## Abstract

Motivated by a traffic model, we study a conservation law whose solutions naturally need to be bounded Radon measures. We investigate the relations between this equation and finite systems of conservation laws that converge to it.

## 1. Introduction

A problem widely considered in the literature is the limiting procedure that allows to pass from individual to collective descriptions. Examples are the convergence of the Boltzmann equation to the Euler equation or the justification of continuum traffic flow models.

The aim of this paper is to consider the opposite limit, see Mischler (1997) and Mieussens (2001). We consider a recently proposed extension (Benzoni & Colombo 2003) of the classical Lighthill–Whitham (Lighthill & Whitham 1955) and Richards (1956) (LWR) equation. In this model, *n* classes of drivers are considered. Here, we let *n*→∞ and obtain a kinetic system whose convective part is *nonlinear* and *non-local*, its collision terms are *zero*.

To this end, we need to introduce a few concepts about conservation laws in Banach spaces. Here, the unknown function attains values in the set of positive Radon measures. Measure valued solutions to conservation were already considered in DiPerna (1985) from an abstract point of view. In the present case, the physical motivation leads us to consider a specific flow with unusual analytical properties, see lemma 2.4. In particular, the present flow does not fit into the framework of DiPerna (1985), is not linear and in a sense *not local*. By this we mean that the flow considered below is defined on and attains values in the same space , while in DiPerna (1985), the flow attains values in .

More precisely, the classical LWR model reads*ρ* being the car density and *v*=*v*(*ρ*) the (average) car speed. A natural generalization (Benzoni & Colombo 2003) to the case of *n* classes of vehicles is the *n*×*n* system,(1.1)here, is the density at (*t*, *x*) of the vehicles of class *i* in the *n*-population model. We assume, as in Benzoni & Colombo (2003), that the speed law of the *i*th population in the *n*-classes model has the form(1.2)where the constants are the maximal possible speeds for the vehicles in the *i*th class. We assume that for all *i*, *n* and for suitable positive fixed constants *V*_{*},*V*^{*}. Above, is a smooth decreasing function representing the attitude of the drivers of the *i*th class to adapt their speed to the total car density. Note that we introduced the simplifying assumption that all drivers have the same attitude, independently from the class to which they belong. For further details on equations (1.1) and (1.2), see Benzoni & Colombo (2003).

A suitable limiting procedure with *n*→+∞ amounts to pass from a multi class macroscopic model to a kinetic model. More precisely, given a solution of equation (1.1), letwhere is the Dirac measure on [*V*_{*},*V*^{*}] supported at . Formally, *μ*_{n} tends to a limit distribution *μ*=*μ*(*t*, *x*) representing the density of drivers having maximal speed *V* at time *t* and position *x*. The measure *μ* solves the following system of conservation laws(1.3)which is the ‘*limit*’ of equation (1.1). The speed law *v* associates to any Radon measure on [*V*_{*},*V*^{*}] a function in **C**^{0}([*V*_{*},*V*^{*}])(1.4)the latter is the ‘*limit*’ of equation (1.2). The above can be seen as an infinite system of conservation laws, the index *i*∈{1, …, *n*} being substituted by the continuous variable *V*∈[*V*_{*},*V*^{*}]. It differs from the usual Boltzmann system both in the convective part and in the collision terms. Here, the former is nonlinear and non-local, while the latter vanishes. The relations between the finite system (1.1) and (1.2) and its continuous counterpart (1.3) and (1.4) is clearer when the two systems are written aswhere in the latter equation, with a slight abuse of notation, we put *ρ* for the measure *μ*.

We rigorously show below that the discrete velocity system (1.1) and its continuous counterpart (1.3) are strictly related. Indeed, the well posedness of equation (1.1) for any *n* implies that of equation (1.3) and vice versa.

## 2. Preliminary results

### (a) Conservation laws in a Banach space

Let be a Banach space with dual . We denote by the space equipped with the strong topology, while (respectively ) is the same space with the weak (respectively weak*) topology.

In the present paper, we use the following notions of measurability and of integral, see Diestel & Uhl (1977, ch. II, §3).

A map is (weak) measurable if for all , the map is measurable in the usual sense. Moreover, a measurable map is integrable if for all , the map is integrable. We set , for some , if and only if for all .

A map is (weak*) measurable if for all , the map is measurable in the usual sense. Moreover, a measurable map is integrable if for all , the map is integrable. We set , for some , if and only if for all .

