Deterministic particle approximation for nonlocal transport equations with nonlinear mobility
M. Di Francesco S. Fagioli E. Radici
math.AP







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definition[theorem]Definition
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Particle approximation for nonlocal transport equations]Deterministic particle approximation for nonlocal transport equations with nonlinear mobility

M. Di Francesco]Marco Di Francesco
Marco Di Francesco - Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell'Aquila. Via Vetoio 1, 67100, Coppito (L’Aquila), Italy
marco.difrancesco@univaq.it


S. Fagioli]Simone Fagioli
Simone Fagioli - Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell'Aquila. Via Vetoio 1, 67100, Coppito (L’Aquila), Italy
simone.fagioli@univaq.it


E. Radici]Emanuela Radici
Emanuela Radici - Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell'Aquila. Via Vetoio 1, 67100, Coppito (L’Aquila), Italy
emanuela.radici@univaq.it






We construct a deterministic, Lagrangian many-particle approximation to a class of nonlocal transport PDEs with nonlinear mobility arising in many contexts in biology and social sciences. The approximating particle system is a nonlocal version of the follow-the-leader scheme. We rigorously prove that a suitable discrete piece-wise density reconstructed from the particle scheme converges strongly in L^1_loc towards the unique entropy solution to the target PDE as the number of particles tends to infinity. The proof is based on uniform BV estimates on the approximating sequence and on the verification of an approximated version of the entropy condition for large number of particles. As part of the proof, we also prove uniqueness of entropy solutions. We also provide a specific example of non-uniqueness of weak solutions and discuss about the interplay of the entropy condition with the steady states. Finally, we produce numerical simulations supporting the need of a concept of entropy solution in order to get a well-posed semigroup in the continuum limit, and showing the behaviour of solutions for large times.


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    January 26, 2018
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§ INTRODUCTION


A wide range of phenomena in biology and social sciences can be described by the combination of classical (local) - linear or nonlinear - diffusion with some nonlocal transport effects. Examples can be found in bacterial chemotaxis <cit.>, animal swarming phenomena <cit.>, pedestrian movements in a dense crowd <cit.>, and more in general in socio-economical sciences <cit.>. In a fairly general setting, a set of N individuals x_1,…,x_N located in a sub-region of the Euclidean space ^d are subject to a drift which is affected by the status of each other individual. In most of the above-mentioned applications, such a “biased drift” can be expressed through a set of first order ordinary differential equations

    ẋ_i(t)=v[(x_1(t),…,x_N(t)],     i=1,…,N,

in which the velocity law v is known. Having in mind a particle system obeying the laws of classical mechanics or electromagnetism, the set of equations (<ref>) is quite unconventional due to the absence of inertia. On the other hand, this choice is very common in the modelling of socio-biological systems, mainly due to the following three reasons. 


  * Inertial effects are negligible in many socio-biological aggregation phenomena. Even in cases in which the system is appropriate for a fluid-dynamical description, a `thinking fluid' model, with a velocity field already adjusted to equilibrium conditions, is often preferable compared to a second order approach. The typical examples are in traffic flow and pedestrian flow modelling. Moreover, it is well known in the context of cells aggregation modelling that the time of response to the chemoattractant signal is, most of the times, negligible. Finally, inertia is almost irrelevant in many contexts of socio-economical sciences, such as opinion formation dynamics.

  * First-order modelling turns out to simulate real patterns in concrete relevant situations arising in traffic flow, pedestrian motion, and cell-aggregation, and such an achievement is satisfactory in many situations, in applied fields often lacking a unified rigorous modelling approach.

  * In several practical problem such as the behaviour of a crowd in a panic situation, the model can be seen as the outcome of an optimization process performed externally, in which the "best strategy" needed to solve the problem under study (reaching the exit in the shortest possible time, in the crowd example) is transmitted to the individuals in real time (e.g. a set of “dynamic" evacuation signals in a smart building).


Further to the `discrete' approach (<ref>), these models are often posed in terms of a “continuum" PDE approach via a continuity equation

    ∂_t ρ + div(ρ v[ρ]) = 0,

in which ρ(·,t) is a time dependent probability measure expressing the distribution of individuals on a given region at a given time, and in which the continuum velocity map v=v[ρ] is detected as a reasonable “cross-grained" version of its discrete counterpart in (<ref>). The modelling of biological movements and socio-economical dynamics are often simulated at the continuum level as the PDE approach is more easy-to-handle in order to analyse the qualitative behaviour of the whole system, in the form e.g. of the emergence of a specific pattern, or the occurrence of concentration phenomena, or the formation of shock waves or travelling waves. In this regard, the descriptive power of the qualitative properties of the solutions in the continuum setting is an argument in favour of the PDE approach (<ref>). On the other hand, the intrinsic discrete nature of the applied target situations under study would rather suggest an `individual based' description as the most natural one. For this reason, the justification of continuum models (<ref>) as a many particle limits of (<ref>) in this context is an essential requirement to validate the use of PDE models.

As briefly mentioned above, the velocity law v=v[ρ] in the PDE approach (<ref>) may include several effects ranging from diffusion effect to external force fields, from nonlinear convection effects to nonlocal interaction terms. We produce here a non-exhaustive list of results available in the literature in which the continuum PDE (<ref>) is obtained as a limit of a system of interacting particles, with a special focus on deterministic particle limits, i.e. in which particles move according to a system of ordinary differential equations (i.e. without any stochastic term). The presence of a diffusion operator has several possible counterparts at the discrete level. The literature on this subject involving probabilistic methods is extremely rich and, by now, well established, see e.g. <cit.> only to mention a few. A first attempt (mainly numerical) to a fully deterministic approach to diffusion equations is due to <cit.>, see <cit.> for the case of nonlinear diffusion. 

Without diffusion and with only a local dependency v=v(ρ), an extensive literature has been produced based on probabilistic methods (exclusion processes), see e.g. <cit.>. A first rigorous result based on fully deterministic ODEs at the microscopic level for a nonlinear conservation law was recently obtained in <cit.>. Nonlocal velocities v=W∗ρ have been considered as a special case of the theory developed in <cit.>, with W a given kernel (possibly singular) using techniques coming from kinetic equations, see <cit.>. In all the above mentioned results, the particle system is obtained as a discretised version of the Lagrangian formulation of the system.

A slightly more difficult class of problems is the one in which the velocity v=v[ρ] depends both locally and non-locally from ρ.
Several results about the mathematical well-posedness of such models are available in the literature, which use either classical nonlinear analysis techniques or numerical schemes. In the paper <cit.> a similar model is studied in the context of pedestrian movements, and the existence and uniqueness of entropy solutions is proven. We also mention <cit.>, which covers a more general class of problems, and <cit.> covering a similar model in the context of granular media. A quite general result was obtained in <cit.> in which the velocity map ρ↦ v[ρ] is required to be Lipschitz continuous as a map from the space of probability measures (equipped with some p-Wasserstein distance) with values in C(^d), and the authors prove convergence of a time-discretised Lagrangian scheme. We also mention <cit.>, in which a special class of local-nonlocal dependencies has been considered, however in a different numerical framework. We also recall at this stage the related results in <cit.> on the overcrowding preventing version of the Keller-Segel system for chemotaxis, in which the existence and uniqueness of entropy solutions is proven. To our knowledge, no papers in the literature provide (so far) a rigorous result of convergence of a deterministic particle system of the form (<ref>) towards a PDE of the form (<ref>) in the case of local-nonlocal dependence v=v[ρ]. Indeed, the result in <cit.> does not apply to this case in view of the Lipschitz continuity assumption on the velocity field, see also a similar result in <cit.>.

In this paper we aim at providing, for the first time, a rigorous deterministic many-particle limit for the one-dimensional nonlocal interaction equation with nonlinear mobility

    ∂_t ρ -∂_x (ρ v(ρ) K∗ρ) = 0,

in which v and K satisfy the following set of assumptions:


  (Av) v ∈ C^1([0,+∞)) is a decreasing function such that v(0)=v_max>0, v(M)=0 for some M>0, v'<0 on interval (0,M], v≡ 0 on [M,+∞).

  (AK) K ∈ C^2(), K(0)=0 (without restriction), K(x)=K(-x) for all x∈, K'(x)>0 for x>0, K”∈Lip_loc().

Also in view of the applications in mind, the unknown ρ=ρ(x,t) in (<ref>) will be assumed to be non-negative throughout the whole paper. The PDE (<ref>) is coupled with an initial condition

    ρ(x,0)=ρ̅(x),    ρ̅∈ L^∞()∩ BV(),   0≤ρ̅(x)≤ M,   supp(ρ̅) compact.

The constant M here plays the role of a maximal density, which is supposed not to be exceeded by the density for all times. Clearly, the property ρ∈ [0,M] has to be proven to be invariant with respect to time. We notice that the total mass of ρ in (<ref>) is formally conserved. For simplicity, throughout the paper we shall set
 We set

    ρ̅_L^1()=1 .

We set [x̅_min, x̅_max] as the closed convex hull of suppρ̅.

Our goal is to approximate rigorously the solution ρ to (<ref>) with initial datum ρ̅ via a set of moving particles. More precisely, we aim to proving that the entropy solution of the Cauchy problem for (<ref>) can be obtained as the large particle limit of a discrete Lagrangian approximation of the form (<ref>). Such a Lagrangian approximation can be introduced as follows as a reasonable generalization of particle approximations considered previously in the literature in <cit.>. For a fixed integer N sufficiently large, we split [x̅_min, x̅_max] into N intervals such that
the integral of the restriction of ρ̅ over each interval equals 1/N. More precisely, we let x̅_0= x̅_min and x̅_N = x̅_max, and define recursively the points x̅_i for i ∈{ 1, …,  N-1} as

    x̅_i = sup{ x ∈ : ∫_x̅_i-1^x ρ̅(x) dx < 1/N} .

