  We study the piston problem of the compressible fluid flow with the
generalized Chaplygin gas. Depending on the inferential critical value of Mach
number, we prove that, there exists an integral weak solution for the
proceeding piston problem, consisting of a shock separating constant states
ahead of the piston if Mach numbers less than this critical value, while a
singular measure solution, with density containing a Dirac measure supported on
the piston, shall be proposed to solve the proceeding piston problem if Mach
numbers greater than or equal to the critical value. For the receding piston
problem, rarefaction wave solution always exists when the piston recedes from
the gas with any constant speed. Moreover, the occurrence of vacuum state and
the convergence of solutions, as well as degeneration of equations are analyzed
in the receding case as Mach number tends to infinity.
