Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux

This paper deals with the derivation of entropy solutions to Cauchy problems for a class of scalar conservation laws with space-density depending fluxes from systems of deterministic particles of follow-the-leader type. We consider fluxes which are product of a function of the density $v(\rho)$ and a function of the space variable $\phi(x)$. We cover four distinct cases in terms of the sign of $\phi$, including cases in which the latter is not constant. The convergence result relies on a local maximum principle and on a uniform $BV$ estimate for the approximating density.


Introduction
The approximation of nonlinear transport equations via follow-the-leader type schemes has attracted a lot of attention in the recent years. As a paradigm, consider Lighthill-Whitham-Richards' equation for traffic flow [21,23] ρ t + (ρv(ρ)) x = 0, where ρ is the density of vehicles and ρ → v(ρ) is a decreasing function modelling the Eulerian velocity of vehicles. As it is well known, in this model instantaneous response to the distance to the preceding vehicle is assumed by neglecting drivers' reaction time, whereas other models [2] take the latter into account. However, (1) is considered as a reliable model in several situations, for instance with low densities, see e.g. the recent book [24] and the references therein. Both approaches in [21,23] and [2] treat the density of cars as a continuum, that is as a medium that can be divided into particles of arbitrary small mass without changing the physical nature of the system. On the other hand, the intrinsic nature of traffic flow is that of a discrete system of agents, each one of non trivial mass. Neglecting the driver's time reaction while adapting the speed to the distance to the preceding vehicle, the simplest (and most reasonable) law for the dynamics of n + 1 drivers is provided by the follow-the-leader system , for i ∈ {0, . . . , n − 1}, where x 0 (t) < . . . < x n (t) denote the positions of the n + 1 vehicles at time t, v is a given non-negative, non-increasing function on [0, +∞) with finite value v max at 0 and ℓ is the (one-dimensional) mass of each vehicle. Typically, a maximum density ρ max is prescribed in the model in order to avoid collisions, and the velocity v satisfies v(ρ max ) = 0. The vehicle x n , called 'leader', travels with maximum speed as no vehicles are ahead of it. The finite dimensional dynamical system (2) is usually coupled with n + 1 initial conditions x i (0) =x i , i = 0, . . . , n.
In a more general framework in which the dependence on ρ in the velocity term v in (1) includes possible diffusion terms, or external force fields, or nonlocal interaction terms, several results are available in the literature. We provide here a partial list of results. A probabilistic approach based on exclusion processes was developed in several works, we mention here [13,14,20]. When diffusion terms are included, we mention here the milestone results in [16,27]. System (2) is a typical example of deterministic particle system, in that no stochastic effects are considered and the position of each particle is exactly computable for all times t ≥ 0. A first attempt to detect diffusion effect via deterministic particles is due to [26]. The result in [15] extends this approach to nonlinear diffusions. A relevant recent result also involving external potentials is contained in [22]. Deterministic particle limits are also relevant in the literature of the modelling of swarming phenomena, see e.g. [5] and the references therein. 1 We observe at this stage that the quantity ℓ xi+1(t)−xi(t) in (2) has the physical dimension of a onedimensional density. Hence, the equation in (2) for i < n can be considered as a Lagrangian discrete counterpart of the continuity equation (1). This motivates the mathematical interest for (2) as a possible many-particle approximation of (1), that is in the limit as n → +∞. While this fact has been largely known in the literature in the spirit of a 'formal limit', the result in [12] proved it as a rigorous result. More precisely, the result in [12] can be be stated as follows: take an arbitrary continuum initial condition ρ ∈ L 1 (R)∩L ∞ (R) with compact support, and consider a suitable atomization of ρ 0 , for instance a set of n+1 particlesx 0 , . . . ,x n ∈ R with the property that´x i+1 xi ρ 0 (x)dx = 1/n for all i = 0, . . . , n − 1. Now, consider the (unique) solution to (2) with initial conditionx i , i = 0, . . . , n and the discrete piecewise reconstruction of the particles' density Then, ρ n converges in L 1 loc (R × R + ) as n → +∞ to the unique entropy solution (in the sense of Kružkov [19]) ρ of the scalar conservation law (1) with initial datum ρ. Such result was later extended to a larger class of initial conditions in [9].
