Electron. J. Diff. Eqns., Vol. 2001(2001), No. 14, pp. 1-18.

$L^1$ stability of conservation laws for a traffic flow model

Tong Li

We establish the $L^1$ well-posedness theory for a system of nonlinear hyperbolic conservation laws with relaxation arising in traffic flows. In particular, we obtain the continuous dependence of the solution on its initial data in $L^1$ topology. We construct a functional for two solutions which is equivalent to the $L^1$ distance between the solutions. We prove that the functional decreases in time which yields the $L^1$ well-posedness of the Cauchy problem. We thus obtain the $L^1$-convergence to and the uniqueness of the zero relaxation limit. We then study the large-time behavior of the entropy solutions. We show that the equilibrium shock waves are nonlinearly stable in $L^1$ norm. That is, the entropy solution with initial data as certain $L^1$-bounded perturbations of an equilibrium shock wave exists globally and tends to a shifted equilibrium shock wave in $L^1$ norm as $t\to \infty$. We also show that if the initial data $\rho_0$ is bounded and of compact support, the entropy solution converges in $L^1$ to an equilibrium $N$-wave as $t\to \infty$.

Submitted June 2, 2000. Published February 20, 2001.
Math Subject Classifications: 5L65, 35B40, 35B50, 76L05, 76J10.
Key Words: Relaxation, shock, rarefaction, L^1-contraction, traffic flows, anisotropic, equilibrium, marginally stable, zero relaxation limit, large-time behavior, L^1-stability.

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Tong Li
Department of Mathematics, University of Iowa
Iowa City, IA 52242, USA
e-mail: tli@math.uiowa.edu
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