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. | Turk J Math (2018) 42: 2735 – 2751 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Delta-shocks and vacuums as limits of flux approximation for the pressureless type system Jinjing LIU1,2∗,, Hanchun YANG1 , Department of Mathematics, Yunnan University, Kunming, . China 2 School of Mathematics, Center for Nonlinear Studies, Northwest University, Xi’an, . China 1 Received: • Accepted/Published Online: • Final Version: Abstract: 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. Key words: Pressureless type system, Riemann problem, delta-shock, vacuum state, flux approximation 1. Introduction The pressureless type system reads as { ρt + (ρf (u))x = 0, (ρu)t + (ρuf (u))x = 0, () where ρ and u represent the density and velocity, and f (u) is given to be a smooth and strictly monotone function. The Riemann solutions of () were obtained in .