Correspondingly, throughout the present paper, the functional spaces **L**^{1} or **L**^{∞} refer to the above weak or weak* notions of measurability. Recall that , see Diestel & Uhl (1977, theorem 1, ch. IV).

Consider a Lipschitz map with . Fix an initial datum . We call *distributional* solution to the Cauchy problem(2.1)any map such that(2.2)for any test function . A *weak* solution to equation (2.1) is a distributional solution in , i.e. in the set of maps such that for a positive *L* independent from *t*,*t*′∈[0,*T*].

Finally, inspired by the definition in Bressan (2000), we say that equation (2.1) generates the valued semigroup *S* if is a map such that

for a suitable

*δ*>0;*S*is a semigroup;For , the orbit is a weak solution to equation (2.1) and

*S*is Lipschitz, i.e. there exists a constant*L*such that for all and*u*,

Above, we used the following definition of total variation, for (2.3)The relation between the integral formulation of a conservation law and its weak formulation is guaranteed by the following lemma.

*Let* *satisfy (i), (ii), (iv), with* . *Then, (iii) is equivalent to require that for all* (2.4)

The proof is deferred to §4.

### (b) The case of traffic flow

Here, we let so that is the Banach space of Radon measures on equipped with the total variation norm(2.5)and . Define bywhere is Lipschitz with Lipschitz constant *L*.

We first investigate the properties of the flow *f*. This nonlinear map is strongly Lipschitz and also weak* continuous. The proofs of the following lemmas are in §4.

*The map* *is Lipschitz continuous and the map* *is weak** *sequentially continuous*.

For any positive *T*, the function induces a map . This map may well fail to be weak* sequentially continuous in , even in the case of a linear *ψ*.

*Fix a positive time T and let ψ*(*r*)=*r*. *Then*, *there exists a sequence μ*^{n} *in* *such that* *in the space* *, while* *does not converge weak** *to* *in* .

To obtain the weak* sequential continuity, a further condition is needed.

*Fix a positive T and let μ*^{n} *be a sequence such that**Then,* *in* .

## 3. Main results

This section is devoted to prove the equivalence between the discrete system(3.1)for *i*=1, …, *n* and its continuous counterpart(3.2)Introduce the assumptions

(

): is non-negative and Lipschitz;**Ψ**(

): for all**V***n*;

and the simplexes

### (a) Distributional solutions

*Fix T*>0. *Let* (* Ψ*)

*and*(

*)*

**V***hold. Choose a sequence*.

*Assume that for all*

*, the Cauchy problem equation*(3.1)

*admits a distributional solution*

*and that the sequence*

*is strongly relatively compact in*.

*Then, there exists a*

*and a solution*

*to equation*(3.2)

*such that, up to a subsequence,*(3.3)

*Moreover*,

*if*

*is uniformly bounded in*

*, then*.

Remark that definition 2.2 also applies to , since .

Define(3.4)By the definition of *Δ*_{n}, the sequence , respectively *μ*_{n}, is weak* relatively compact in , respectively . Therefore, eventually passing to subsequence, we assume in the sequel that in and in

Since is a distributional solution (Bressan 2000, definition 4.2) of equation (3.1), for any , we haveIntroduce a ,Therefore, for any ,proving that for any *n*, *μ*_{n} is a distributional solution to equation (3.2). Now apply lemma 2.6 and passing to the weak* limit, we also prove that *μ* is a distributional solution to equation (3.2).

Observe thathence in .

Assume now that is uniformly bounded in . Then, .

The convergence strongly in , ensures that for a.e. *t*∈[0,*T*], in . Then, also for a.e. *t*∈[0,*T*], *μ*_{n}()(*t*) converges pointwise a.e. to *μ*()(*t*). By Fatou lemma the proof is completed. ▪

The next corollary about the propagation speed of the solution is motivated by the uniform bound on the characteristic speeds proved in Benzoni & Colombo (2003).

*Fix T*>0. *Let* (* ψ*)

*and*(

*)*

**V***hold. Choose a sequence*

*as in*

*theorem*3.1

*and assume that there exists a Λ*>0

*such that if*

*for a.e.*

*, then*

*for a.e. t*∈[0,

*T*]

*and*.

*Then, also*

*vanishes for a.e.*

*and μ*(

*t*,

*x*)=0

*for a.e.*

*and*.

The proof is immediate, for weak* convergence preserves the support.

### (b) Continuity with respect to time

In the sequel, we choose the **L**^{1} norm in , i.e. for all (3.5)Let (* V*) hold. If

*μ*∈

*Δ*

_{∞}, then for any

*n*, the

*n*-tuple

*ρ*

_{n}defined by(3.6)belongs to

*Δ*

_{n}.

