,   ) ∈[[1;]] defined on (  ,   ) (0 ≤   <   ).Introduce the notations: We suppose that for all  ∈ [[1; ]],   ∈ W 1,∞ ((  ,   )) and   ∈ L ∞ ((  ,   ); R + ).This notation means that what can be seen as crossing points between interfaces will be considered as endpoints of the interfaces; for instance, given two crossing lines, we split them into four interfaces having a common endpoint.We denote by (  ) 1≤≤ the set of all endpoints of the interfaces Γ  ,  ∈ [[1; ]].

Reduction to a single interface
Let us denote by  the compact support of .

Definition of solutions and uniqueness
Following Section 2 and Definition 2.6, we give the following definition of solution.
Proof.We split the reasoning in two steps.
Step 2. Consider now  ∈ C ∞ c (Ω). Fix  ∈ N * .From the first step, a classical approximation argument allows us to apply (4.4) with the Lipschitz test function where for all  ∈ [[1;  ]], where, by analogy with the proof of Lemma 2.10, dist 1 denotes the R 2 distance associated with the norm ‖ • ‖ 1 .We let  → +∞, keeping in mind that: Straightforward computations lead to (4.4) with  ∈ C ∞ c (Ω), concluding the proof.).Then for all  > 0, we have In particular, Problem (1.2) admits at most one -entropy solution.
Proof.Estimate (4.7) follows from Kato inequality (4.4) with a suitable choice of test function and in light of the inequality: see Theorem 2.11.

Proof of existence
Following the reasoning of Sections 2 and 3, we introduce a second definition of solutions, more suitable to prove existence.
We now turn to the proof of existence for admissible entropy solutions of (1.2).We make use of the precise study of Section 3 in the case of a single trajectory and build a finite volume scheme.We keep the notations of Section 3 when there is no ambiguity.

Construction of the mesh, definition of the scheme
For the sake of clarity, suppose that we only have two trajectories/constraints (  ,   ) (1 ≤  ≤ 2) defined on [0,  ], which cross at time  .We denote by  this crossing point.Suppose also that this crossing point results in two additional trajectories/constraints (  ,   ) (3 ≤  ≤ 4) defined on [,  ], and which do not cross, as represented in Figure 4.
Let us fully make explicit the steps of the reasoning leading to the construction of our scheme in that situation.Suppose that  = ∆/∆ is fixed and verifies the CFL condition 2 (4.10) Set  ∈ N such that  ∈ [  ,   +1 ).We divide the discussion in four parts.
Part 1. Introduce the number The definition of  1 ensures that for all  ∈ [[0;  1 − 1]], we can independently modify the mesh near the two trajectories  1 Δ and  2 Δ , as presented in Figure 5. Consequently, we can simply define the approximate solution  Δ on R × [0,  1−1 ] as the finite volume approximation of a conservation law, with initial data  0 , with flux constraints on two non-interacting trajectories, using the recipe of Section 3 for each trajectory/constraint.one cannot modify the mesh in the neighbourhood of one of them without affecting the other.However, the scheme has to be defined globally so we proceed as described below.
-First, introduce the mean trajectory and the new constraint: For  ∈ [[ ;  2 ]], we are in the same situation as Part 2. We proceed to the same construction, mutatis mutandis.
-As in Part 2, define the mean trajectory and the new constraint: ∀ ∈ [,  ],  34 () =  3 () +  4 () 2 ;  34 () = min{ 3 (),  4 ()}, represented in purple in Figure 5, after the crossing point.Remark 4.6.Let us stress out that the details of the treatment done in Parts 2-3 do not play any significant role in the convergence proof below thanks to the choice of test functions vanishing at neighbourhood of the crossing points, see Proposition 4.5.Consequently, taking the mean trajectory and the minimum of the constraint is merely an example aiming at preserving some consistency while keeping the scheme simple to understand and implement.
The general case of a finite number of interfaces (locally finite number can be easily included) is treated in the same way, leading to a pattern with the uniform rectangular mesh adapted to each of the interfaces Γ  ,  ∈ [[1; ]] except for small (in terms of the number of impacted mesh cells) neighbourhoods of the crossing points   ,  ∈ [[1;  ]].
Proof.We make use of the fact that in Definition 4.4, we only need to consider test functions that vanish at a neighbourhood of the crossing points (this is the key observation leading to Rem. 4.6 hereabove).
(i) Proof of the entropy inequalities.
, using the appropriate partition of unity, see Section 4.1.Since  0 vanishes along all the interfaces,  Δ verifies inequality (3.12) with ℛ ≡ 0 on the domain Ω 0 and with test function  0 .Indeed, for a sufficiently small ∆ > 0, the scheme we constructed in the previous section reduces to a standard finite volume in Ω 0 .Fix now  ∈ [[1; ]].Since   vanishes along all the interfaces but Γ  ,  Δ verifies inequality (3.12) with reminder term ℛ   Δ (,   Δ ) along the trajectory   Δ on the domain Ω  and with test function   , due to the analysis of Section 3; indeed, in the support of the test function, our scheme for the multi-interface problem reduces to the scheme for the single-interface problem.By summing these previous inequalities, we obtain an approximate version of (4.

