Critical thresholds in a relaxation system with resonance of characteristic speeds. (English) Zbl 1171.35428
The paper deals with critical threshold phenomena in a dynamic continuum traffic flow model. The Cauchy problem for the hyperbolic system \(\rho _t+(\rho u)_x=0,\) \(u_t+uu_x+\frac{1}{\rho}p(\rho)_x=\frac{1}{\tau}(v_e(\rho)-u),\) where \(p(\rho)=c_0^2\rho,\) \( v_0(\rho)=c_0\ln \frac{1}{\rho},\) \( 0<\rho \leq 1,\) is considered. The authors derive a priori estimates of the derivatives of the Riemann invariants in order to identify the upper threshold for global existence of smooth solutions and lower threshold for finite time singularity formation.
Reviewer: Marie Kopáčková (Praha)
MSC:
35L60 | First-order nonlinear hyperbolic equations |
35B45 | A priori estimates in context of PDEs |
35L45 | Initial value problems for first-order hyperbolic systems |
35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
35L67 | Shocks and singularities for hyperbolic equations |
90B20 | Traffic problems in operations research |