Piston Problem in Common-Velocity Idealized Three-Component Models

This paper studies the common-velocity idealized three-component compressible model with pressure law \(P=\rho^\gamma\ (0<\gamma\le1)\) and the constant-pressure model with \(P=1\). We systematically analyze the piston problem for these two models. When the piston velocity is lower than the initial fluid velocity, fluid blockage or severe blockage occurs, corresponding to a shock or delta shock wave solution. When the piston velocity exceeds the initial fluid velocity, the flow expands, corresponding to a rarefaction wave or vacuum solution. Furthermore, we prove that, as \(\gamma\to0^+\), the conservative variables \((l,m,q,\rho u)\) of the solutions to the piston problem for the compressible model converge in the sense of distributions to those for the constant-pressure model, and we present numerical simulations for finite values of \(\gamma\).