The cavitation and concentration of Riemann solutions for the isentropic Euler equations with isothermal dusty gas

In this paper, we are mainly concerned with the phenomena of cavitation and concentration to the isentropic Euler equations with isothermal dusty gas as the pressure vanishes with double parameters. Firstly, we solve the Riemann problem by analyzing the properties of the elementary waves due to the existence of the inflection points. Secondly, we investigate the limiting behaviors of the Riemann solutions as the pressure vanishes and observe the cavitation and concentration phenomena. Finally, some numerical simulations are performed and the results are consistent with the theoretical analysis. The highlight of this paper is that we extend the restriction of rho theta << 1 in the previous works to rho theta < 1, which makes the wave curve from convex to non-convex. And we prove that the limit of the Riemann solutions of isothermal dusty gas equations is the Riemann solutions of the limit of that equations as pressure vanishes, while the limiting process to vacuum state is different from the previous works.