\(L^1\) stability of conservation laws for a traffic flow model. (English) Zbl 0964.35095

Li, Tong

Electron. J. Differ. Equ. 2001, Paper No. 14, 18 p. (2001).

The author establishes \(L^1\) well-posedeness of the Cauchy problem for a hyperbolic system of conservation laws with relaxation \[ \rho_t+(\rho v)_x=0,\qquad v_t+\biggl({v^2\over 2}+g(\rho)\biggr)_x={v_e(\rho)-v\over\tau}, \] where \(g'(\rho)=\rho(v'_e(\rho))^2\), \(\tau>0\) is a relaxation parameter. The systems of such kind arise in traffic flows. The continuous dependence of the solution on its initial data in \(L^1\) topology is proved. On the base of this result the zero relaxation limit is justified. It is proved that a sequence of solutions of the relaxed system converges to a solution of the equilibrium scalar equation \[ \rho_t+(\rho v_e(\rho))_x=0. \] The author also studies the large-time asymptotic behavior of the equilibrium entropy solutions. In particular the \(L^1\) stability of the equilibrium shock waves is established.

Reviewer: Evgeniy Panov (Novgorod)

MSC:

35L65	Hyperbolic conservation laws
35B40	Asymptotic behavior of solutions to PDEs
76L05	Shock waves and blast waves in fluid mechanics
35B50	Maximum principles in context of PDEs
35L45	Initial value problems for first-order hyperbolic systems

Keywords:

\(L^1\) well-posedness; relaxation; continuous dependence of the solution on its initial data; equilibrium entropy solutions; zero relaxation limit; large-time behavior; equilibrium shock waves