Said results provide an abstract validation of the continuum approach (1) as a good approximation of the discrete model (2). In particular, the emergence of typical patterns such as rarefaction waves and shocksthat are easily computable in (1) -is established as a phenomenon that occurs also in the discrete setting in a 'coarse-grained' sense, that in the proper scaling regime in which the domain is large enough to include a very large number of vehicles and the total mass of the vehicles is normalized. We stress that said patterns are not detectable analytically in (2) for finite n. Moreover, this set of results is relevant also from the numerical point of view, as it allows to follow the movement of each vehicle unlike standard approaches such as classical Godunov type methods. Finally, these results hold without prescribing the initial condition to be far from the vacuum state (a restriction which would contradict the fact that the inertia-free approach of (1) is more suitable for low densities). We mention at this stage that the literature contains several results about the derivation of the second order ARZ model via deterministic follow-the-leader systems, see e.g. [1,3].
Although traffic flow is a motivating example to justify (1) as a many-particle limit for (2), the results in [12,9] hold under more general assumptions on the velocity map v: it suffices to assume that v is monotone (decreasing or increasing) and the monotonicity of v determines the proper upwind direction for the discrete density on the right-hand side of (2) (an increasing velocity v requires the use of the backward density ℓ xi(t)−xi−1(t) ). Indeed, this deterministic particle approach to solving nonlinear continuity equations was later on extended to other models. In [11] the same approach was used to approximate solutions to (1) on a bounded domain with Dirichlet type conditions. In [10] a suitable modification of (2) was proven to converge in the many-particle limit to weak solutions to the Hughes model for pedestrian movements in one space dimension. Finally, a nonlocal version of (1) was considered in [8] as the many-particle limit of a suitable variant of (2) considering nonlocal interactions with all particles.
The present paper contributes to this line of research by considering the case of a scalar conservation law with space-dependent flux where v is monotone and φ is a given external drift term depending on the position x. Besides being well motivated in the modelling context of traffic flow -for example in situations in which the speed of the vehicles is also affected by external factors (such as temporary road maintenance, or sudden turns or rises) -the equation (4) has a pretty wide range of potential applications in sedimentation processes [4], flow of glaciers [17], formation of Bose-Einstein condensates [28]. For a more general description of the applications of nonlinear scalar conservation laws we refer to [7] and the references therein.
Similarly to the approach of [12] and later results, we will assume throughout this paper that v : [0, +∞) → [0, +∞) is monotone non-increasing and non-negative, with v(0) < +∞. A symmetric result could be stated in case of a non-decreasing v, we shall omit the details. As for the potential φ, we consider four cases: (P3) xφ(x) ≥ 0 for all x ∈ R (repulsive movement); (P4) xφ(x) ≤ 0 for all x ∈ R (attractive movement). We refer to section 2 for the precise statement of all assumptions on v and φ. For each of the above four cases we shall provide an ad-hoc many-particle approximation result in the spirit of (2). For example, case (P1) requires the use of the forward follow-the-leader scheme The distinction between case (P1) and case (P2) is relevant in that it implies a change in the upwind direction of the scheme. More in detail, if φ ≤ 0 then all particles are subject to a drift directed towards the negative direction. Hence, it is reasonable to assume that each particle adjusts its speed by considering the distance to its left nearest neighbor, and the leftmost particle will be the leader travelling with v = v(0). In case (P3), the drift direction changes at the origin x = 0, with a positive direction on x ≥ 0 (non-negative φ) and a negative one on x ≤ 0 (non-positive φ). We shall refer to this case as repulsive movement, since it implies a drift of all particles away from the origin. Two leaders (leftmost and rightmost particle) will travel with speed v = v(0). Symmetrically, in case (P4) particles move towards the positive (negative respectively) direction on x ≤ 0 (on x ≥ 0 respectively). This implies that no actual 'leader' exists in the sense of the previous cases, and particles adapt their speed with respect to the relative position with their right (left respectively) nearest neighbor on x ≤ 0 (on x ≥ 0 respectively). This situation implies an attractive movement towards the origin, a phenomenon that could potentially imply collision between the two particles nearest to the origin in a finite time. To see this, consider the example v(ρ) = (1 + ρ) −1 , φ(x) = −|x| α with α ∈ (0, 1). Setting two particles at initial positions −x 0 , x 0 with x 0 > 0, one can easily show that the two particles −x(t) and x(t) obeyingẋ reach the origin in a finite time. Other significant examples originate in the continuum setting in the study of Bose-Einstein condensates, see [6], with φ(x) = −x and v(ρ) = ρ 2 , in which the finite time blow-up in L ∞ of the density is proven. In order to bypass this problem, we shall require an additional assumption for case (P4), namely that the velocity map v(ρ) vanishes at some prescribed maximal density value R max and is equal to zero on [R max , +∞). Such assumption is reasonable in contexts such as traffic flow in a single lane, in which overtaking of vehicles is not allowed. Cases (P3) and (P4) are paradigmatic of sign changing φ: more general situations in which φ changes sign at more than one point can be treated via minor modifications with the strategy outlined above, we omit the details for simplicity. In all the aforementioned four cases we are able to prove a convergence result in the spirit of [12]: given an initial conditionρ ∈ L ∞ (R) ∩ BV (R) non-negative and with compact support, we atomizeρ by a set of n + 1 particlesx 0 , . . . ,x n , we consider the piecewise constant density ρ n as in (3) with x 0 (t), . . . , x n (t) solution to a suitable follow-the-leader scheme ((5) in case (P1) as an example) with initial datumx 0 , . . . ,x n , and prove that ρ n converges locally in L 1 x,t towards the unique entropy solution to (4) withρ as initial condition. Such result requires as crucial steps: • A local maximum principle showing that ρ n L ∞ (R) is uniformly bounded with respect to n on arbitrary time intervals [0, T ]; • BV compactness estimates; • Consistency with the definition of entropy solutions (in the Kružkov's sense [19]) in the n → +∞ limit.
Such a strategy requires an L 1 ∩L ∞ setting. This is why we cannot consider case (P4) in presence of blow-up or concentration phenomena. This issue will be tackled in a future paper. The paper is structured as follows. In section 2 we define our four approximation schemes and prove their main properties, including the maximum principle for all of them. We highlight that cases (P1)-(P2)-(P3) feature a maximum principle in terms of the initial L ∞ norm, whereas in case (P4) the uniform bound for ρ n is provided in terms of the maximal density R max . In section 3 we prove the needed uniform BV estimate, as well as an equicontinuity property with respect to the Wasserstein distance that provide local L 1 compactness in space and time. Finally, in section 4 we state and prove our main result in Theorem 4.1, that collects the convergence of the scheme in all four cases.

Statement of the problem and maximum principles
Let us consider the following Cauchy problem for a one-dimensional conservation law where we assume that the function v and the initial datum ρ satisfy respectively is a non-negative, compactly supported function.
Concerning the potential φ, we shall deal with four different cases In all these cases, we assume the basic condition and in the last one we add the following condition on the function v For the sake of simplicity, we suppose that the initial mass is normalised, that is Moreover, let us denote with [x min , x max ] = Conv(supp(ρ)) the convex hull of the support of ρ. Our next goal is to provide an initial condition for the follow-the-leader systems. To perform this task, we split the interval [x min , x max ] into n sub-intervals having equal mass ℓ n := 1/n. So, for a fixed n ∈ N sufficiently large, we set x n 0 := x min , x n n := x max and we define recursively From the previous definition we immediately have that x n 0 < x n 1 < · · · < x n n and Next we introduce the follow-the-leader systems describing the evolution of the n + 1 particles with initial positions x n i , i = 0, . . . , n. The definition of the particle system depends on the cases (P1)-(P4) introduced above, hence we should introduce four different approximation schemes, nevertheless, as we will see in a moment, the latter two are strictly related to the former two. Cases (P1) and (P2) are the simplest ones, as the constant sign of φ does not affect the monotonicity of the velocity field v(ρ)φ(x). Consistently with the homogeneous case [12], when φ is non-negative the velocity field decreases with respect to the density ρ. Therefore, in case (P1) the follow-the-leader scheme should consider a forward finite-difference approximation of the density. Symmetrically, (P2) implies a backward approximation. Therefore, with the notation and in case (P2) we use For cases (P3) and (P4) we consider a sort of combination of the previous two cases. With the notation whereas in case (P4) we use For the sequel, we define the quantities which are, thanks to assumption (P), two positive constant and, moreover, we drop the n-dependence for simplicity, whenever there is no ambiguity. We remark that the Lipschitz conditions on v and φ ensure the local existence and uniqueness of solution to (8), (9), (10) and (11). In order to safeguard global existence, we need to prove three properties: a) particles have a finite position and velocity on bounded time intervals, b) particles always move in the same direction in (10) and (11), c) particles never collide, consequently they always maintain the same order.

