Delta-shocks and vacuums as limits of flux approximation for the pressureless type system

In this paper, we investigate the phenomena of concentration and cavitation and the formation of delta-shocks and vacuum states in solutions of the pressureless type system with flux approximation. First, the Riemann problem of the pressureless type system with a flux perturbation is considered. A parameterized delta-shock and generalized constant density solution are obtained. Then we rigorously prove that, as the flux perturbation vanishes, they converge to the delta-shock and vacuum state of the pressureless type system, respectively. Secondly, by adding an artificial pressure term in the pressureless type system, we solve the Riemann problem of the system with a double parameter flux approximation including pressure. It is shown that, as the flux perturbations vanish, any two-shock Riemann solution tends to a delta-shock solution to the pressureless type system; any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution to the pressureless type system and the intermediate nonvacuum state in between tends to a vacuum state.