It is clear from the construction that ∫_x̅_N-1^x̅_Nρ̅(x) dx = 1/N and x̅_0 < x̅_1 < …  < x̅_N-1 < x̅_N.
Consider then N+1 particles located at initial time at the positions x̅_i and let them evolve accordingly to the following system ODEs

    ẋ_i(t) =   - v(R_i(t))/N∑_j > i K'(x_i(t) - x_j(t)) - v(R_i-1(t))/N∑_j < i K'(x_i(t) - x_j(t)) ,

with i∈{0,…,N}, where the discrete density R_i(t) is defined as follows

    R_i (t):= 1/N(x_i+1(t) - x_i(t)),     i=0,…, N-1.

In (<ref>), each particle x_i has mass 1/N. We are then in position to define the N-discrete density

    ρ^N(t, x):= ∑_i=0^N-1  R^N_i (t) χ_[x_i(t), x_i+1(t))(x).

We observe that ρ^N(t,·) has total mass equal to 1 for all times. We refer to system (<ref>) as non-local Follow-the-leader scheme, as in fact this system is a non-local extension of the classical Follow-the-leader scheme previously considered in the literature. More in detail, system (<ref>) is motivated as follows. The right-hand side of (<ref>) represents the velocity of each particle. Therefore, it has to be reminiscent of a discrete Lagrangian formulation of the Eulerian velocity -v(ρ)K'∗ρ in the continuity equation (<ref>). Now, since we are in one-space dimension, the discrete density R_i is a totally reasonable replacement for the continuum density ρ, except that one has to decide whether the discrete density should be constructed in a forward, backward, or centred fashion. Our choice of splitting the velocity ẋ_i into a backward and forward term is motivated by the sign of the nonlocal interaction K'(x_i-x_j), which is concordant with the sign of x_i - x_j. Hence, since K'(x) is negative on x<0, particles labelled by x_j with x_j>x_i yield a drift on x_i oriented towards the positive direction. Since the role of the nonlinear mobility term ρ v(ρ) is that of preventing overcrowding at high densities (consistently with the assumption of v being monotone decreasing), such a drift term should be “tempered" by the position of the (i+1)-th particle. This motivates the use or v(R_i) in the sum with x_j>x_i. A symmetric argument justifies the use of v(R_i-1) in the remaining part of the sum with x_j<x_i.

Our main results concerns with the study of the many particle limit as N →∞ for the discrete density ρ^N defined above. Apart from the above mentioned assumptions on v and K and ρ̅, we shall also assume that ρ̅∈ BV(). Such a condition is crucial in order to prove the needed estimate which guarantee that ρ^N converge (up to a subsequence) to some limiting density ρ in a strong enough topology. As a minimal requirement, the limit ρ should satisfy (<ref>) in a distributional sense. On the other hand, the presence of a nonlinear convection in (<ref>) suggests the possibility of multiple weak solutions for fixed initial data. A notion of entropy solution in the sense of Kruzkov <cit.> is therefore needed to secure uniqueness. Motivated by this remark, we shall actually prove that the limit density ρ of the above particle scheme is an entropy solution to (<ref>) with initial condition ρ̅, in the sense of the following definition.


Let ρ̅∈ L^∞()∩ L^1_+(). Denoting f(z):= zv(z), we say that ρ:[0,+∞)×→ [0,+∞) is an entropy solution of (<ref>) with initial condition ρ̅ if ρ∈ L^∞([0,∞), L^1(,[0,1])) and, for all constants c≥ 0 and for all φ∈𝒞^∞_c ([0,+∞)×) with φ≥ 0 one has

    0 ≤∫_ |ρ̅(x) -c|φ(0,x) dx+∫_0^+∞∫_ |ρ -c|φ_t - sign(ρ - c)[(f(ρ) - f(c))K' ∗ρφ_x - f(c)K”∗ρφ] dxdt.



We are now ready to state the main result of our paper.


Assume v and K satisfy (Av) and (AK) respectively. Let ρ̅∈ BV()∩ L^1_+() be a compactly supported function with total unit mass and such that ρ̅≤ M.
Then, for all T>0, the discrete density ρ^N constructed in (<ref>) converges almost everywhere and in L^1([0, T] ×) to the unique entropy solution ρ of the Cauchy problem

    {[ ∂_t ρ = ∂_x(ρ v(ρ) K' ∗ρ)       (t, x) ∈ (0, T] × ,;           ρ(0, x) = ρ̅(x)                     x ∈ . ].



As a by-product, the above result also imply existence of entropy solutions for (<ref>), a task which has been touched in other papers previously <cit.>.
Implicitly, our results also asserts the uniqueness of entropy solutions for (<ref>), a side result that we shall prove as well in the paper, similarly to what done in <cit.>.

The need of the entropy condition to define a suitable notion of solution semigroup for (<ref>) is not only motivated by the possibility of proving its uniqueness. We actually prove in the paper that a mere notion of weak solution does not infer the well-posedness of the semigroup as multiple weak solution can be produced with the same initial condition. 

Our paper is structured as follows. In Section <ref> we introduce the nonlocal follow-the-leader particle scheme and prove that it satisfies a discrete maximum principle, a crucial ingredient in order to deal with the particle approximation in the sequel of the paper. In Section <ref> we prove all the estimates needed in order to detect strong L^1 compactness for the approximating sequence ρ^N. The main ingredient of this section is the BV estimate proven in Proposition <ref>. We emphasize that the presence of an attractive interaction potential in the particle system implies most likely a growth w.r.t. time of the total variation. Therefore, one has to check that the blow-up in finite time of the total variation is avoided. In Section <ref>, we prove that the limit of the approximating sequence is an entropy solution in the sense of Definition <ref>. This task is quite technical as it requires checking a discrete version of Kruzkov's entropy condition. In Section <ref> we provide an explicit example of non uniqueness of weak solutions, which has links with the admissibility of steady states. Finally, in Section <ref> we complement our results with numerical simulations.



§ THE NON-LOCAL FOLLOW-THE-LEADER SCHEME


In this section we introduce and analyse in detail our approximating particle scheme (<ref>). Here the macroscopic variable ρ does not need to be labelled by N, as N is supposed to be fixed throughout the whole section. The regularity assumptions on v and K in (Av) and (AK) imply that the right-hand side of (<ref>) is locally Lipschitz with respect to the N+1-tuple (x_0,x_1,…,x_N) as long as we can guarantee that the denominator in R_i does not vanish. Such a property is a consequence of the following  Discrete Maximum Principle, ensuring that the particles cannot touch each other at any time. This implies both the (global-in-time) existence of solutions of the system (<ref>) for all times t>0, and the conservation of the initial particle ordering during the evolution.


Let N∈ be fixed and assume that (Av) and (AK) hold. In particular, let M>0 be as in assumption (Av). Let x̅_0<x̅_1<…<x̅_N be the initial positions for (<ref>), and assume that

    x̅_i+1-x̅_i ≥1/MN

Then every solution x_i(t) to the system (<ref>) satisfies

    1/MN≤  x_i+1(t) - x_i(t)     for all i ∈{0, …, N-1} and for all t ∈ [0, +∞).

Consequently, the unique solution (x_0(t),…,x_N(t)) to (<ref>) with initial condition (x̅_0,…,x̅_N) exists globally in time.



Let T_max>0 be the maximal existence time for (<ref>). Due to the assumptions (Av) and (AK), the local-in-time solution (x_0(t),…,x_N(t)) is C^1 on [0,T_max). If we prove that (<ref>) holds on [0,T_max), this will automatically prove global existence by a simple continuation principle. Arguing by contradiction, assume that t_1< T_max is the first instant where two consecutive particles are the distance 1/MN and get closer afterwards, i.e.

    t_1=inf{ t ∈ [0, T] :   there exists  i  : x_i+1(t) - x_i(t) = 1/MN },

and there exists t_2 ∈ (t_1, T] such that

    x_i+1(t) - x_i(t) < 1/MN    ∀ t ∈ (t_1,  t_2] .

Notice that the minimality of t_1 ensures that all particles maintain their initial order for all t ∈ [0, t_1). At time t_1 we have R_i(t_1)=0 due to (Av). Substituting this value in the equation (<ref>) for x_i, we easily see that only the terms j<i survive in the nonlocal part, thus yielding ẋ_i(t_1) ≤ 0. Similarly, we get ẋ_i+1(t_1) ≥ 0. For similar reasons, if ẋ_i+1(t_1) = 0 then the ODE for x_i+1 implies that at time t_i we have R_i+1(t_1)=M, or equivalently x_i+2(t_1) - x_i+1(t_1)= 1/MN. Similarly, if ẋ_i(t_1)=0 then x_i(t_1) - x_i-1(t_1)= 1/MN. Let us now assume for the moment that x_i+2(t_1) - x_i+1(t_1)= x_i(t_1) - x_i-1(t_1) = 1/MN. Then, with similar arguments as above one can show that ẋ_i-1(t_1)≤ 0 and ẋ_i+2(t_1)≥ 0, and we can repeat the same argument above to obtain that ẋ_i-1(t_1)= 0 implies x_i-1(t_1) - x_i-2(t_1) = 1/MN and ẋ_i+2(t_1)=0 implies x_i+3(t_1) - x_i+2(t_1)= 1/MN. Such a procedure can be iterated to conclude that there exists either some index k≥ i with ẋ_k+1(t_1)>0 or some index h≤ i such that ẋ_k(t_1)<0, otherwise any two consecutive particles would be placed at distance 1/MN and the system would be static for all t ∈ (t_1, T], which would contradict the existence of t_2.

