The singular wave in a pressureless hydrodynamic model

Abstract

In this paper, we investigate the non-classical wave for a pressureless hydrodynamic model with the flux perturbed term by the Riemann problem and a singularity formation. All the possible Riemann solutions, the combination of two contact discontinuities $J_1+J_2$, and a delta shock wave $\delta S$, are constructed in fully explicit forms. It should be mentioned that the delta shock wave appears in the solution if and only if the flux perturbed parameter $\varepsilon$ satisfies some specific condition. Due to the particularity of the delta shock wave in the Riemann solutions, we investigate the formation of singularity, namely, the traffic density blowing up under certain data. Moreover, its result gives an example of the conjecture proposed by Majda [Springer New York, 1984]: “If a hyperbolic system of conservation laws is totally linearly degenerate, then the system has smooth global solutions when the initial data are smooth, unless the solution itself blows up in a finite time.” We further explore and discuss their asymptotic behaviors to analyze the effect of $\varepsilon$, in which the delta shock wave and vacuum state solutions for a pressureless hydrodynamic model can be obtained by $J_1+J_2$ as $\varepsilon$ tends to 0. In addition, we offer some typical numerical simulations that are identical well to our theoretical results and provide a more intuitive way to observe the singular wave.