Remark 2.1 (Finite position and velocity on bounded time intervals).
In all the four cases we have that the respective solution satisfies, for t ≥ 0 and i ∈ {0, . . . , n}, Indeed, due to assumption (P) and (V) we immediately get |ẋ i (t)| ≤ L for all t ≥ 0. Furthermore, integrating the ODE defining the evolution of x i (t), it follows that Remark 2.2 (Upper bound for the distance of two consecutive particles). In all the four cases there exists a positive constant c, depending only on φ and v, such that the corresponding solution satisfies indeed the previous remark implies that Now we prove the following technical result regarding cases (P3)-(P4).
Proposition 2.1 (Preservation of the particles' sign in cases (P3)-(P4)). Assume (V), (I) and (P) are satisfied and, moreover, assume (V * ) is satisfied in case (P4). Then, as long as the solution to (10) and (11) exists, we have Proof. Concerning the case (P3), let us show the property for a fixed i ∈ {0, . . . , k n } (for the remaining indices we can follow a symmetric reasoning): we have to prove that t * := inf{t ≥ 0 : It follows thatẋ i (t) ≤ 0 for all t < T 1 and thus Reasoning as before, we have the existence of a time T 2 > T 1 such that x i (t) ≤ 0 for all t < T 2 and hence, since we can repeat this argument arbitrarily, we get t * = +∞ as we claimed.
As for the case (P4), let us consider as before only the indices i ∈ {0, . . . , k n }. Then x i < 0 for i < k n , while x kn can be either negative or zero and, in the latter case, we remark thatẋ kn (0) = 0, so by continuity there exists T 1 > 0 such that x kn (t) = 0 for all t < T 1 . Thereforeẋ kn (t) = 0 for all t < T 1 , so we get and hence, arguing as before, by continuity we have that x kn (t) = 0 for all t ≥ 0. As a consequence, we can assume without restriction that x i < 0 and, under this assumption, by continuity there exists T > 0 such that which is a contradiction.
The next proposition ensures that particles never collide in all the four cases: this gives the global existence of the solution for (8), (9), (10) and (11). Proposition 2.2 (Discrete maximum principle). Assume that (V), (I) and (P) are satisfied.
• If one of (P1), (P2) or (P3) holds, then the respective solution to (8), (9) and (10) satisfies • If (V * ) and (P4) hold, then the solution to (11) satisfies Proof. We first observe that the statement is true for t = 0, indeed by (7) and (I) we have Now we should consider the four cases separately, but we are going to exploit in all the cases a recursive argument with suitable differences. Let us suppose (P1) holds and let us take as basis of our recursive argument the index i = n − 1. Since the first order Taylor's expansion of φ at x n−1 (t) is given by Therefore, applying Gronwall lemma, we get that and finally, since the assumption (P) implies that φ ′ is bounded, from the previous inequality we easily get (14) for i = n − 1.
Concerning the remaining indices, we argue by contradiction and assume (without restriction) the existence of an index j ∈ {0, . . . , n − 2} and of two times 0 ≤ t 1 < t 2 satisfying Using as before the first order Taylor's expansion of φ at x j (t), it holds On the other hand, from the contradictory assumption it follows that Using these estimates in (17), for all t 1 < t ≤ t 2 we get that d dt and finally, using again Gronwall lemma on the time interval (t 1 , t) with t ≤ t 2 , it follows that Concerning case (P2), we can reason in a symmetric way, taking the index i = 0 as basis of the recursive argument and assuming the existence of a 'first index j' at which the statement fails on a certain time interval (we omit the details).
In case (P3) holds, the proof of (14) for i ∈ {0, . . . , k n − 1} and i ∈ {k n + 1, . . . , n} is straightforward, indeed we can simply apply (rearranging the indices properly) the same arguments used in cases (P2) and (P1) respectively. Turning to the remaining case i = k n , we first remark that (14) is valid for i = k n − 1 and i = k n + 1. As a consequence we have and, using the first order Taylor's expansion of φ, it follows that d dt for somex(t) ∈ x kn (t), x kn+1 (t) . Finally, applying again Gronwall lemma we get and this concludes the proof for case (P3).