The above considerations imply that we can assume, without loss of generality, that

    ẋ_i+1(t_1) >0,    ẋ_i(t_1) ≤ 0 .

Let _i+1>0 be small enough such that t_1+ _i+1 < t_2, then by Taylor expansion one has

    x_i+1(t) = x_i+1(t_1) +  ẋ_i+1(t_1)(t-t_1) + o(|t-t_1|) ,

where, up to taking _i+1 even smaller, the contribute o(t-t_1) does not affect the sign of ẋ_i+1(t_1)(t-t_1). As a consequence, x_i+1(t) > x_i+1(t_1) for all t∈ (t_1,t_1 +_i+1) and a symmetric argument gives also x_i(t) ≤ x_i(t_1) for all t∈ (t_1,t_1+_i). In particular, we deduce that

    x_i+1(t) - x_i(t) ≥  x_i+1(t_1) - x_i(t_1) = 1/MN  ∀ t ∈ (t_1,  t_1 + min{_i, _i+1})

and this contradicts the existence of t_2. This argument ensures both the validity of (<ref>) and the existence of solutions for all times t>0.


Let us consider the discrete density

    ρ(t, x):= ∑_i=0^N-1  R_i (t) χ_[x_i(t), x_i+1(t))(x).

A straightforward consequence of Lemma <ref> is that

    ρ(t,x)≤ M    for all (t,x)∈ [0,+∞)×.

Moreover, we observe that ρ has unit mass on  for all times.

As already mentioned before, a straightforward consequence of the above Maximum Principle is that the particles can never touch or cross each other. In particular, the particle x_0 will have no particles at its left for all times, which means that the ODE for x_0 will only feature terms with j>0 on the nonlocal sum. A symmetric statement holds for x_N. As a consequence of that ẋ_0(t) ≥ 0 and ẋ_N(t) ≤ 0 for all t, thus the support of ρ^N(t, ·) is bounded by ℓ uniformly in N and t.
We summarize this property in the next lemma.


Under the same assumptions of Lemma <ref>, the support of ρ(t,·) is contained in the interval [x̅_0,x̅_N] for all times t∈ [0,+∞).




§ CONVERGENCE OF PARTICLE SCHEME


We now focus on the converge of the particle scheme (<ref>), where the initial condition (<ref>) is constructed from an L^∞() initial density ρ̅ having compact support and finite total variation.

The proof of Theorem <ref> relies on two main steps: the first one consists in proving that the discrete density (ρ^N) defined in (<ref>) is strongly convergent (up to a subsequence) to a limit ρ in L^1([0,T] ×), the second one is to show that the limit ρ is a weak entropy solution of (<ref>) according to Definition <ref>. In this section we take care of the former step. As we will show in Propositions <ref> and <ref> below, the sequence (ρ_N)_N∈ satisfies good compactness properties with respect to the space variables but, on the other hand, we cannot reach a uniform L^1 control on the time oscillations. In our case, we are only able to prove a uniform time continuity estimate with respect to the 1-Wasserstein distance (see <cit.>), which nevertheless will suffice to achieve the required compactness in the product space. Such a strategy recalls the one used in <cit.> for the case of a scalar conservation law. The main result of this section is the content of the following


Under the assumptions of Theorem <ref>, the sequence ρ^N is strongly relatively compact in L^1([0,T] ×)


The proof of Theorem <ref> relies on a generalized statement of the celebrated Aubin-Lions Lemma (see <cit.>) that we recall here for the reader's convenience. In what follows, d_1 is the 1-Wasserstein distance.


Let τ>0 be fixed. Let η^N be a sequence in L^∞((0, τ); L^1()) such that
η^N(t, ·) ≥ 0 and η^N(t, ·) _L^1()=1 for every N∈ and t∈ [0, τ].
If the following conditions hold


  I) sup_N∫_0^τ[η^N(t, ·)_L^1()dt + TV[ η^N(t, ·)]+ meas(supp[η^N(t,·)])]dt < ∞,

  II) there exists a constant C>0 independent from N such that d_W^1( η^N(t, ·), η^N(s, ·) ) < C |t-s| for all s, t ∈ (0, τ),

then η^N is strongly relatively compact in L^1([0, τ]×).


In view of Theorem <ref>, the result in Theorem <ref> will follow as a consequence of the following two propositions.


Let ρ̅, v, K and T be as in the statement of Theorem <ref>. Then, there exists a positive constant C>0 (only depending on K, v, and on supp(ρ̅)) such that for every N ∈ one has

    TV[ρ^N(t, ·)] ≤ TV[ρ̅] e^Ct    for all t ∈ [0,T] .




Let ρ̅, v, K and T be as in the statement of Theorem <ref>. Then, there exists a positive constant C>0 (only depending on K) such that

    d_W^1( ρ^N(t, ·), ρ^N(s, ·) ) < C |t-s|   for all s, t ∈ (0, T), and for all N ∈ .


















The remaining part of this section is devoted to prove Propositions <ref> and <ref>.
For future use we compute

    Ṙ_i(t) =    -N(R_i)^2 (ẋ_i+1 - ẋ_i) = -N(R_i)^2 [ -2v(R_i)1/NK'(x_i+1 - x_i) 
        -(v(R_i+1) - v(R_i))1/N∑_j > i+1 K'(x_i+1 - x_j)
        - v(R_i) 1/N∑_j > i+1( K'(x_i+1 - x_j) - K'(x_i - x_j) ) 
         -(v(R_i) - v(R_i-1))1/N∑_j<i K'(x_i - x_j) - v(R_i)1/N∑_j<i(K'(x_i+1 - x_j) - K'(x_i - x_j) ) ] .






Proposition <ref>
It is easy to see that TV[ρ^N(0, ·)] ≤ TV[ρ̅]. Then estimate (<ref>) follows by Gronwall Lemma as soon as we show that

    d/dt TV[ρ^N(t, ·)] ≤ C  TV[ρ^N(t, ·)],

for a suitable constant C>0. The total variation of ρ^N at time t is given by

    T   V[ρ^N(t, ·)] = R_0(t) + R_N(t) + ∑_i=0^N-1 |R_i+1(t) - R_i(t)| 
       =∑_i=1^N-1R_i [sign(R_i - R_i-1) - sign(R_i+1-R_i)] - R_0 (sign(R_1-R_0) -1)
         + R_N( sign(R_N - R_N-1) + 1) 
       =μ_0(t)R_0(t)+μ_N(t)R_N +  ∑_i=1^N-1R_i μ_i,

where we set for brevity

    μ_i(t):=sign(R_i(t) - R_i+1(t)) - sign(R_i-1(t) - R_i(t))     i=1,…,N-1,
       μ_0(t)= ( 1- sign(R_1 -R_0) ),
       μ_N(t)=(1+ sign(R_N -R_N-1) ).

Then we can compute

    d/dt TV[ρ^N(t, ·)]    = Ṙ_0(t) +Ṙ_N(t) + ∑_i=0^N-1sign(R_i+1(t) - R_i(t))( Ṙ_i+1(t) - Ṙ_i(t) ) 
       = μ_0(t)Ṙ_0(t) + μ_N(t) Ṙ_N(t) + ∑_i=1^N-1μ(R_i(t))Ṙ_i(t) .

The value of the coefficient μ_i(t) clearly depends on the positions of the consecutive particles, it is easy to see that for i ∈{ 1, …, N-1}

    μ_i(t)= {[ -2  ,;  2  ,;  0    ].

moreover

    μ_0(t)={[ 0; 2 ].
        μ_N(t)= {[ 0; 2 ].

Recalling (<ref>), we can rewrite

    d/dt TV[ρ^N(t, ·)] = μ_0(t)Ṙ_0(t) + μ_N(t) Ṙ_N(t) - ∑_i=1^N-1μ_i(t)(R_i(t))^2 I_i -  ∑_i=1^N-1μ_i(t)R_i(t) II_i ,

where

    I_i = -(v(R_i+1(t)) - v(R_i(t)))∑_j > i+1 K'(x_i+1(t) - x_j(t)) - (v(R_i(t)) - v(R_i-1(t)))∑_j < i K'(x_i(t) - x_j(t)) ,

and

    II_i = -R_i(t)v(R_i(t)) ∑_j ≠ i, i+1( K'(x_i+1(t) - x_j(t)) - K'(x_i(t) - x_j(t)) )
           - 2R_i(t)v(R_i(t))K'(x_i+1(t)-x_i(t)) .



Let us first estimate -∑_i=1^N-1μ_i(t)(R_i(t))^2I_i in (<ref>). Clearly, the only relevant contributions in the sum come from the particles x_i for which μ_i(t)≠ 0. However, if the index i is such that μ_i(t)=-2, then R_i+1, R_i-1 > R_i and the monotonicity of v implies

    v(R_i+1(t)) - v(R_i(t))  <0 ,      v(R_i(t))-v(R_i-1(t)) >0 .

The assumption (AK) on K ensures that I_i < 0, thus, on the other hand, μ_i(t)(R_i (t))^2 I_i  <0.
An analogous argument implies that, if i such that μ_i(t)=2, then I_i > 0 and 2(R_i (t))^2 I_i  >0. These considerations lead immediately to

    -∑_i=1^N-1μ_i(t)(R_i(t))^2I_i < 0 .