Numerical experiment with crossing trajectories
In this section, we perform a numerical test to illustrate the scheme analyzed in Section 3 and Section 4.3.We take the GNL flux  () = (1 − ).
We model the following situation.A vehicle breaks down on a road and reduces by half the surrounding traffic flow, which initial state is given by  0 = 0.8 × 1 [1,3] .At some point, a tow truck comes to move the immobile vehicle.We summarized this situation in Figure 6.Notice the time interval in which  3 ≡ 0.1.This corresponds to the time needed for the tow truck to move the vehicle.Note also that the value of the constraint on this time interval is smaller than the one when only the broken down vehicle was reducing the traffic flow.The evolution of the numerical solution is represented in Figure 7. Let us comment on the profile of the numerical solution.
-At first (0 ≤  ≤ 5.80), the solution is composed of traveling waves separated by a stationary nonclassical shock located at the immobile vehicle position.-When the tow truck catches up with the vehicle (6.30 ≤  ≤ 8.0), the profile of the numerical solution is the same, but the greater value of the constraint in this time interval changes the magnitude of the nonclassical shock; at this point the combined presence of both the tow truck and the immobile vehicle clogs the traffic flow even more.-Finally, once the tow truck starts again ( > 8.0), the traffic congestion is reduced.
Notice at time  = 7.44 the small artefact (circled in red in Fig. 7) created by Parts 2 and 3 in the construction of the approximate solution and reproduced by the scheme.This highlights the fact that even if the treatment of the crossing points brings inconsistencies or artefacts to the numerical solution, these undesired effects are not amplified by the scheme, and become negligible when one refines the mesh.

Appendix A. Proof of the OSL bound
We prove in this appendix Lemma 3.6.All the notations are taken from Sections 3.1 and 3.4.The proof is a simple rewriting of the proof of Lemma 4.2 of [25].
, so that for all  ∈ N, when  ∈ ̂︀ Z +1 , the scheme (3.4) can be rewritten as: Lemma A.1.For all  ∈ N and  ∈ Z, we have Proof.Indeed, using first the uniform convexity of  and then the CFL condition (3.2), we can write: Proof.We divide the proof in three steps.
Step Step 2. We assume that and we are going to prove that (A.3) holds.Using the uniform convexity assumption of  , we can write that A similar inequality holds for  − as well.Using (A.1), we obtain:

Figure 2 .
Figure 2. Illustration of the modification to the mesh.

Figure 4 .
Figure 4. Illustration of the configuration.

Figure 5 .𝜕
Figure 5. Illustration of the local modifications of the mesh.

Figure 6 .
Figure 6.A tow truck comes moving an immobile vehicle.

Figure 7 .
Figure 7.The numerical solution at different fixed times; for an animated evolution of the solution, follow: https://utbox.univ-tours.fr/s/YLpAgfHJHzNWYBB.
Using the monotonicity of , we get: Taking  = 0, then  = 1 in (2.2), from the condition (, ) ∈ [0, 1] a.e.we deduce that any admissible weak solution to Problem (1.3) is also a distributional solution to the conservation law   +   () = 0.If  is a regular enough solution, then for all test functions  ∈ C ∞ c (Ω),  ≥ 0, we have A. SYLLA Remark 2.2.