Deterministic particle approximation for nonlocal transport equations with nonlinear mobility
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 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.
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 [27, 32], animal swarming phenomena [22, 7], pedestrian movements in a dense crowd , and more in general in socio-economical sciences [36, 1]. In a fairly general setting, a set of individuals located in a sub-region of the Euclidean space 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
in which the velocity law is known. Having in mind a particle system obeying the laws of classical mechanics or electromagnetism, the set of equations (1.1) 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 (1.1), these models are often posed in terms of a “continuum” PDE approach via a continuity equation
in which 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 is detected as a reasonable “cross-grained” version of its discrete counterpart in (1.1). 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 (1.2). 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 (1.2) as a many particle limits of (1.1) in this context is an essential requirement to validate the use of PDE models.
As briefly mentioned above, the velocity law in the PDE approach (1.2) 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 (1.2) 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. [37, 23, 12] only to mention a few. A first attempt (mainly numerical) to a fully deterministic approach to diffusion equations is due to , see  for the case of nonlinear diffusion.
Without diffusion and with only a local dependency , an extensive literature has been produced based on probabilistic methods (exclusion processes), see e.g. [18, 19]. A first rigorous result based on fully deterministic ODEs at the microscopic level for a nonlinear conservation law was recently obtained in . Nonlocal velocities have been considered as a special case of the theory developed in , with a given kernel (possibly singular) using techniques coming from kinetic equations, see . 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 depends both locally and non-locally from . 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  a similar model is studied in the context of pedestrian movements, and the existence and uniqueness of entropy solutions is proven. We also mention , which covers a more general class of problems, and  covering a similar model in the context of granular media. A quite general result was obtained in  in which the velocity map is required to be Lipschitz continuous as a map from the space of probability measures (equipped with some -Wasserstein distance) with values in , and the authors prove convergence of a time-discretised Lagrangian scheme. We also mention , 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 [5, 6] 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 (1.1) towards a PDE of the form (1.2) in the case of local-nonlocal dependence . Indeed, the result in  does not apply to this case in view of the Lipschitz continuity assumption on the velocity field, see also a similar result in .
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
in which and satisfy the following set of assumptions:
is a decreasing function such that , for some , on interval , on .
, (without restriction), for all , for , .
The constant here plays the role of a maximal density, which is supposed not to be exceeded by the density for all times. Clearly, the property has to be proven to be invariant with respect to time. We notice that the total mass of in (1.3) is formally conserved. For simplicity, throughout the paper we shall set We set
We set as the closed convex hull of .
Our goal is to approximate rigorously the solution to (1.3) with initial datum via a set of moving particles. More precisely, we aim to proving that the entropy solution of the Cauchy problem for (1.3) can be obtained as the large particle limit of a discrete Lagrangian approximation of the form (1.1). Such a Lagrangian approximation can be introduced as follows as a reasonable generalization of particle approximations considered previously in the literature in [17, 14, 15, 16]. For a fixed integer sufficiently large, we split into intervals such that the integral of the restriction of over each interval equals . More precisely, we let and , and define recursively the points for as
It is clear from the construction that and . Consider then particles located at initial time at the positions and let them evolve accordingly to the following system ODEs
with , where the discrete density is defined as follows
In (1.6), each particle has mass . We are then in position to define the -discrete density
We observe that has total mass equal to for all times. We refer to system (1.6) 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 (1.6) is motivated as follows. The right-hand side of (1.6) represents the velocity of each particle. Therefore, it has to be reminiscent of a discrete Lagrangian formulation of the Eulerian velocity in the continuity equation (1.3). Now, since we are in one-space dimension, the discrete density is a totally reasonable replacement for the continuum density , 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 into a backward and forward term is motivated by the sign of the nonlocal interaction , which is concordant with the sign of . Hence, since is negative on , particles labelled by with yield a drift on oriented towards the positive direction. Since the role of the nonlinear mobility term is that of preventing overcrowding at high densities (consistently with the assumption of being monotone decreasing), such a drift term should be “tempered” by the position of the -th particle. This motivates the use or in the sum with . A symmetric argument justifies the use of in the remaining part of the sum with .
