Piston problem for the pressureless hydrodynamic traffic flow model

The hydrodynamic model can be used to describe traffic problems in transport. When the speed of the first car is less than the speed behind it, it leads to traffic jams. When the first car's speed is faster than the cars behind it, it leads to traffic evacuation. If we consider the first car to be a piston, then the speed of the piston will cause traffic jams and traffic evacuation. In this paper, we study the piston problem for the hydrodynamic model. The formation and propagation of shock wave, rarefaction wave, delta-shock wave, and vacuum can describe the phenomena of traffic jams, traffic evacuation, severe traffic jams, and traffic evacuation with traffic volume of zero, respectively. Therefore, for different traffic phenomena, we prove the existence of shock solution, rarefaction solution, delta shock solution, and vacuum solution. In addition, we perform some representative numerical simulations.