In case (P4), we prove (15) using a different recursive argument with respect to the previous cases. Let us first consider the indices i ∈ {0, . . . , k n − 1}, let us take as base case the index i = k n − 1 and suppose by contradiction the existence of 0 ≤ t 1 < t 2 such that Integrating the ODE in (11), for all t 1 < t ≤ t 2 it follows that where, due to the contradictory assumption, we have that R kn−1 (s) > R max and this implies, together with (V * ), that v(R kn−1 (s)) = 0. Hence we get, for and so it follows that which is a contradiction. For the remaining indices i ∈ {0, . . . , k n − 2}, we can repeat a recursive argument similar to case (P1), while the validity of (15) for i ∈ {k n + 1, . . . , n − 1} can be proved in a symmetric way with respect to the previous indices: it is sufficient to take as base case the index i = k n + 1 and then proceeding by contradiction (the details are left to the reader). As a consequence, it remains to show (15) for i = k n and for this purpose we argue again by contradiction, supposing the existence of 0 ≤ t 1 < t 2 such that On the other hand, for all t 1 < t ≤ t 2 it holds that where the contradictory assumption implies that R kn (s) > R max and hence, due to (V * ), that v(R kn (s)) = 0 for all t 1 < t ≤ t 2 . From this it follows that which is a contradiction and this concludes the proof for case (P4).
Remark 2.3. The exponential rate in (14) is not optimal: using the same strategy, we can indeed prove that Remark 2.4. The discrete maximum principle proved above allows us to improve the upper bound for the distance of two consecutive particles exposed in remark 2.2 which becomes, for t ≥ 0 and i ∈ {0, . . . , n − 1}, where c is the same as in remark 2.2, unless in case (P4) where c = 0. More precisely, since x i+1 (t)−x i (t) ≥ 0 for all t ≥ 0 and i ∈ {0, . . . , n − 1}, then it holds and so we get Finally, the fact that c = 0 in case (P4) is a consequence of the attractive movement of the two particles at the endpoints of the support.
Remark 2.5. We can slightly weaken our assumption (P) in three of the cases examined, indeed it is sufficient to take φ ∈ W 2,∞ ([x min , +∞)) in case of forward movement, φ ∈ W 2,∞ ((−∞, x max ]) in case of backward movement and φ ∈ W 2,∞ ([x min , x max ]) in case of attractive movement.
According to our construction and the previous propositions and remarks, the evolution of the n + 1 particles x i (t) is well defined for all t ≥ 0, hence we can introduce a time-depending piecewise constant density on the interval [x 0 (t), x n (t)]. Therefore, we set

BV estimate, time continuity and compactness
We first show a uniform control of the total variation of ρ n which plays a key role in the proof of the convergence of our particle scheme. In the sequel, since we are interested on large values of n ∈ N, without loss of generality we suppose n sufficiently large such that 2 ≤ k n ≤ n − 3.
Proof. We start observing that Defining for brevity and d dt TV[ρ n (·, t)] := A(t) + B(t) + C(t), with A(t) := µ 0 (t)Ṙ 0 (t), B(t) := µ n−1 (t)Ṙ n−1 (t) and C(t) := Now we need to determine an upper bound for the functions A(t), B(t), C(t) and we should treat as before the four cases separately. Let us consider the first case (P1). For A(t) and B(t) we have, due to (P) and (14), the following three sub-cases: Regarding the sum C(t), we notice that, using (8) and the definition ofṘ i (t), we can split it as .

Furthermore we remark that each
Therefore it follows that where, since the assumption (P) implies Moreover, we can rewrite the sum in the right-hand side of (21) as where we immediately remark that Turning to the remaining sum, from the second order Taylor's expansion of φ at x i+1 (t), we get Hence we can rewrite the sum as We first notice that D 2 (t) satisfies Concerning D 1 (t), since the Lagrange mean value theorem implies for someρ i,i+1,i+2 (t) in the interval (R i (t), R i+1 (t)) or (R i+1 (t), R i (t)), then from (14) it follows that Turning to the last sum D 3 (t), defining for brevity where D 1 3 (t) satisfies the same inequality as D 2 (t), while (13) implies that Putting together all the previous estimates, we finally get the following differential inequality where c i , i = 1, . . . , 5, are five positive constants depending only on φ, v and ρ.
