Free Piston Problem In One-Dimensional Zero-Pressure Compressible Euler Flow

This paper established the equivalence of free piston problem and delta shock of Riemann problem,for the one-dimensional pressureless compressible Euler equations.The delta shock appearing in the latter is exactly the piston that may move freely forward or backward in a straight tube,driven by the pressureless Euler flows on two sides of it in the tube.This result not only helps to understand the physics of the delta shocks,but also provides a way to reduce the fluid-solid interaction problem to a Cauchy problem.This paper shows the equivalence from three di?erent perspectives.The first one is from the sticky particles,and derives the ordinary di?erential equation(ODE)of the trajectory of the piston by a straightforward application of conservation law of momentum,which is physically simple and clear.The second one is to study a coupled initial-boundary value problem of pressureless Euler equations,with the piston as a moving boundary following the Newton's second law.It depends on a new concept of Radon measure solutions of initial-boundary value problems of the compressible Euler equations which enables us to calculate the force on the piston given by the flow.The third one is to solve directly the Riemann problem and obtain the ODE by the generalized Rankine-Hugoniot conditions of delta shock.All the three methods lead to the same ODE.