A class of piecewise constant Radon measure solutions to Riemann problems of compressible Euler equations with discontinuous fluxes: pressureless flow versus Chaplygin gas

We investigate the wave structure and new phenomena of the Riemann problems of isentropic compressible Euler equations with discontinuous flux in momentum caused by different equations of states, including pressureless flow and Chaplygin gas. Specifically, we focus on solutions within the class of Radon measures. To resolve the discontinuous flux, we introduce a delta shock that admits mass concentration between the pressureless flow on the left and Chaplygin gas on the right. By exploring both the classical and singular Riemann problems, we find that a global delta shock solution exists, satisfying the over-compressing condition. This finding is a generalization of classical theories on Riemann problems. In particular, we demonstrate that a vacuum left state and right Chaplygin gas can always be connected by a global delta shock satisfying the over-compressing condition. For singular Riemann problems, influenced by initial velocity, we observe that for some initial data, the composite wave comprises contact discontinuities, vacuum, and a local delta shock satisfying the over-compressing condition. Through a detailed analysis of the intricate interactions between contact discontinuities and delta shocks, we show that this local solution can be extended globally.