In case (P2) we can proceed in a symmetric way and we get also here (22) (the details are omitted). Turning to case (P3), we first remark that the bounds for A(t) and B(t) are the same already pointed out in cases (P2) and (P1) respectively. Regarding C(t), we split this sum as where C 1 (t) and C 2 (t) can be estimated, rearranging the indices properly, using the same reasoning as in cases (P2) and (P1) respectively, while for C 3 (t) we need to consider three different sub-cases: , then µ kn (t) = 0 and hence C 3 (t) = 0; (2) If R kn (t) > R kn−1 (t) and R kn (t) > R kn+1 (t), then µ kn (t) = 2 and so, since φ(x kn (t)) ≤ 0 and φ(x kn+1 (t)) ≥ 0, it holds ) and therefore, since φ(x kn (t)) ≤ 0 and φ(x kn +1 (t)) ≥ 0, we get As a consequence, we hence have that (22) is valid also in case (P3). In the last case (P4), we can estimate A(t) and B(t) in the same way as in cases (P1) and (P2) respectively, substituting Re L ′ t with R max whenever it appears. Concerning C(t), we can rewrite this sum using the same splitting (23) seen in case (P3), where C 3 (t) now satisfies C 3 (t) ≤ 2L ′ R max . For C 1 (t) and C 2 (t), rearranging the indices properly and substituting again Re L ′ t with R max , we can follow the same reasoning as in cases (P1) and (P2) respectively: in this way, we obtain all the previous estimates with a slight difference only on the term D 2 3 (t). For this sum it holds, due to remark 2.4, that and therefore we have that (22) is valid in this case with c 4 = c 5 = 0. Applying now Gronwall lemma to (22), we get where, since (7) and the mean value theorem imply then the total variation of ρ n (·, 0) satisfies Moreover, after a simple calculation it follows that therefore, combining together the previous three estimates in (24), we finally get (20).
Remark 3.1. The previous proposition gives us the needed compactness of the sequence {ρ n } n∈N with respect to space. Concerning the time variable, we are not able to obtain a L 1 uniform continuity estimate which would provide a sufficient control on the time oscillation. Nevertheless, we are going to prove a uniform time continuity estimate with respect to the 1-Wasserstein distance which is sufficient to get the required strong L 1 compactness with respect to space and time.
We now recall the main properties on the one-dimensional 1-Wasserstein metric. Let µ be a probability measure on R with finite first moment, and let us denote with X µ the pseudo-inverse of its cumulative distribution function, that is We first notice that X µ ∈ L 1 ([0, 1]) (see [29]). Moreover, the 1-Wasserstein distance between two probability measures µ and ν on R can be defined as the L 1 distance of X µ and X ν , that is In particular, starting from the definition (19) of ρ n , we can explicitly compute the pseudo-inverse function X ρ n and we get After this short preamble, we can prove the uniform time continuity estimate with respect to the 1-Wasserstein distance, which is stated in the following Proof. Let 0 < s < t fixed. From (25), (26) and the triangular inequality, it follows that A(s, t) and B(s, t) separately. Due to (12), we have that

Now we should estimate
while, turning to B(s, t), we first notice that For clarity we should treat the four cases separately from now on. In case (P1), substituting the definition ofṘ i (t) we get where, for all τ ≥ 0 and i ∈ {0, . . . , n − 2}, it holds This implies that B(s, t) ≤ L 2 (t − s) and hence that W 1 ρ n (·, t), ρ n (·, s) ≤ 3 2 L(t − s), which concludes the proof of (27) in case (P1), since the calculation is still valid interchanging s and t.
In case (P2) we can reason in a symmetric way with respect to the previous case and we get the same estimate for B(s, t). The details are left to the reader.
Before passing to the main result of this paper, we recall a generalised version of Aubin-Lions lemma (see [25], [12], [9]) which has a key role in the sequel. Furthermore, let us report an adapted version of the L 1 contraction property proved by Karlsen and Risebro in [18], which will be crucial in the proof of the uniqueness of the entropy solution to (6).
Theorem 3.2. Let T > 0 fixed arbitrarily, let f a locally Lipschitz function on R, let ψ ∈ W 1,1 loc (R) ∩ C(R) such that ψ, ψ ′ ∈ L ∞ (R) and consider the problem If u, v ∈ L ∞ ((0, T ); BV (R)) are two entropy solutions of (28) with respective initial datum u 0 and v 0 , both in L 1 (R) ∩ L ∞ (R) ∩ BV (R), then, for almost every t ∈ (0, T ), it holds In particular, this implies the existence of at most one entropy solution of (28).