*Let the hypotheses of* *theorem* 3.1 *hold*. *Moreover*, *assume that for an α∈*]0,1]*, the solutions ρ*_{n} *to equation* (3.1) *are in* *with Hölder constant L independent from n*. *Then, there exists a* *and a solution* *to equation* (3.2) *with Hölder constant L such that equation* (3.3) *holds, up to a subsequence*.

Above, the Hölder continuity of has to be understood in the strong sense, i.e.Recall that is separable, so that it admits a countable dense subset which allows the introduction in of the metric(3.7)that induces on the Fréchet space its weak* topology. In the following proof, we use also the weak* Hölder continuity, namely

Introduce and *μ*_{n} as in equation (3.4). By theorem 3.1, we may assume that equation (3.3) holds for a suitable and a corresponding . The uniform Hölder condition and the compactness of the sections allow to apply Ascoli theorem (Kelley 1975, theorem 17, ch. 7). By possibly passing to a subsequence, we obtain that and that

By the uniform Hölder continuity of the *ρ*_{n}, also the *μ*_{n} are weakly uniformly Hölder continuous, in the sense that for all *t*_{1}, *t*_{2}∈[0,*T*] and all . The norm of the space is lower semicontinuous with respect to the Fréchet topology induced by d_{F}, therefore for all *t*_{1}, *t*_{2}∈[0,*T*]. Passing to the limit *n*→∞ we obtain the Hölder continuity of *μ*. ▪

Below, we need the following assumption, slightly stronger than (* V*):

(* VV*): for all

*n*and moreover .

*Let* (* VV*)

*hold*.

*Then*,

*the sequence ρ*

_{n}

*defined by equation*(3.6)

*satisfies*.

Fix *a*, *b*∈ with *a*<*b*. By (* VV*), there exist sequences

*i*

_{n}and

*j*

_{n}such that ; for all

*n*, and ; and . Thenby the continuity of bounded measures from below. ▪

*Let* (**VV**) *and* (* ψ*)

*hold*.

*Fix a datum*

*for equation*(3.2).

*Define the sequence*

*through*

*and assume that the corresponding Cauchy problem*(3.1)

*admits a distributional solution*

*with Hölder constant uniform in n. If the sequence*

*is relatively compact in*.

*Then*,

*equation*(3.2)

*admits a distributional solution*.

The proof follows directly from lemmas 3.3 and 3.4.

### (c) The semigroup

Throughout this section, we write for brevity *Δ*_{n} meaning and, similarly, *ρ*_{n} for . We keep using below the norm (3.5) in . Define(3.8)(3.9)The latter set above is the closure with respect to the norm (2.5) of the convex hull of Dirac's deltas centred at some . Note that . Note that the choice equation (3.8) satisfies (* VV*).

*Let equation* (3.8) *and* (*ψ*) *hold*. *Assume that for fixed L and for all n*, *there exists a semigroup* *such that*

1_{n} *for all* *, the map* *is a distributional solution to equation* (3.1);*Then*, *there exists a semigroup* *such that*

1_{∞} *for all* *, the map* *is a distributional solution to equation* (3.2);

Choose two initial data . Then, there exist two sequences of finite linear combinations of Dirac deltas centred at the such that and strongly in . Let be such that and . Then, .

Define and . By 1_{n}, *μ*_{n} and *v*_{n} are distributional solutions of equation (3.2) with initial datum . Moreover, *2*_{n} and the choice (3.5) ensure thatPassing to the strong limit in we obtain *2*_{∞}. Indeed, both sequences *μ*_{n} and *v*_{n} satisfy the Cauchy condition. In the case of *μ*_{n}, ▪

We remark here that if the semigroups *S*^{n} in the proposition above are assumed to be defined on domains of the type for a given *M*>0, then the valued semigroup *S*^{∞} turns out to be defined correspondingly on the domain .

### (d) Entropy

This paragraph is devoted to show that the presence of a superlinear entropy avoids the concentration phenomena: if the initial data is absolutely continuous with respect to the Lebesgue measure and possesses a finite initial entropy, then the corresponding solution *μ*(*t*) is also absolutely continuous with respect to the Lebesgue measure with a finite entropy for all *t*≥0. However, the formal extension to conservation laws in Banach spaces of the various definitions and properties of entropies and entropic solutions is not within the scope of the present work.

