The free piston problem for pressureless Euler equations under the gravity

Abstract

In this paper, the exact solutions of the free piston problem for the one-dimensional pressureless Euler equations under gravity are constructed in fully explicit forms. It is found that delta shock wave and vacuum state are contained in the solutions with appropriate initial condition. Furthermore, the trajectory function of the piston can be obtained in the combination of the generalized Rankine-Hugoniot conditions and Newton's second law, and it can be uniquely determined by the generalized entropy condition. We observe and analyze carefully some interesting nonlinear phenomena. For instance, delta shock wave absorbs the vacuum state and forms a new delta shock wave. In addition, some representative numerical simulations are presented to confirm the correctness of theoretical analysis.