Let us now focus on -∑_i=1^N-1μ_i(t)R_i(t)II_i. In this case, we would like to obtain an upper bound in terms of TV[ρ^N(t, ·)] and for this purpose we need to estimate | II_i |. We recall that K' is locally Lipschitz and that v(ρ)∈ [0,v_max]. The former in particular implies that K' has finite Lipschitz constant on the compact interval [-2meas(supp(ρ̅)),2meas(supp(ρ̅))], we name such a constant L=L(ρ̅). We get

    |II_i|    =   R_i(t)|v(R_i(t))| | -∑_j ≠ i, i+1(K'(x_i+1(t) - x_j(t)) - K'(x_i(t) - x_j(t))  )  -2K'(x_i+1(t) - x_i(t))| 
       ≤ R_i(t) L v_maxN-2/N1/R_i(t) + 2v_max L1/N≤ L v_max ,

and this gives

    |  -∑_i=1^N-1μ_i(t)R_i(t)II_i  | ≤ L v_max| ∑_i=1^N-1μ_i(t)R_i(t) | ≤ L v_max TV[ρ^N(t, ·)].

We can now focus on Ṙ_0 and Ṙ_N. Since the setting is symmetric, we only present the argument for μ_0(t)Ṙ_0 and leave the one for μ_N(t)Ṙ_N to the reader. Since μ_0(t)≠ 0 only if R_1(t)>R_0(t), without restriction we can assume (v(R_1) - v(R_0)) ≤ 0 and can compute

    μ_0Ṙ_0    = μ_0R_0 [R_0v(R_1)∑_j>1(K'(x_1-x_j) - K'(x_0-x_j)) +2R_0v(R_0)K'(x_1-x_0)] 
          + μ_0(R_0)^2(v(R_1) - v(R_0))∑_j>1 K'(x_0 - x_j) 
       ≤μ_0R_0 [R_0v(R_1)∑_j>1(K'(x_1-x_j) - K'(x_0-x_j)) +2R_0v(R_0)K'(x_1-x_0)] .

Moreover,

    | R_0v(R_1)∑_j>1(K'(x_1-x_j) - K'(x_0-x_j)) +2R_0v(R_0)K'(x_1-x_0)| ≤ v_max LN-1/N + 2v_max L/N .

In particular, μ_0Ṙ_0 ≤ (3CL) R_0 and

    μ_0Ṙ_0 + μ(R_N)Ṙ_N ≤ 3v_max L  (R_0 + R_N) ≤ 3v_max L  TV[ρ^N(t, ·)] .

By putting together (<ref>), (<ref>) and (<ref>) we get estimate (<ref>) and (<ref>) follows as a consequence of Gronwall Lemma.


We now prove the equi-continuity w.r.t. time with respect to the 1-Wasserstein distance for ρ^N.

Proposition <ref>
Assume without loss of generality that 0< s<t < T.
Our goal then is to investigate the continuity in time of the discrete density ρ^N with respect to the 1-Wasserstein distance. We exploit the well known relation between the 1-Wasserstein distance of two probability measures and the L^1 distance of their respective pseudo inverse functions. More precisely, for any two probability measures μ, ν the following identity holds

    d_1 (μ, ν)  =   X_μ - X_ν_L^1([0, 1]),

where X_μ and X_ν are the pseudo inverses of the cumulative distribution functions of μ and ν respectively.
The assertion of the proposition will follow once we prove that there exists a constant C>0 independent of N such that

    X_ρ^N(t,·) - X_ρ^N(s, ·)_L^1([0,1]) < C|t-s|,

for all s, t ∈ (0,T).
By the definition of ρ^N we can explicitly compute

    X_ρ^N(t, ·)(z) = ∑_i=0^N-1(x_i^N(t) + (z-i1/N) 1/R_i^N(t)) 1_[i1/N, (i+1)1/N)(z) .

Therefore,

    d_1( ρ^N(t, ·), ρ^N(s, ·) )    =  X_ρ^N(t, ·) - X_ρ^N(s, ·)_L^1([0, 1])
       ≤∑_i=0^N-1∫_i/N^(i+1)/N| x_i^N(t) - x_i^N(s) + (z - i/N) (1/R_i^N(t) - 1/R_i^N(s))  | dz 
       ≤∑_i=0^N-11/N |x_i^N(t) - x_i^N(s)| +  ∑_i=0^N-1|1/R_i^N(t) - 1/R_i^N(s)|  ∫_i/N^(i+1)/N(z - i/N)dz 
       = ∑_i=0^N-11/N |x_i^N(t) - x_i^N(s)|  + ∑_i=0^N-11/2N^2∫_s^t | d/dτ1/R_i^N(τ)| dτ
       ≤ 3 ∑_i=0^N1/N∫_s^t |ẋ_i^N(τ)| dτ ,

where in the last inequality we used that

    | d/dτ1/R_i^N(τ)| = N |ẋ_i+1^N(τ) - ẋ_i^N(τ)| ≤ N |ẋ_i+1^N(τ)| + N|ẋ_i^N(τ)| .

Notice that we can control |ẋ_i^N(τ)| uniformly in N and in τ. Indeed, recalling the assumption (AK), setting L as the Lipschitz constant of K' on the interval [-2meas(supp(ρ̅)),2meas(supp(ρ̅))] as in the proof of Proposition <ref>, we have

    |ẋ_i^N(τ)| = 1/N|-v(R_i(t)) | ∑_j>i K'(x_i - x_j) - v(R_i-1)∑_j<i K'(x_i - x_j) | ≤2LN v_max/N = 2L v_max,,

which gives

    d_1( ρ^N(t, ·), ρ^N(s, ·) ) ≤ 6L v_max |t-s| ∑_i=0^N 1/N≤ 12L v_max |t-s|,

and (<ref>) is proven.















































































































































§ CONSISTENCY OF THE MANY PARTICLE SCHEME: CONVERGENCE TO ENTROPY SOLUTION


In this section we show that the limit ρ obtained in Section <ref> satisfies the entropy condition in the sense of Definition <ref>. Moreover, we can prove that ρ is the unique entropy solution of the Cauchy problem

    {[ ∂_t ρ = ∂_x(ρ v(ρ) K' ∗ρ)   t ∈ (0,T],;                           ρ(0,·) = ρ̅. ].

The first step consists in showing that the discrete densities satisfy an analogue version of the entropy condition. In order to do that, we need to introduce another sequence of approximations of the solution ρ, namely the N-empirical measure

    ρ̂^N(t,x) := 1/N∑_i=0^N δ_x_i(t)(x).

In the next lemma we show that ρ̂^N and ρ^N are arbitrarily close in the 1-Wasserstein distance, which implies that ρ̂^N converge up to a subsequence to the same limit ρ obtained in the previous section.


  For all N∈ℕ, we have
  
    d_1(ρ^N(t,·),ρ̂^N(t))≤C/N,

  for some constant C only depending on ρ̅.



  In view of the standard isometric mapping between the 1-Wasserstein space of probability measures and the convex cone of non-decreasing functions in L^1([0,1]), similarly to the proof of proposition <ref>, we have

    d_1(ρ^N(t,·),ρ̂^N(t))≤∑_i=0^N-1∫_i/N^(i+1)/N| (z-i1/N) 1/R_i^N(t)|  dz
        = 1/2N∑_i=0^N-1(x^N_i+1(t)-x^N_i(t)) = 1/2N(x^N_N(t)-x^N_0(t))
        ≤1/2Nmeas(supp(ρ̅)),

  which proves the assertion.



  Let W∈ C() be even and locally Lipschitz. Then, there exists a constant C>0 depending only on ρ̅ such that
  
    sup_t≥ 0W∗ρ^N(t,·)-W∗ρ̂^N(t)_L^1≤C/N,

  for all N∈ℕ. To prove this, let γ^N_o(t) be an optimal plan between ρ^N(t,·) and ρ̂^N(t,·) with respect to the cost c(x)=|x|. We then estimate, for all t≥ 0,
  
    W∗ρ^N-W∗ρ̂^N_L^1() = ∫_|∫_ W(x-y)  dρ^N(t,·)(y)-∫_ W(x-y) dρ̂^N(t)(y)|  dx
        =∫_|∬_^2(W(x-y)-W(x-z))  dγ_0^N(t)(y,z)|  dx
        ≤ C ∫_∬_^2|y-z|dγ_0^N(t)(y,z),dx,

  where we have used that the supports of ρ^N and ρ̂^N are contained in supp(ρ̅) which is bounded and independent of time. By definition of 1-Wasserstein distance we therefore have
  
    W∗ρ^N-W∗ρ̂^N_L^1()≤ C d_1(ρ^N(t,·),ρ̂^N(t)) ≤C̃/N,

  for some suitable constant C̃>0 in view of Lemma <ref>.


Our next goal is to prove that the entropy inequality

    0 ≤∫_0^T ∫_ |ρ^N - c|φ_t -  sign(ρ^N - c)[(f(ρ^N) - f(c))K' ⋆ρ̂^N φ_x - f(c)K”∗ρ̂^N φ]dxdt

holds for every non-negative test function φ with compact support in 𝒞^∞_c ((0,+∞)×), every constant c≥ 0 and every N large enough. Such a goal, which requires some tedious calculations, is however not enough to prove that the limit ρ of the previous section is an entropy solution because the of the discontinuity of the sign function in the above inequality, which does not allow to pass to the limit for ρ^N→ρ almost everywhere and in L^1. To bypass this problem we shall then introduce a δ-regularization of the sign function in order to first let N→ +∞ and then δ↘ 0. In the last part of the section we prove the uniqueness of entropy solutions, which allows to conclude that the whole approximating sequence ρ^N converges to ρ, thus completing the proof of our main Theorem <ref>.




