  In mathematical physics, the pressure function is determined by the equation
of state. There are two existing barotropic state equations: the state equation
for polytropic gas with adiabatic index greater than or equal to 1 and the
state equation for generalized Chaplygin gas in cosmology. In this paper, a
logarithmic pressure is derived naturally with the coexistence of the two
existing state equations through an equivalent symmetric hyperbolic
transformation. On the study of the logarithmic pressure, global existence of
solutions with small initial data to the one-dimensional compressible Euler
equations with time-dependent damping is established.
