Concentration and cavitation phenomena of Riemann solutions for the isentropic Euler system with the logarithmic equation of state under the flux approximation

In this paper, we investigate the concentration and cavitation phenomena of Riemann solutions for the isentropic Euler system with the logarithmic equation of state under the flux approximation. The concentration and cavitation are fundamental and physical phenomena in fluid dynamics, which can be mathematically described by delta shock waves and vacuums, respectively. The main objective of this paper is to rigorously investigate the formation of delta shock waves and vacuums and observe the concentration and cavitation phenomena. Firstly, the Riemann problem for the isentropic Euler system with the logarithmic equation of state under the flux approximation is solved analytically. Secondly, it is rigorously proved that, as the pressure and flux perturbation all vanish, any two-shock-wave Riemann solution tends to a δ-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure which form the δ-shock; while any two-rarefaction-wave Riemann solution tends to a solution consisting of four contact discontinuities together with vacuum states with three different virtual velocities in the limiting scenario.