  This paper is concerned with the large time behavior of the Cauchy problem
for Navier-Stokes/Allen-Cahn system describing the interface motion of
immiscible two-phase flow in 3-D. The existence and uniqueness of global
solutions and the stability of the phase separation state is proved under the
small initial perturbations. Moreover, the optimal time decay rates are
obtained for higher-order spatial derivatives of density, velocity and phase.
Our results implies that if the immiscible two-phase flow is initially located
near the phase separation state, then under small perturbation conditions, the
solution exists globally and decays algebraically to the complete separation
state of the two-phase flow, that is, there will be no interface fracture,
vacuum, shock wave, mass concentration at any time, and the interface thickness
tends to zero as the time $t\rightarrow+\infty$.
