\(L^1\) stability of conservation laws for a traffic flow model (Q2703369)

The author establishes \(L^1\) well-posedeness of the Cauchy problem for a hyperbolic system of conservation laws with relaxation NEWLINE\[NEWLINE\rho_t+(\rho v)_x=0,\qquad v_t+\biggl({v^2\over 2}+g(\rho)\biggr)_x={v_e(\rho)-v\over\tau},NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE \rho_t+(\rho v_e(\rho))_x=0. NEWLINE\]NEWLINE 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.