Radon measure-valued solutions to Riemann problems for isentropic compressible Euler equations with a discontinuous flux function

We study a class of discontinuous-flux Riemann problems with mass concentration in gas dynamics. Specifically, we consider the singular Riemann problem for the one-dimensional unsteady isentropic compressible Euler equations, where the flux has a jump across a discontinuous curve x = x(t) that separates two different polytropic gases. The main result establishes the existence of Radon measure-valued solutions to this problem. We also show that for certain initial data, the problem admits multiple solutions consisting of piecewise constant states connected by delta shocks satisfying the over-compressing condition. The essential difficulty stems from the flux discontinuity along the discontinuity curve combined with the distinct polytropic gas properties on either side, which leads to more complex nonlinearity and makes the construction of solutions challenging. This work extends the existing theory of Radon measure-valued solutions for compressible Euler systems with a discontinuous flux to a general case. We conclude that for conservation laws featuring discontinuous flux functions with mass concentration, genuine nonlinearity does not guarantee the uniqueness of piecewise constant solutions containing delta shocks under the over-compressing condition.