For every non negative φ∈ C^∞_c ((0,+∞)×), c≥ 0 and N ∈ the following inequality holds

    lim inf_N→ +∞∫_0^T ∫_ |ρ^N - c| φ_t - sign(ρ^N -c)[(f(ρ^N) - f(c))K' ∗ρ̂^N φ_x - f(c)K”∗ρ̂^N φ] dx dt ≥ 0.



Let T>0 such that suppφ⊂[0,T]×. The basic idea of the proof is rather simple, although the computations are quite technical: we need to rewrite the left hand side of the inequality so that it is possible to isolate a term with positive sign and then show that the remaining terms give negligible contributions as N →∞.
By definition of ρ^N and ρ̂^N we obtain

    ∫_0^T ∫_ |ρ^N - c| φ_t - sign(ρ^N -c)[(f(ρ^N) - f(c))K' ∗φ_x - f(c)K”∗ρ^N φ] dx dt = B.T._1 + ∑_i=0^N-1 I_i + ∑_i=0^N-1 II_i,

where

    I_i := ∫_0^T ∫_x_i^x_i+1 |R_i^N -c| φ_t dxdt ,
        II_i := - ∫_0^T ∫_x_i^x_i+1sign(R_i^N - c)(f(R_i^N) - f(c))K' ∗ρ̂^N φ_x dxdt  
       + ∫_0^T ∫_x_i^x_i+1 f(c) sign(R_i^N - c) K”∗ρ̂^N φ  dxdt, 
        B.T._1 := ∫_0^T ∫_-∞^x_0 cφ_t - f(c)[K' ∗ρ̂^N φ_x + K”∗ρ̂^N φ] dxdt  
       + ∫_0^T ∫_x_N^∞ cφ_t - f(c)[K' ∗ρ̂^N φ_x + K”∗ρ̂^N φ] dxdt.

For simplicity of notation we set S^N_i : = sign(R_i^N - c) and we omit the dependence on N and t wherever it is clear from the context.
Integrating by parts and recalling the definition of ρ̂^N and the expression for Ṙ_i, we can rewrite I_i as

    I_i=   ∫_0^T S_i R_i (ẋ_i+1 - ẋ_i) (_x_i^x_i+1φ(t,x) dx   - φ(t,x_i+1))dt  
       + ∫_0^T S_i [R_i (ẋ_i+1 - ẋ_i) φ(t,x_i+1)  - (R_i -c) (ẋ_i+1φ(t,x_i+1) - ẋ_i φ(t,x_i))]dt,

and II_i as

    II_i =    -∫_0^T S_i(f(R_i) - f(c))/N∑_j=0^N (K'(x_i+1 - x_j)φ(t,x_i+1) - K'(x_i - x_j)φ(t,x_i))dt 
       + ∫_0^T S_i f(R_i)/N∑_j=0^N ∫_x_i^x_i+1 K”(x-x_j)φ(t,x)dxdt .

Then the sum I_i + II_i becomes

    I_i + II_i = A^1_i + A^2_i + Z_i,

where we set

    A^1_i    = ∫_0^T S_i R_i (ẋ_i+1 - ẋ_i) (_x_i^x_i+1φ(t,x) dx   - φ(t,x_i+1))dt, 
    
    A^2_i    =  ∫_0^T S_i f(R_i)/N∑_j=0^N ∫_x_i^x_i+1 K”(x-x_j)φ(t,x)dxdt,

and

    Z_i =    - ∑_i=0^N-1∫_0^T S_iφ(t,x_i+1)[R_i ẋ_i + f(R_i)/N∑_j=0^N K'(x_i+1-x_j)]dt 
       + ∑_i=0^N-1∫_0^T S_iφ(t,x_i+1)[c ẋ_i+1 + f(c)/N∑_j=0^N K'(x_i+1-x_j)]dt 
       + ∑_i=0^N-1∫_0^T S_iφ(t,x_i)[R_i ẋ_i + f(R_i)/N∑_j=0^N K'(x_i-x_j)]dt 
       - ∑_i=0^N-1∫_0^T S_iφ(t,x_i)[c ẋ_i + f(c)/N∑_j=0^N K'(x_i-x_j)]dt.

By performing a summation by parts, we get

    ∑_i=0^N-1 Z_i    = B.T._2 + ∑_i=1^N-1∫_0^T φ(t,x_i) S_i(R_i ẋ_i +f(R_i)/N∑_j=0^N K'(x_i -x_j) )dt 
          - ∑_i=1^N-1∫_0^T φ(t,x_i) S_i-1(R_i-1ẋ_i-1 + f(R_i-1)/N∑_j=0^N K'(x_i-x_j) )dt 
          +∑_i=1^N-1∫_0^T φ(t,x_i)(S_i-1- S_i)(cẋ_i + f(c)/N∑_j=0^N K'(x_i-x_j) )dt 
       = B.T._2 + B.T._3 + ∑_i=1^N-2 (A_i^3 + A_i^4) + ∑_i=1^N-1B_i.

where B.T._2 and B.T._3 regard the external particles. More precisely, B.T._2= B.T._21+ B.T._22, where

    B.T._21 =    c ∫_0^T φ(t,x_N) S_N-1v(c)-v(R_N-1)/N∑_j=0^N K'(X_N - x_j)dt 
         -c ∫_0^T φ(t,x_0) S_0v(c)-v(R_0)/N∑_j=0^N K'(X_0 - x_j)dt, 
    
    B.T._22 =   ∫_0^T φ(t,x_0)S_0R_0 (ẋ_0 + v(R_0)/N∑_j=0^N K'(X_0 - x_j))dt 
       - ∫_0^T φ(t,x_N)S_N-1 R_N-1(ẋ_N-1 + v(R_N-1)/N∑_j=0^N K'(X_N - x_j))dt,

and B.T._3 corresponds to

    B.T._3 =   ∫_0^T φ(t,x_N-1) S_N-1(R_N-1ẋ_N-1 + f(R_N-1)/N∑_j=0^N K'(x_N-1-x_j))dt 
       - ∫_0^T φ(t,x_0)S_0 ( R_0ẋ_0 + f(R_0)/N∑_j=0^N K'(x_1-x_j))dt.

The terms A_i^3, A_i^4 and B_i regards, instead, the internal particles and they are defined as follows

    A_i^3 =   ∫_0^T φ(t,x_i) S_i f(R_i)/N∑_j=0^N [K'(x_i-x_j)-K'(x_i+1-x_j)]dt, 
    
    A_i^4 =   ∫_0^T (φ(t,x_i) - φ(t,x_i+1))S_i R_i ( ẋ_i + v(R_i)/N∑_j=0^N K'(x_i+1-x_j) )dt,
    
    B_i =   ∫_0^T φ(t,x_i)(S_i-1- S_i))(cẋ_i + f(c)/N∑_j=0^N K'(x_i-x_j) )dt.

Summarizing, we can rewrite B.T._1 + ∑_i=0^N-1 (I_i + II_i) as

    B.T._1 + B.T_21 + B.T._22 + B.T._3 + ∑_i=0^N-1 (A^1_i + A^2_i) + ∑_i=1^N-2 (A^3_i + A^4_i) + ∑_i=1^N-1 B_i,

then estimate (<ref>) follows if we prove that such sum is non negative when N ≫ 1, and this can be done by showing that

    B.T._1 + B.T._21 + ∑_i=1^N-1B_i > 0,

while

    | B.T._22 + B.T._3 + ∑_i=0^N-1(A_i^1 + A_i^2) + ∑_i=1^N-2 (A_i^3 + A_i^4) | ≤C/N

for a positive constant C= C(φ,K,ρ̅,v,T).
The remaining part of the proof is devoted to show the validity of (<ref>) and (<ref>).
We focus first on (<ref>). Integrating by parts, recalling that φ(0,·)=φ(T,·)=0, φ(t,·) ≥ 0 and the assumption (AK), we immediately obtain

    B.T._1 = -f(c)/N∫_0^T (φ(t,x_N)∑_j=0^N K'(x_N-x_j) + φ(t,x_0)∑_j=0^N K'(x_0-x_j)) > 0.

Because of the monotonicity of v (see (Av)), for all times t we know that

    S_0(t)(v(c) - v(R_0(t))) ≥ 0,     S_N-1(t)(v(c) - v(R_N-1(t))) ≥ 0

thus, recalling again (AK), we deduce

    B.T._21≥ 0.

Let us now consider the generic term B_i. Substituting the expression of ẋ_i, we get

    B_i = ∫_0^Tφ(t,x_i) (S_i-1 - S_i)[v(c)-v(R_i)/N∑_j>i K'(x_i-x_j) + v(c)-v(R_i-1)/N∑_j<i K'(x_i-x_j) ]dt.

Now, if R_i(t),R_i-1(t) are both strictly bigger than c or strictly smaller than c, then S_i-1(t) - S_i(t)=0. Otherwise, since v is decreasing (and we assume sign(0)=0), we get

    R_i(t) ≥ c ≥ R_i-1(t)      ⇒  -2 ≤ S_i-1(t) - S_i(t)≤ 0,    v(c)-v(R_i(t)) ≥ 0,    v(c)-v(R_i-1) ≤ 0 
    
    R_i-1(t) ≥ c ≥ R_i(t)      ⇒  0 ≤ S_i-1(t) - S_i(t) ≤ 2,    v(c)-v(R_i(t)) ≤ 0,    v(c)-v(R_i-1) ≥ 0

and, recalling that K' is symmetric and K'(x)>0 if x>0, for all times holds

    (S_i-1- S_i)[v(c)-v(R_i)/N∑_j>i K'(x_i-x_j) + v(c)-v(R_i-1)/N∑_j<i K'(x_i-x_j) ] ≥ 0.

