Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 23, pp. 1-11.

Title: L^1 singular limit for relaxation and viscosity 
       approximations of extended traffic flow models

Authors: Christian Klingenberg (Wurzburg Univ., Germany)
         Yun-guang Lu (Uiv. Nacional, Bogota Colombia)
	 Hui-jiang Zhao (Chinese Acad. of Sci., Wuhan, China)

Abstract: 
 This paper considers the Cauchy problem for an extended 
 traffic flow model with $L^1$-bounded initial data. 
 A solution of the corresponding equilibrium equation
 with $L^1$-bounded initial data is given by the limit of
 solutions of viscous approximations of the original
 system as the dissipation parameter $\epsilon$ tends to zero 
 more slowly than the response time $\tau$. The proof of 
 convergence is obtained by applying the Young measure to 
 solutions introduced by DiPerna and, based on the estimate
 $$
 |\rho(t,x)| \leq \sqrt {|\rho_0(x)|_1/(ct)}
 $$
 derived from one of Lax's results and Diller's idea, 
 the limit function $\rho(t,x)$ is shown to be a $L^1$-entropy 
 week solution. A direct byproduct is that we can get the 
 existence of $L^1$-entropy solutions for the Cauchy problem  
 of the scalar conservation law with $L^1$-bounded initial data
 without any restriction on the growth exponent of the flux 
 function provided that the flux function is strictly convex.  
 Our result shows that, unlike the weak solutions of the 
 incompressible fluid flow equations studied by DiPerna 
 and Majda in [6], for convex scalar conservation laws with 
 $L^1$-bounded initial data, the concentration phenomenon will 
 never occur in its global entropy solutions.
 
Submitted October 25, 2002. Published March 7, 2003.
Math Subject Classifications: 35B40, 35L65.
Key Words: Singular limit; traffic flow model; relaxation and 
           viscosity approximation.