Blow-up of solutions to the initial–boundary value problem for quasilinear hyperbolic systems of conservation laws

This work is a continuation of our previous work [Z.-Q. Shao, D.-X. Kong, Y.-C. Li, Shock reflection for general quasilinear hyperbolic systems of conservation laws, Nonlinear Anal. 66 (1) (2007) 93–124]. In the present paper, we study the global structure instability of the self-similar solution u=U(xt) containing shocks, contact discontinuities, and at least one rarefaction wave of the initial–boundary Riemann problem for general n×nn×n quasilinear hyperbolic systems of conservation laws on the quarter plane {(t,x)∣t≥0,x≥0}{(t,x)∣t≥0,x≥0}. We prove the nonexistence of global piecewise C1C1 solution to a class of the mixed initial–boundary value problem for general n×nn×n quasilinear hyperbolic systems of conservation laws on the quarter-plane {(t,x)∣t≥0,x≥0}{(t,x)∣t≥0,x≥0}. Our result indicates that the kind of Riemann solution u=U(xt) mentioned above is globally structurally unstable. As an application of our result, we consider the model proposed by Aw and Rascle [A. Aw, M. Rascle, Resurrection of “second order” models of traffic flow, SIAM J. Appl. Math. 60 (2000) 916–938] describing traffic flow on a road network. Following the work of Garavello and Piccoli [M. Garavello, B. Piccoli, Traffic flow on a road network using the Aw–Rascle model, Comm. Partial Differential Equations 31 (2) (2006) 243–275], we prove the nonexistence of global piecewise C1C1 solution containing one rarefaction wave of the initial–boundary value problem for this model.