LAUSR

Abstract The Riemann problem for the two-dimensional zero-pressure Euler equations is considered. The initial data are constant values in each quadrant, which satisfy an assumption that each initial discontinuity projects only one two-dimensional wave. The phenomenon of two-dimensional delta shock wave with a Dirac delta function in both density and internal energy is identified. Both generalized Rankine–Hugoniot relation and entropy condition for this type of two-dimensional delta shock wave are proposed. The qualitative behavior of entropy solutions to this relation with certain special initial data is established. Based on these preparations, we obtain twenty-three explicit solutions and their corresponding criteria. In particular, the Mach-reflection-like patterns arise in the exact solutions.