In particular, B_i ≥ 0 and

    ∑_i=1^N-1 B_i ≥ 0.

Then estimate (<ref>) is a direct consequence of (<ref>), (<ref>) and (<ref>). Let us consider now (<ref>). First of all, observe that Lemma <ref> ensures that the support of ρ^N is always contained in the support of ρ̅, then since we assume K' locally Lipschitz then there exists a constant L>0 such that

    L=sup{ |K”(x)| ,   x∈[-(x̅_max-x̅_min),(x̅_max-x̅_min)]}.

Since the argument is quite technical, it is more convenient to split the left hand side of (<ref>) in three parts:

    Γ_1 = B.T._22 + B.T._3 + A_0^2 + A_N-1^2,  Γ_2 = ∑_i=0^N-1 A_i^1 + ∑_i=1^N-2 A_i^4,  Γ_3 = ∑_i=1^N-2 (A_i^2 + A_i^3).

Recalling that K', φ and v are uniformly bounded and Lipschitz, we get

    |Γ_1| ≤    4 L φ_L^∞v_L^∞∫_0^T (R_0(x_1 - x_0) + R_N-1(x_N-x_N-1))dt  
       + 2 L v_L^∞Lip[φ] ∫_0^T R_N-1(x_N - x_N-1)dt ≤C(φ,v,L,T)/N

Then, inserting the expression of ẋ_i, we can rearrange Γ_2 in such a way that

    |Γ_2|    ≤ 3∑_i=0^N-1∫_0^T R_i |_x_i^x_i+1φ(t,x) - φ(t,x_i+1)||v(R_i+1)-v(R_i)|/N∑_j=0^N |K'(x_i+1-x_j)| dt 
          + ∑_i=0^N-1∫_0^T R_i |_x_i^x_i+1φ(t,x) - φ(t,x_i+1)| v(R_i)/N∑_j=0^N |K'(x_i-x_j)- K'(x_i+1-x_j)|dt 
          + ∑_i=1^N-2∫_0^T R_i |φ(t,x_i) - φ(t,x_i+1)||v(R_i-1)-v(R_i)|/N∑_j=0^N |K'(x_i-x_j)|dt 
          + ∑_i=1^N-2∫_0^T R_i |φ(t,x_i) - φ(t,x_i+1)|v(R_i)/N∑_j=0^N |K'(x_i-x_j)- K'(x_i+1-x_j)|dt

and using the Lipschitz and the uniform regularity of K', φ, v, estimate (<ref>) and the uniform bound on the support of ρ^N, it is easy to see that

    |Γ_2| ≤    4L  Lip[φ]Lip[v] TV[ρ̅] ∫_0^T e^Ct∑_i=0^N-1 R_i (x_i+1-x_i)dt 
        + 2 L v_L^∞ Lip[φ] ∫_0^T ∑_i=0^N-1 R_i (x_i+1-x_i)^2 dt ≤C(φ,v,K,ρ̅,T)/N.

It remains to show that also Γ_3 vanishes as N →∞. In this case, the uniform bound on K” implies

    |Γ_3|    ≤∑_i=1^N-2∫_0^T |f(R_i)|/N∫_x_i^x_i+1|φ(t,x)-φ(t,x_i)| ∑_j=0^N |K”(x-x_j)|dxdt 
       ≤ L v_L^∞ Lip[φ] ∫_0^T R_i ∫_x_i^x_i+1 (x-x_i)dxdt ≤C(φ,v,K,ρ̅,T)/N.

Finally, by putting together (<ref>),(<ref>) and (<ref>), we obtain (<ref>) and, recalling also (<ref>), (<ref>).


We are now in position to prove that the large particle limit ρ that we obtained in the previous section is an entropy solution for the PDE.


Let ρ be the limit of ρ^N up to a subsequence. For every non negative φ∈ C^∞_c ([0,+∞)×) and c≥ 0, one has

    0 ≤∫_ |ρ̅- c|φ(0,x)dx + ∫_0^+∞∫_ |ρ - c| φ_t - sign(ρ -c)[(f(ρ) - f(c))K' ∗ρ φ_x - f(c)K”∗ρ φ] dx dt.



Let T>0 be such that supp(φ)⊂ [0,T). Roughly speaking, the statement holds provided we can show that it is possible to pass to the limit as N →∞ in the inequality (<ref>). More precisely,
we need to prove the following

    lim_N →∞∫_ |ρ^N(0,x) - c|φ(0,x)dx = ∫_ |ρ̅- c|φ(0,x)dx
       lim_N →∞∫_0^T ∫_ |ρ^N-c| φ_t  dx dt = ∫_0^T ∫_ |ρ-c| φ_t   dx dt 
       lim_N →∞∫_0^T ∫_sign(ρ^N-c)(f(ρ^N)-f(c))K'∗ρ̂^N φ_x   dx dt
            = ∫_0^T ∫_sign(ρ-c)(f(ρ)-f(c))K'∗ρ φ_x   dx dt
       lim_N →∞∫_0^T ∫_ f(c) sign(ρ^N - c)K”∗ρ̂^N φ  dx dt = ∫_0^T ∫_ f(c)   sign(ρ - c)K”∗ρ φ  dx dt

The first two limits are immediate in view of the strong L^1-convergence of ρ^N(0,x) to ρ̅ and to the convergence of ρ^N to ρ almost everywhere in L^1([0,T] ×) respectively.




Notice now that the continuity of f ensures the continuity of the function h(μ):= sign(μ - c)(f(μ)-f(c)). We have

    ∫_0^T ∫_    [sign(ρ^N-c)(f(ρ^N)-f(c))K'∗ρ̂^N - sign(ρ-c)(f(ρ)-f(c))K'∗ρ ] φ_x dx dt
       = ∫_0^T ∫_(h(ρ)-h(ρ^N))K'∗ρ φ_x dx dt + ∫_0^T ∫_  h(ρ^N) K'∗(ρ -ρ̂^N) φ_x dx dt,

then the regularity of h and K' required in the assumptions (Av) and (AK), the convergence of ρ^N to ρ almost everywhere in [0,T] × and the strong L^1-convergence of K' ∗ρ̂^N to K' ∗ρ established in Remark <ref> imply that

    ∫_0^T ∫_|[(h(ρ)-h(ρ^N))K'∗ρ +  h(ρ^N) K'∗(ρ -ρ̂^N)] φ_x| dx dt → 0

as N tends to +∞. Concerning the fourth limit, instead, we can see that

    ∫_0^T ∫_    f(c)[sign(ρ^N-c)K”∗ρ̂^N - sign(ρ-c)K”∗ρ] φ dx dt 
       = ∫_0^T ∫_ f(c)sign(ρ^N-c)K”∗ (ρ̂^N - ρ) φ dx dt 
          + ∫_0^T ∫_ f(c)(sign(ρ^N-c) -sign(ρ-c))K”∗ρ φ dx dt.

The first of the two terms can be handled as above. By using Remark <ref> and Lemma <ref>, we get 

    ∫_0^T ∫_ |f(c)sign(ρ^N-c)K”∗ (ρ̂^N - ρ) φ dx| dt ≤C(K”,f_∞,φ)/N .

On the other hand, passing to the limit in the terms including the difference sign(ρ^N-c)- sign(ρ-c) is less straightforward because of the discontinuity of the sign function.
Let us then focus on the proof of

    lim_N →∞∫_0^T ∫_ f(c)(sign(ρ^N-c)-sign(ρ-c))K”∗ρ φ dxdt = 0 .

In order to get rid of the discontinuity, we need to introduce two smooth approximations of the sign function, we call them η_δ^±, so that

    sign(z) - η_δ^+(z) ≥ 0     sign(z)-η_δ^-(z) ≤ 0.

Let us recall that the regularity of K ensures the existence of a constant L>0 such that |K”(z)| ≤ L for every z ∈ [-2meas(supp(ρ^N)), 2meas(supp(ρ^N))] and every N.
Then we can estimate

    ∫_0^T∫_f(c)sign   (ρ^N-c) K”∗ρφ
       = ∫_0^T∫_f(c)sign(ρ^N-c)(K”-L)∗ρ φ + ∫_0^T∫_f(c)sign(ρ^N-c)L∗ρ φ
       ≤∫_0^T∫_f(c)η_δ^+(ρ^N-c)(K”-L)∗ρ φ + ∫_0^T∫_f(c)η_δ^-(ρ^N-c)L∗ρ φ .







where the inequality holds because

    (sign(ρ^N-c)-η_δ^+(ρ^N-c))(K”-L)∗ρ≤ 0 
       (sign(ρ^N-c)-η_δ^-(ρ^N-c)) L∗ρ≤ 0.