Our main results concerns with the study of the many particle limit as for the discrete density defined above. Apart from the above mentioned assumptions on and and , we shall also assume that . Such a condition is crucial in order to prove the needed estimate which guarantee that converge (up to a subsequence) to some limiting density in a strong enough topology. As a minimal requirement, the limit should satisfy (1.3) in a distributional sense. On the other hand, the presence of a nonlinear convection in (1.3) suggests the possibility of multiple weak solutions for fixed initial data. A notion of entropy solution in the sense of Kruzkov  is therefore needed to secure uniqueness. Motivated by this remark, we shall actually prove that the limit density of the above particle scheme is an entropy solution to (1.3) with initial condition , in the sense of the following definition.
Definition 1.1 (Entropy solution).
Let . Denoting , we say that is an entropy solution of (1.3) with initial condition if and, for all constants and for all with one has
We are now ready to state the main result of our paper.
Assume and satisfy (Av) and (AK) respectively. Let be a compactly supported function with total unit mass and such that . Then, for all , the discrete density constructed in (1.7) converges almost everywhere and in to the unique entropy solution of the Cauchy problem
As a by-product, the above result also imply existence of entropy solutions for (1.8), a task which has been touched in other papers previously [9, 11, 5, 3]. Implicitly, our results also asserts the uniqueness of entropy solutions for (1.3), a side result that we shall prove as well in the paper, similarly to what done in [5, 6].
The need of the entropy condition to define a suitable notion of solution semigroup for (1.3) 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 2 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 3 we prove all the estimates needed in order to detect strong compactness for the approximating sequence . The main ingredient of this section is the estimate proven in Proposition 3.3. 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 4, we prove that the limit of the approximating sequence is an entropy solution in the sense of Definition 1.1. This task is quite technical as it requires checking a discrete version of Kruzkov’s entropy condition. In Section 5 we provide an explicit example of non uniqueness of weak solutions, which has links with the admissibility of steady states. Finally, in Section 6 we complement our results with numerical simulations.
2. The non-local Follow-the-leader scheme
In this section we introduce and analyse in detail our approximating particle scheme (1.6). Here the macroscopic variable does not need to be labelled by , as is supposed to be fixed throughout the whole section. The regularity assumptions on and in (Av) and (AK) imply that the right-hand side of (1.6) is locally Lipschitz with respect to the -tuple as long as we can guarantee that the denominator in 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 (1.6) for all times , and the conservation of the initial particle ordering during the evolution.
Lemma 2.1 (Discrete Maximum Principle).
Let be fixed and assume that (Av) and (AK) hold. In particular, let be as in assumption (Av). Let be the initial positions for (1.6), and assume that
Then every solution to the system (1.6) satisfies
Consequently, the unique solution to (1.6) with initial condition exists globally in time.
Let be the maximal existence time for (1.6). Due to the assumptions (Av) and (AK), the local-in-time solution is on . If we prove that (2.2) holds on , this will automatically prove global existence by a simple continuation principle. Arguing by contradiction, assume that is the first instant where two consecutive particles are the distance and get closer afterwards, i.e.
and there exists such that
Notice that the minimality of ensures that all particles maintain their initial order for all . At time we have due to (Av). Substituting this value in the equation (1.6) for , we easily see that only the terms survive in the nonlocal part, thus yielding . Similarly, we get . For similar reasons, if then the ODE for implies that at time we have , or equivalently . Similarly, if then . Let us now assume for the moment that . Then, with similar arguments as above one can show that and , and we can repeat the same argument above to obtain that implies and implies . Such a procedure can be iterated to conclude that there exists either some index with or some index such that , otherwise any two consecutive particles would be placed at distance and the system would be static for all , which would contradict the existence of .
