  In this paper, we consider the Cauchy problem of the multi-dimensional
compressible Navier-Stokes-Euler system for two-phase flow motion, which
consists of the isentropic compressible Navier-Stokes equations and the
isothermal compressible Euler equations coupled with each other through a
relaxation drag force. We first establish the local existence and uniqueness of
the strong solution for general initial data in a critical homogeneous Besov
space, and then prove the global existence of the solution if the initial data
is a small perturbation of the equilibrium state. Moreover, under the
additional condition that the low-frequency part of the initial perturbation
also belongs to another Besov space with lower regularity, we obtain the
optimal time-decay rates of the global solution toward the equilibrium state.
These results imply that the relaxation drag force and the viscosity
dissipation affect regularity properties and long time behaviors of solutions
for the compressible Navier-Stokes-Euler system.
