An inverse piston problem for the system of one-dimensional adiabatic flow

Abstract

In this paper, we first consider the Cauchy problem with Cauchy data given on a semibounded initial axis for quasilinear hyperbolic systems. Under suitable assumptions, we obtain the global existence and uniqueness of C1 solution on the corresponding maximum determinate domain and a sharp point-wise decay estimate of the solution. By means of this result, we globally solve an inverse piston problem for one-dimensional gas dynamics as follows: suppose that the original state of the gas on the right side of the piston and the position of the forward shock are known, then we can globally determine the speed of the piston in a unique manner. In particular, the speed of the piston is estimated sharply.