Relaxation Limit for Aw-Rascle System

DELACRUZGUERRERORichardA.,JUAJIBIOYJuanC.and REND′ON LeonardoDepartamento de Matem′aticas,Universidad Nacional de Colombia,Bogot′a D.C., Colombia.

2Escuela de Matem′aticas y Estad′?stica,Universidad Pedag′ogica y Tecnol′ogica de Colombia,Tunja,Colombia.

Relaxation Limit for Aw-Rascle System

DELACRUZGUERRERORichardA.1,2,JUAJIBIOYJuanC.1,?and REND′ON Leonardo11Departamento de Matem′aticas,Universidad Nacional de Colombia,Bogot′a D.C., Colombia.

2Escuela de Matem′aticas y Estad′?stica,Universidad Pedag′ogica y Tecnol′ogica de Colombia,Tunja,Colombia.

Received 21 November 2013;Accepted 8 April 2014

.We study the relaxation limit for the Aw-Rascle system of traffic flow.For this weapplythetheoryofinvariantregionsandthecompensatedcompactnessmethod to get global existence of Cauchy problem for a particular Aw-Rascle system with source,where the source is the relaxation term,and we show the convergence of this solutions to the equilibrium state.

Aw-Rascle system;relaxation term;compensated compactness;invariant regions.

1 Introduction

In[1]the authors introduce the system

as a model of second order of traffic flow.It was proposed by the author to remedy the deficiencies of second order model or car traffic pointed in[2]by the author.The system(1.1)models a single lane traffic where the functions ρ(x,t)and v(x,t)represent the density and the velocity of cars on the road way and P(ρ)is a given function describing the anticipation of road conditions in front of the drivers.In[1]the author solves the

Riemann problem for the case in which the vacuum appears and the case in which the vacuum does not.Making the change of variable

the system(1.1)is transformed in to the system

Multiplying the second equation in(1.2)by ρ we have the system

Now making the substitution m=wρ,system(1.3)is transformed in to system

where φ(ρ,m)=m/ρ-P(ρ),this is a system of non symmetric Keyfitz-Kranzer type. In[3],the author,using the Compensate Compactness Method,shows the existence of global bounded solutions for the Cauchy problem for the homogeneous system(1.4).In this paper we are concerned with the Cauchy problem for the following Aw-Rascle system

with bounded measurable initial data

The Riemann’s invariants are given by

Now as

we see that the second wave family is always linear degenerate and the behavior of the second family wave depends to the values of ρP(ρ).In fact,if θ(ρ)=ρP(ρ)is concave or convex then the second family wave is genuinely non linear,see[5]to the case in which the two families wave are linear degenerate.

2 The positive invariant regions

In this section we show the theorem for invariant regions for find a estimates a priori of the parabolic system(3.3).

Proposition 2.1.LetO???R2be a compact,convex region whose boundary consists of a finite number of level curves γjof Riemann invariants,ξj,such that

with initial data

exists in[0,∞)×R and(u?(x,t),v?(x,t))∈O.

Proof.It is sufficient to prove the result for

be the unique solution of the Cauchy problem

where F=(f,g)and P(U)=|U-Y|2for some fixed Y∈O.If we suppose that U?,δ/∈Ofor all(x,t),then there exist some t0＞0 and x0such as

Now by(2.1)we have that

and

The characterization of(x0,t0)implies

Replacing(2.6)-(2.8)in(2.5),we have that

which is a contradiction.Now we show that U?,δ→Uδas δ→0.For this let W?,δ,σbe the solution of

where

Multiplying by W?,δ,σin(2.9),and integrating over R,we have

Then,integrating respect to variable t over interval(0,t)we have that

Finally by applying Gronwall’s inequality we obtain

then,for σ=0 and δ→0 we have that U?,δ→U?as δ→0.

3 Relaxation limit

Based in the Theory of Invariant Regions and Compensated Compactness Method we can obtain the following result.

Theorem 3.1.Let h(ρ)∈C(R).Suppose that there exists a region

where C1＞0,C2＞0.Assume that Σ is such that the curve m=h(ρ)as 0≤ρ＜ρ1and the initial data(1.5)are inside Σ and(ρ1,m1)is the intersection of the curve W=C1with Z=C2.Then, for any fixed ?＞0,τ＞0 the solution(ρ?,τ(x,t),m?,τ(x,t))of the Cauchy problem(1.5),(1.6) globally exists and satisfies

Moreover,if τ=o(?)as ?→0 then there exists a subsequence(ρ?,τ,m?,τ)converging a.e.to(ρ,m) as ?→0,where(ρ,m)is the equilibrium state uniquely determined by:

I.The function m(x,t)satisfies m(x,t)=h(ρ(x,t))for almost all(x,t)∈[0,∞)×R.