Now, observe that

    lim_N→∞    f(c) ∫_0^T ∫_ (η_δ^+(ρ^N -c) - η_δ^+(ρ -c)) (K”-L)∗ρ φ
       ≤lim_N→∞ f(c)∫_0^T∫_ |η_δ^+(ρ^N -c) -η_δ^+(ρ -c)| |(K”-L)∗ρ φ| 
       ≤lim_N→∞ f(c) 2L φ_∞ Lip[η_δ^+] ∫_0^T∫_ |ρ^N-ρ| 
       ≤ C(L, φ, η_δ^+) lim_N→∞ρ^N - ρ_L^1([0,T]×)= 0

and in a similar way we get also

    lim_N →∞∫_0^T∫_ f(c)(η_δ^-(ρ^N-c) - η_δ^-(ρ-c))L∗ρ φ = 0 ,

thus implying that

    lim sup_N→∞∫_0^T∫_f(c)sign(ρ^N-c) K”∗ρ φ≤∫_0^T∫_ f(c)[η_δ^+ (ρ-c)(K”-L)∗ρ + η_δ^-(ρ-c)L∗ρ] φ




Once here, the dominated convergence Theorem ensures that we can pass to the limit in δ to get

    lim sup_N →∞ f(c)∫_0^T∫_sign(ρ^N-c) K”∗ρ φ≤∫_0^T∫_sign(ρ-c) K”∗ρ φ.


A symmetric argument provides the inverse inequality with lim inf replacing lim sup, hence we obtain


    lim_N →∞ f(c)∫_0^T∫_ (sign(ρ^N-c)-sign(ρ-c)) K”∗ρ φ = 0.

The above argument, together with (<ref>)-(<ref>), implies estimate (<ref>), and the proof is complete.


We now tackle another crucial task for our result, namely the uniqueness of the entropy solution for a fixed initial datum. To perform this task we rely on a stability result due to Karlsen and Risebro <cit.>, that we report here for sake of completeness in an adapted version.


Let f,P,Q be such that

    f        P,Q ∈ W^1,1() ∩𝒞(),      P_x,Q_x ∈ L^∞(),

and let p,q ∈ L^∞([0,T]; BV()) be respectively entropy solutions to

    {[                                      p_t = (f(p)P(x))_x    p(0,x)=p_0(x),;                                      q_t = (f(q)Q(x))_x    q(0,x)=q_0(x), ].

where the initial data (p_0,q_0) are in L^1() ∩ L^∞() ∩ BV().
Then for almost every t ∈ (0,T) one has

    p(t) - q(t)  _L^1()≤ p_0 - q_0  _L^1() + t(C_1  P-Q_L^∞() + C_2 P-Q_BV())

where C_1 = Lip[f] min{P_BV(), Q _BV()} and C_2 =  f_L^∞.


We are now ready to prove our main theorem.

Theorem <ref>
The results in Theorem <ref> and Lemma <ref> imply that there exist a subsequence of ρ^N converging almost everywhere on [0,+∞)× and in L^1_loc to an entropy solution ρ to (<ref>) in the sense of Definition <ref>. Therefore, the proof of Theorem <ref> is concluded once we show that ρ is the unique entropy solution. We argue by contradiction. Assume that there exist two different functions ρ and ϱ satisfying Definition <ref> with ρ(0,·) = ϱ(0,·) = ρ̅, then we can define two vector fields P(x) = K' ∗ρ (x) and Q(x)= K'∗ϱ(x).
In order to apply Theorem <ref> to P and Q, let us check that all assumptions therein are satisfied.
First of all, P and Q are locally Lipschitz in  thanks to the assumption (AK), thus P_x, Q_x ∈ L^∞_loc(). Then, we observe that

    |P(t,x) - Q(t,x)|    = | ∫_ K'(x-y)ρ(t,y)dy - ∫_ K'(x-y)ϱ(t,y)dy | 
       ≤∫_ |K'(x-y)(ρ(t,y) - ϱ(t,y))|dy ≤ L_ρ̅ρ - ϱ_L^∞([0,T];L^1()),

and

    ∫_ |P_x(s,x) - Q_x(s,x)|dx    = ∫_ |K”∗ρ(t,x) - K”∗ϱ(t,x)| dx 
       = ∫_ |K”∗(ρ - ϱ)(t,x)|dx  ≤ L_ρ̅ρ - ϱ_L^∞([0,T];L^1()),

where L_ρ̅=max{K'_L^∞(I_ρ̅),K”_L^1(I_ρ̅)}, and I_ρ̅=[-2meas(supp(ρ̅)), 2meas(supp(ρ̅))].
As a consequence

    P-Q_L^∞([0,T] ×)≤ L_ρ̅ρ - ϱ_L^∞([0,T];L^1())
        P-Q_L^∞([0,T] ; BV())≤ L_ρ̅ρ - ϱ_L^∞(0,T;L^1()).

By applying Theorem <ref> to ρ,ϱ, P and Q we obtain

    ρ(t) - ϱ(t)  _L^1()≤ C(K,ρ̅) t ρ(t) - ϱ(t) _L^1().

Assume that there exists an open interval (t_1,t_2)⊂ [0,T] such that ρ(t,·) and ϱ(t,·) differ in L^1() on t∈ (t_1,t_2). Then, due to the fact that (<ref>) is invariant with respect to time-translations, the inequality   (<ref>) implies

    ρ(t,·) - ϱ(t,·)  _L^1()≤ C(K,ρ̅) (t-t_1) ρ(t,·) - ϱ(t,·) _L^1()  ∀ t ∈ (t_1,t_2).

Clearly can always consider t∈ (t_1,t_2) small enough such that C(K,ρ̅) (t-t_1)< 1, but this is in contradiction with (<ref>). In conclusion, ρ(t,·)≡ϱ(t,·) on [0,T] and the proof is complete.




§ NON-UNIQUENESS OF WEAK SOLUTIONS AND STEADY STATES


The use of the notion of entropy solution in the present context is not merely motivated by the technical need of identifying a notion of solution (stronger than weak solutions) allowing to prove uniqueness. Similarly to what happens for scalar conservation laws, we prove that there are explicit examples of initial data in BV for which there exists two weak solutions to the Cauchy problem (<ref>). 

For simplicity, we use

    v(ρ)=(1-ρ)_+.

Consider the initial condition

    ρ̅(x)=1_[-1,-1/2]+1_[1/2,1].

Clearly, the stationary function

    ρ_s(t,x)=1_[-1,-1/2]+1_[1/2,1]

is a weak solution to (<ref>) with initial condition ρ̅. To see this, let φ∈ C^1_c([0,+∞)×). We have

    ∫_0^+∞∫_[ρ_sφ_t + ρ_s v(ρ_s)K'∗ρφ_x] dx dt + ∫_ρ̅φ(0,x) dx 
        = ∫_0^+∞d/dt(∫_[-1,-1/2]∪ [1/2,1]φ dx)dt +∫_[-1,-1/2]∪ [1/2,1]φ(0,x) dx = 0.


We now prove that ρ_s is not an entropy solution, in that it does not satisfy the entropy condition in Definition <ref>. Let ψ∈ C^∞_c() be a standard non-negative mollifier supported on [-1/4,1/4]. Let T>0 and consider the test function φ(t,x)=ϕ(x)ξ(t) with

    ϕ(x)=
    ψ(x+1/2)    if -3/4≤ x≤ -1/4
    ψ(x-1/2)    if 1/4≤ x≤ 3/4
    
    0    otherwise,

and ξ∈ C^∞([0,+∞)) with ξ(t)=1 for t≤ T, ξ(t)=0 for t≥ T+1 and ξ non-increasing. Let us set c=1/2, I=[1/4,3/4], and compute

    ∫_ |ρ_s -c|ϕ dx +∫_0^+∞∫_[|ρ_s-c|ϕ(x)ξ'(t) -sign(ρ_s-c)(f(ρ)-f(c))K'∗ρ_sϕ'(x)ξ(t) .
           .-f(c)K”∗ρ_s ϕ(x)ξ(t)] dxdt
       ≤ 2∫_I φ dx + 1/4∫_0^T+1ξ(t)dt[∫_(-I)∩(-∞,1/2]K'∗ρ_s φ_x dx - ∫_(-I)∩[1/2,+∞)K'∗ρ_s φ_x dx .
         .- ∫_I∩(-∞,1/2]K'∗ρ_s φ_x dx +∫_I∩[1/2,+∞)K'∗ρ_s φ_x dx- ∫_(-I)∪ IK”∗ρ_s φ dx]
        = 2∫_I φ dx - 1/4∫_0^T+1ξ(t)dt[∫_(-I)∩(-∞,1/2]K”∗ρ_s φ dx - ∫_(-I)∩[1/2,+∞)K”∗ρ_s φ dx .
         .- ∫_I∩(-∞,1/2]K”∗ρ_s φ dx +∫_I∩[1/2,+∞)K”∗ρ_s φ dx+∫_(-I)∪ IK”∗ρ_s φ dx].

Now, since K” and ρ_s are even, the same holds for K”∗ρ_s. Therefore we get

    ∫_ |ρ_s -c|φ dx +∫_0^T∫_[|ρ_s-c|φ_t -sign(ρ_s-c)(f(ρ)-f(c))K'∗ρ_sφ_x -f(c)K”∗ρ_s φ] dxdt
        ≤ 2∫_I φ dx - 1/2∫_0^T+1ξ(t)dt∬_I× I(K”(x-y)+K”(x+y)) φ(x)dy dx.

Let us now require for simplicity the following additional assumption:

    K”(x)>0    for all x∈.

Actually, such assumption can be relaxed, see remark <ref> below. Then, the last integral in (<ref>) is clearly positive, and recalling that ξ(t)=1 on t∈[0,T], we can choose T large enough so that the whole right-hand side of (<ref>) is strictly negative, thus contradicting Definition <ref>.

The above argument shows that ρ_s is a weak solution but not an entropy solution. On the other hand, the initial condition ρ_s is L^∞ and BV, therefore it must generate an entropy solution according to our main Theorem <ref>. Clearly, such solution cannot coincide with ρ_s. We have therefore proven the following theorem.


