The Riemann problem for isentropic compressible Euler equations with discontinuous flux

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux, more specifically, for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x = x(t). We prove that this problem admits global Radon measure solutions for all kinds of initial data. The over-compressing condition on the discontinuity x = x(t) is not enough to ensure the uniqueness of the solution. However, there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x = x(t) + 0, in addition to the full adhesion condition on its left-side. As an application, we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas. In particular, we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas. This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.