II.The function ρ(x,t)is the L∞entropy solution of the Cauchy problem

The proof of this theorem is postponed for later,first we collect some preliminary estimates in the following lemmas.

Lemma 3.1.Let(ρ?,m?)be solutions of the system(1.5),with bounded measurable initial data (1.6),and the following stability condition

it holds.Then(ρ?,m?)is uniformly bounded in L∞with respect to ? and τ. Proof.First,we show that the region

is invariant for the parabolic system(see Fig.1)

If γ1is given for m(ρ)=C1ρ and γ2is given for m(ρ)=ρC2+ρP(ρ),it is easy to show that if u=(ρ,m)∈γ1and y=(ρ,m)∈Σ it then holds

Figure 1:Riemann Invariant Region I.

Figure 2:Riemann Invariant Region II.

and if u=(ρ,m)∈γ2and y=(ρ,m)∈Σ then we have

Using Proposition 2.1,we have that Σ is an invariant region for(3.3).For the case in which the system(1.5)contains relaxation term we use the ideas of the authors in[6]

the system(1.5)is transformed into the system

which does not depend on τ,and taken C1=W(1,0),C2=Z(1,0)the curves M=Q(R), W=C1,Z=C2intersect with the R axis at the same point in R=0,R=1(see Fig.2). Using the stability conditions(3.2),it is easy to show that the vector(0,h(R)-M)points inwards the region Σ2and from[7]it follows that Σ2is an invariant region.

Multiplying the system(1.5)by(Qρ,Qm)we have

Adding terms and applying the mean value theorem in the m variable to the functionsand Q(ρ,m)we have that

where

Putting φ2(ρ,m)=mφ(ρ,m)and proceeding as above we have that

where

Now replacing the values of Q and φ1in T2we have that

then

where C=max(|h′′(ρ)m-C1|,|h′(ρ)|).Using the Young’s δ-inequality

we have that

For T3and3we have

Let us introducing the following

and R(ρ,m)=T1+bT2.Then substituting(3.8),(3.9)in(3.7)and using(3.12)-(3.15),and δ=1/8 we have

For ? sufficiently small we can choose C2,C4such that C4τ≤(C2-T)? for T＞0.Let K be a compact subset of R×R+and Φ(x,t)∈D(R×R+),such that Φ=1 in K,0≤Φ≤1. Then,multiplying(3.19)by Φ(x,t)and integrating by parts we have

The proof is complete.

Lemma 3.3.If(η(ρ),q(ρ))is any entropy-entropy flux pair for the scalar equation

then

is compact in H-1(R×R+).

Proof.Adding ψ(ρ)=ρφ(ρ,h(ρ))in the first equation of(1.5)we have

and multiplying by η′in(3.22)we have that

η(ρ)t+q(ρ)x=?η(ρ)xx-?η2(ρ)ρxx+?η′(ρ)(ψ(ρ)-ρφ(ρ,m))?-η2(ρ)(ψ(ρ)-ρφ(ρ,m))ρx.

and

The proof is complete.

Now we prove Theorem 3.1.By the Lemma 3.1 we have the a priori bounds(3.1),and we also have that there is a subsequence of(ρ?,m?)such as

Let us introduce the following

Then by the weak convergence of determinant[9,p.15],we have that

by direct calculations,replacing ρ?in(3.24)-(3.27)we have that

an since by(3.23)

we have that

Now,using Minty’s argument[10]or arguments of author in[11]it is finished the proof of the Theorem 3.1.?

Acknowledgments

We would like to thanks Professor Juan Carlos Galvis by his observation and many valuable suggestions,and to the professor Yun-Guang Lu by his suggestion this problem.

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[11]Cheng Zhixin,Relaxation limit for a symmetrically hyperbolic system.Nonl.Anal.:TMA,72 (2010),555-561.

10.4208/jpde.v27.n2.7 June 2014

?Corresponding author.Email addresses:richard.delacruz@uptc.edu.co(R.De la cruz),jcjuajibioyo@ unal.edu.co(J.Juajibioy),lrendona@unal.edu.co(L.Rend′on)

AMS Subject Classifications:35L65

Chinese Library Classifications:O175.27