The asymptotic problem for the springlike motion of a heavy piston in a viscous gas

Abstract

This paper treats the classical problem for the longitudinal motion of a piston separating two viscous gases in a closed cylinder of finite length. The motion of the gases is governed by singular initial-boundary-value problems for parabolic-hyperbolic partial differential equations depending on a small positive parameter $\varepsilon$, which characterizes the ratios of the masses of the gases to that of the piston. (The equation of state giving the pressure as a function of the specific volume need not be monotone and the viscosity may depend on the specific volume.) These equations are subject to a transmission condition, which is the equation of motion of the piston. The specific volumes of the gases are shown to have a positive lower bound at any finite time. This bound leads to the theorem asserting that (under mild smoothness restrictions) the initial-boundary-value problem has a unique classical solution defined for all time. The main emphasis of this paper is the treatment of the asymptotic behavior of solutions as $\varepsilon \searrow 0$. It is shown that this solution admits a rigorous asymptotic expansion in $\varepsilon$ consisting of a regular expansion and an initial-layer expansion. The reduced problem, for the leading term of the regular expansion (which is obtained by setting $\varepsilon = 0$), is typically governed by an equation with memory, rather than by an ordinary differential equation of the sort governing the motion of a mass on a massless spring. The reduced problem nevertheless has a 2-dimensional attractor on which the dynamics is governed precisely by such an ordinary differential equation.