  Assume (Av), (AK), and (<ref>) are satisfied. Then, there exists an initial condition ρ̅∈ L^∞()∩ BV() such that the Cauchy problem (<ref>) has more than one distributional weak solution.



The assumption (<ref>) can be relaxed to include also Gaussian kernels K(x)=- A e^-B x^2 with A,B>0. Indeed, in order to fulfil 

    ∬_I× I(K”(x-y)+K”(x+y)) φ(x)dy dx>0

one has to choose the size of the interval I small enough. We omit the details.



The fact that the initial condition ρ_s will not give rise to a stationary solution can also be seen intuitively by using the result in Theorem <ref>. Let us approximate ρ̅ with 2(N+1) particles with mass 1/(2(N+1)), with N integer, and with the particles located at x̅_i, i=1,…,2(N+1), with

    x̅_i=-1+i/2(N+1) ,     i=0,…,N
       x̅_i=1/2+i-N/2(N+1) ,     i=N+1,…,2N+1.

Let now evolve the particles' positions with the usual ODE system

    ẋ_i=-v(R_i)/N∑_j>i(x_i-x_j) -v(R_i-1)/N∑_j<i(x_i-x_j).

It can be easily proven (we omit the details) that the solution to the particle system preserves the even symmetry of the initial condition.
Moreover, the particle x_N - i.e. the leading particle of the left bump of the initial condition - has a positive initial speed which can be controlled from below by a constant provided that, for example, K' is supported on  and is strictly monotone on (0,+∞). Indeed, as all particles x_i with i<N are posed at minimal distance at t=0 and the initial distance x_N+1-x_N=1, we have

    ẋ_N(0)=v(1/N)1/N∑_j>NK'(x_j(0)-x_N(0)) ≥ v(1/N)N+1/NK'(2)>v(1/2)K'(2)>0.

Similarly, one can show that all particles i=0,…,N-1 `move' from their initial position, although their initial speed is zero. A numerical simulation performed in Section <ref> actually show that for large N the discrete density tends to form a unique bump for large times. Hence, since Theorem <ref> shows that the particle solution is arbitrarily close in L^1_loc to the entropy solution, this argument supports the evidence that the entropy solution is not stationary. 


Apart from producing an explicit example of non-uniqueness of weak solutions, the above example shows that there are stationary weak solutions that are not entropy solutions, and therefore cannot be considered as stationary solutions to our problem according to Definition <ref>. This raises the following natural question: what are the steady states of (<ref>) in the entropy sense? Before asking this question, it will be useful to tackle another task: as the approximating particle system converges to the entropy solutions, detecting the steady states of (<ref>) will give us a useful insight about the steady states at the continuum level.

Let us restrict, for simplicity, to the case of an even initial condition ρ̅, such that ρ̅_L^1=1  and N∈ fixed. We assume here that K' is supported on the whole . Consider the following particle configuration,

    x̃_1=-1/2+1/2N,
    x̃_i+1=x̃_1+i/N,   i=1,...,N-2,
    x̃_N=x̃_1+N-1/N=1/2-1/2N.

With this choice we get

    R_i = 1/N(x̃_i+1-x̃_i)=1,    v(R_i)=0,   ∀ i=1,...,N-1,

and it is easy to show that this configuration is a stationary solution for system (<ref>). Actually, up to space translations, this is the only possible stationary solution. In order to prove that, assume that we have a particle configuration as in (<ref>) but with only one particle labelled I such that

    x̃_I=x̃_1+I-1/N,   x̃_I+1=x̅>x̃_I+1/N.

For such a configuration

    R_I = m/N(x̃_I+1-x̃_I)<1,    v(R_I)>0,   v(R_i)=0   ∀ i≠ I,

and the I particles evolves according to

    ẋ̃̇_I = -v(R_I)/N∑_j>IK'(x̃_I-x̃_j)  =-v(R_I)/N∑_j>IK'(1/N(I-j))>0,

and then x̃_I moves with positive velocity.

We observe that, as N→∞, the piecewise constant density reconstructed by configuration (<ref>) converges in L^1 to the step function

    ρ_S=1_[-1/2,1/2].

The above discussion suggests that all initial data with multiple bumps only attaining the values 0 and 1 are (weak solutions but) not entropy solutions except ρ_S. Actually, this statement can be proven exactly in the same way as we proved Theorem <ref>, as it is clear that the position of the decreasing discontinuity at x=-1/2 and of the increasing discontinuity at x=1/2 do not play an essential role. By choosing the test function φ suitably, one can easily show that the entropy condition can be contradicted by suitably centring φ around the non-admissible discontinuities. We omit the details. As a consequence, we can assert that ρ_S is the only stationary solution to (<ref>) in the sense of Definition <ref>.



§ NUMERICAL SIMULATIONS

The last section of the paper is devoted to present some numerical experiments based on the particle methods presented in the paper, supporting the results in the previous sections. The qualitative property that emerges more clearly in the simulations below is that solutions tend to aggregate and narrow their support. However, the maximal density constraint avoids the blow-up, and the density profile tends for large times towards the non-trivial stationary pattern presented at the end of the previous section. We compare our particle method with a classical Godunov method for (<ref>).



 §.§ Particle simulations


We first test the particle method introduced in Section <ref>. We proceed as follows: we set the number of particles as N and we reconstruct the initial distribution according to (<ref>) (for step functions we simply set the particles initially at distance ℓ/N from each other where ℓ is the length of the support). Once we have defined the initial distribution, we solve the system (<ref>) with a MATLAB solver and we reconstruct the discrete density as

    R_i(t)=m/2N(x_i+1(t)-x_i-1(t)),    i=2,N-1.


The choice of central differences does not effect the particle evolution, since in solving system (<ref>) we define R_i with forward differences. The choice in(<ref>) is only motivated by the symmetry of the patterns we expect to achieve for large times.

In the construction of the discrete densities we get the problem of giving density to the first and the last particles (or only to the last one if we use forward differences). Among all the possible choices we set at zero this two densities, namely

    R_1(t)=R_N(t)=0.

This is a natural choice if we are dealing with step functions but it is not suitable with more general initial conditions, see   <ref>.


In all the simulations we set

    v(ρ)=1-ρ,    K(x)=C/√(2π)e^-x^2/2 N=300.

In the particles evolution we don't fix any time step that is automatically determined by the solver.

The first example we furnish is the case of a single step function with symmetric support,

    ρ̅(x) = 0.3    x∈[-1,1].

For this initial condition m=0.6, so the final configuration will be a step function of value ρ=1 supported in [-0.3,0.3]. In   <ref> we plot initial (left) and final (right) configurations, while in   <ref> evolution in time is plotted.



Next we show the evolution corresponding to the following initial condition,

    ρ̅(x) = 3/4(1-x^2),    x∈[-1,1].

Even in this case the function is symmetric with respect to the origin so it will converge to the unitary step function supported in [-0.5,0.5] since ρ̅ has normalized mass. As in the previous example initial and final configurations and time evolution of the solution are plotted in  <ref>.


We conclude with step functions with disconnected support. We first study the case

    ρ̅(x) = 
      	                            0.2    x∈[-0.5,0]
    
                                    0.6    x∈[0.5,1]
                       	,

showing that the two bumps merge into a single step. Since symmetry is lost, it is not straightforward to determine where this final configuration will stabilize, but in  <ref> we can see that they still aggregate in a step of unitary density and support of length m.



More interesting is the case of the following initial condition:

    ρ̅(x) = 
      	                            1    x∈[-0.5,0]
    
                                    1    x∈[0.5,1]
                       	.

Note that this is a stationary weak solution to (<ref>) but it is not an entropy solution.
In  <ref> we plot the time evolution of this initial configuration.













 §.§ Comparison with classical Godunov method

In order to validate the previous simulations we compare the results with a classical Godunov method. The main issue in this case is to dealing with the two directions in the transport term. More precisely, since the kernel K is an even function, we can rephrase (<ref>) as

    ∂_t ρ = ∂_x(ρ v(ρ) )K_ρ^+(x)+∂_x(ρ v(ρ)) K_ρ^-(x)+ρ v(ρ) K”∗ρ,

where

    K_ρ^+(x)=∫_x≥ yK'(x-y)ρ(y)dy≥ 0,
           K_ρ^-(x)=∫_x < yK'(x-y)ρ(y)dy≤ 0.

 The evolution of ρ is driven by two transport fields: K_ρ^+ pushing the density from left to right K_ρ^- pushing the density from right to left. The third term on the r.h.s. in (<ref>) plays the role of a source term. Following the standard finite volume approximation procedure on N cells [x_j-1/2,x_j+1/2], the discrete equation reads as

    d/dtρ̃_j = K_ρ^+(x_j)F_j+1/2^+-F_j-1/2^+/Δ x + K_ρ^-(x_j)F_j+1/2^--F_j-1/2^-/Δ x + ρ̃_j v(ρ̃_j)dK_j

where F_j+1/2^+ and F_j+1/2^- are the Godunov approximations of the fluxes and dK_j is an approximation of the convolution in the reaction term obtained via a quadrature formula. We integrate in time with a time step satisfying the CFL condition of the method.  In  <ref> we compare the solutions obtained with the two methods in all the examples illustrated above.


 



§ ACKNOWLEDGEMENTS

The authors acknowledge support from the EU-funded Erasmus Mundus programme `MathMods - Mathematical models in engineering: theory, methods, and applications' at the University of L'Aquila, from the Italian GNAMPA mini-project `Analisi di modelli matematici della fisica,
della biologia e delle scienze sociali', and from the local fund of the University
of L’Aquila `DP-LAND (Deterministic Particles for Local And Nonlocal Dynamics).

 
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