Proof of the main results
Now we are ready to prove the main result of the paper. For clarity, in the sequel we drop the time dependence whenever it is clear from the context. Proof. We first show that {ρ n } n∈N converges, up to a subsequence, almost everywhere and in L 1 on R×[0, T ]. We notice that the support of ρ n (·, t) is contained, for every n ∈ N and t ∈ [0, T ], in the closed interval Therefore, taking as I any open interval of the type (a − c, b + d) with c and d arbitrary positive constants, we can apply theorem 3.1, indeed assumption (A) is valid due to the above construction, while assumptions (B) and (C) are a direct consequence of propositions 3.1 and 3.2 respectively. As a result, it follows that {ρ n } n∈N converges, up to a subsequence that we still denote in the sequel with {ρ n } n∈N , almost everywhere and in L 1 on R × [0, T ] to a certain function ρ. Now we show that {ρ n } n∈N satisfies, for every k ≥ 0 and every non- where we denote f (η) := ηv(η). Let us omit from now on also the x dependence whenever it is clear from the context. We first remark that, since supp[ϕ] is compact in R × (0, T ), then it holdŝ Recalling that d dtˆβ and analogously that 15 while we can rewrite A as where xi φϕ x dx dt and Integrating by parts and since supp[ϕ(x, ·)] ⊆ (0, T ) for every x ∈ R, we get that A 1 satisfies while A 2 has a different expression in the four cases, since we need to approximate the function φ differently according to its sign. In particular we have: (P1) For case (P1), using the first order Taylor's expansion of φ at x i in the interval (x i , x i+1 ), which is given, for all x ∈ (x i , x i+1 ), by we can rewrite A 2 as (P2) In case (P2), we use instead the first order Taylor's expansions of φ at x i+i , that is and in this way we get that (P3) For case (P3), we use the Taylor's expansions (30) and (31) of φ respectively for i ∈ {k n +1, . . . , n−1} and i ∈ {0, . . . , k n − 1}, in order to have that (P4) In case (P4), we combine again cases (P1) and (P2), namely we use the Taylor's expansions (30) and (31) of φ respectively for i ∈ {0, . . . , k n − 1} and i ∈ {k n + 1, . . . , n − 1}. Then it follows that From now on, let us consider the four cases separately. Putting together all the previous identities, we get that in case (P1) it holdŝ Moreover, using the definition ofṘ i (t), the previous identity becomeŝ Regarding E 1 , since ϕ(x, ·) is a Lipschitz function for every x ∈ R and due to (18), we have that while, using again the Lipschitz condition of ϕ and due to (18) and (20), we get that E 2 satisfies Putting together all the previous estimates, we hence get that E 1 + E 2 ≥ −c ℓ n for some constant c independent on n and so, since the right-hand side of this inequality tends to zero as n → +∞, to conclude the proof it is sufficient to show that D is non-negative. From a direct calculation we remark that therefore D can be rewritten as where and , . . . , n − 1}. Concerning D 0 and D n , we have two different sub-cases, namely Turning to D i with i ∈ {1, . . . , n − 1}, since v is non-increasing and φ, ϕ are non-negative, after a simple calculation we get that D i = 0 if R i−1 > k and R i > k or R i−1 < k and R i < k or R i = k, ≥ 0 otherwise and this concludes the proof of (29) in case (P1). For case (P2), using instead the first order Taylor's expansion (31), we can proceed in a symmetric way to get the validity of (29) also in this case (the details are left to the reader).
Let us now consider the case (P3). Proceeding as in the previous cases we get x kn φ(x)ϕ x (x)dx dt.
Turning to the last case (P4), we combine again cases (P1) and (P2), namely we use the Taylor's expansions (30) and (31) of φ respectively for i ∈ {0, . . . , k n − 1} and i ∈ {k n + 1, . . . , n − 1}. This implies, arguing as before, that T 0ˆR |ρ n − k|ϕ t + sign(ρ n − k) f (ρ) − f (k) φϕ x − sign(ρ n − k)f (k)φ ′ ϕ dxdt := kD + E 4 1 + E 4 2 + F, while F is the same term defined in case (P3). Arguing as in the previous cases, we can prove that E 4 1 and E 4 2 satisfy respectively (32) and (33), so it follows that E 4 1 + E 4 2 + F ≥ −c ℓ n for some constant c independent 20 on n and hence to conclude the proof we need to show as before that D is non-negative. Using again (34) we get that and hence it follows that D ≥ 0 also in this last case, since we can estimate each term as we did before. Now it remains to prove that ρ satisfies the entropy condition, that iŝ