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.