Note first that each discrete problem (3.1) admits the entropy–entropy flux pair(3.10)*ψ* being a primitive of *ψ*. We remark that this choice *differs* from that in Benzoni & Colombo (2003). Lemma 4.1 explains why the choice therein has to be abandoned.

Assume that the solution *ρ*_{n} to the finite system (1.1) and (1.2) converges as *n*→+∞ to a measure *μ* absolutely continuous with respect to the Lebesgue measure with the density . Then, formally, the above pair (3.10) for the discrete system becomesfor the infinite system. To formalize the absence of concentration we introduce the subsetof the densities of absolutely continuous measures in *Δ*_{∞} having finite entropy. The next proposition states that if the initial datum to equations (1.3) and (1.4) is absolutely continuous and has finite entropy, then the solution also enjoys the same properties.

*Let the assumptions of* *theorem* 3.1 *hold. Assume moreover that the initial data* *converge to an absolutely continuous measure* *with density* *belonging to* . *Then, the corresponding solution μ is an absolutely continuous measure with density ρ in* .

Note that *E*_{n} is non-negative and computed along the solutions to equations (1.1) and (1.2) is non-increasing. Therefore, the map is bounded uniformly in time.

Now, let . Note that on one hand, , *μ* being the solution to equation (1.3) with data constructed in theorem 3.1. On the other hand, , so that the *r*_{n} are uniformly integrable in *L*^{1}. Therefore, using the lower semicontinuity of convex functionals with respect to the weak convergence, *μ* is absolutely continuous with respect to the Lebesgue measure. ▪

## 4. Technical proofs

Let *S* be a semigroup generated by equation (2.1). Note that for all *u*∈, for all *t*. Fix *t*_{1},*t*_{2}∈[0, +∞[ with *t*_{1}<*t*_{2} and with *x*_{1}<*x*_{2}. Let *Χ*_{n} be the characteristic function of . Denote by *η*_{n} a sequence of mollifiers converging to Dirac delta. Choose *η*_{n}*Χ*_{n} as test function in equation (2.2) and pass to the limit *n*→+∞, obtaining equation (2.2).

Assume now that *S* satisfies (i), (ii), (iv) and equation (2.4). Then, *S* satisfies (iii) by essentially the same techniques in Vol'pert & Hudjaev (1985), ch. 5. ▪

Fix with . Consider first the Lipschitz property: for all *μ*, *v*∈*Ω*, computeSimilar computations yieldshowing Frechet differentiability.

Now let *μ*_{n} be a sequence in *Ω* such that , for a suitable *μ*∈*Ω*. Then, following the same computations as above, we obtainThe quantity *ψ*(*μ*_{n} ()) is bounded uniformly in *n*. For any , the map is also in **C**^{0}(), so that vanishes as *n*→+∞. Concerning the latter summand above, the continuity of *ψ* ensures that and that is bounded uniformly in *n*, completing the proof. ▪

Fix a compact set . Choose a sequence *ρ*_{n} in such that spt and in for a suitable with spt . Assume, moreover, that the *ρ*^{n} do not converge strongly in . Fix and define , so that . For any measurable with ,By lemma 2.4, the condition in then implies that in . By Brezis (1983, proposition III. 30), the convergence *ρ*^{n}→*ρ* also needs to be strong, against the choice of *ρ*^{n}. ▪

The tensor products with and *b*∈**C**^{0}() are dense in . Thenand both the last two lines above vanish. The former one by the convergence , while the latter by . ▪

*Let* (* V*)

*hold and fix R*=1.

*Consider the entropy–entropy flux pair*(4.1)

*introduced in*

*Benzoni*&

*Colombo*(2003).

*There exists a sequence of initial data*

*, with*,

*such that*

*problem*(3.1)*admits a weak solution*;*ρ*_{n}*is entropic with respect to the entropy–entropy flux pair**equation*(3.10);*the sequence**satisfies the assumptions of**theorem*3.1*for any T*and*for all t*,*x*.

For with *n*>1, define . Then, problem (3.1) admits the constant and stationary solution . Note that for all *n*, *t*, *x*. Moreover,completing the proof. ▪

## Acknowledgments

This work was completed while P.G. was at the Brescia Department of Mathematics, supported by the HYKE-project, EU (RTN), HPRN-CT-2002-00282. Support from the Brescia University fund for international activities is also acknowledged. The authors thank the referees for their valuable and constructive criticisms.

## Footnotes

- Received September 21, 2004.
- Accepted December 21, 2005.

- © 2006 The Royal Society