The above considerations imply that we can assume, without loss of generality, that
Let be small enough such that , then by Taylor expansion one has
where, up to taking even smaller, the contribute does not affect the sign of . As a consequence, for all and a symmetric argument gives also for all . In particular, we deduce that
and this contradicts the existence of . This argument ensures both the validity of (2.2) and the existence of solutions for all times . ∎
Let us consider the discrete density
A straightforward consequence of Lemma 2.1 is that
Moreover, we observe that 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 will have no particles at its left for all times, which means that the ODE for will only feature terms with on the nonlocal sum. A symmetric statement holds for . As a consequence of that and for all , thus the support of is bounded by uniformly in and . We summarize this property in the next lemma.
Under the same assumptions of Lemma 2.1, the support of is contained in the interval for all times .
3. Convergence of particle scheme
The proof of Theorem 1.2 relies on two main steps: the first one consists in proving that the discrete density defined in (1.7) is strongly convergent (up to a subsequence) to a limit in , the second one is to show that the limit is a weak entropy solution of (1.8) according to Definition 1.1. In this section we take care of the former step. As we will show in Propositions 3.3 and 3.4 below, the sequence satisfies good compactness properties with respect to the space variables but, on the other hand, we cannot reach a uniform control on the time oscillations. In our case, we are only able to prove a uniform time continuity estimate with respect to the -Wasserstein distance (see ), which nevertheless will suffice to achieve the required compactness in the product space. Such a strategy recalls the one used in  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 1.2, the sequence is strongly relatively compact in
The proof of Theorem 3.1 relies on a generalized statement of the celebrated Aubin-Lions Lemma (see [34, 13, 14]) that we recall here for the reader’s convenience. In what follows, is the -Wasserstein distance.
Theorem 3.2 (Generalized Aubin-Lions Lemma).
Let be fixed. Let be a sequence in such that and for every and . If the following conditions hold
there exists a constant independent from such that for all ,
then is strongly relatively compact in .
Let and be as in the statement of Theorem 1.2. Then, there exists a positive constant (only depending on , , and on ) such that for every one has
Let and be as in the statement of Theorem 1.2. Then, there exists a positive constant (only depending on ) such that
Proof (of Proposition 3.3)..
It is easy to see that . Then estimate (3.1) follows by Gronwall Lemma as soon as we show that
for a suitable constant . The total variation of at time is given by
where we set for brevity
Then we can compute
The value of the coefficient clearly depends on the positions of the consecutive particles, it is easy to see that for
Recalling (3.3), we can rewrite
Let us first estimate in (3.5). Clearly, the only relevant contributions in the sum come from the particles for which . However, if the index is such that , then and the monotonicity of implies
The assumption (AK) on ensures that , thus, on the other hand, . An analogous argument implies that, if such that , then and . These considerations lead immediately to
Let us now focus on . In this case, we would like to obtain an upper bound in terms of and for this purpose we need to estimate . We recall that is locally Lipschitz and that . The former in particular implies that has finite Lipschitz constant on the compact interval , we name such a constant . We get
and this gives
We can now focus on and . Since the setting is symmetric, we only present the argument for and leave the one for to the reader. Since only if , without restriction we can assume and can compute
In particular, and
We now prove the equi-continuity w.r.t. time with respect to the -Wasserstein distance for .
Proof (of Proposition 3.4)..
Assume without loss of generality that . Our goal then is to investigate the continuity in time of the discrete density with respect to the -Wasserstein distance. We exploit the well known relation between the -Wasserstein distance of two probability measures and the distance of their respective pseudo inverse functions. More precisely, for any two probability measures the following identity holds
where and are the pseudo inverses of the cumulative distribution functions of and respectively. The assertion of the proposition will follow once we prove that there exists a constant independent of such that
for all . By the definition of we can explicitly compute
where in the last inequality we used that
Notice that we can control uniformly in and in . Indeed, recalling the assumption (AK), setting