$\delta-$shocks and vacuum states in the Riemann problem for isothermal van der Waals dusty gas under the flux approximation

In this paper, we analyze the Riemann problem for concentration and cavitation phenomena to the isothermal Euler equations for van der Waals gas with dust particles in the presence of flux approximation. By Riemann problem, we mean that it is an initial value problem for the system of conservation laws supplemented by a constant discontinuous initial data involving one breaking. The existence of $\delta-$shock and vacuum state in the Riemann problem for the system under consideration are examined. Further, it is shown that as the flux approximation and pressure vanish, the Riemann solution containing two shock waves to the isothermal Euler equation for van der Waals gas with dust particles converge to $\delta-$shock solution and the Riemann solution containing two rarefaction waves tend to the vacuum state solution of the transport equations. Numerical simulations presenting the formation of $\delta-$ shocks and vacuum states for different flux approximations are also shown.