  In this paper, we present a new framework for the global well-posedness and
large-time behavior of a two-phase flow system, which consists of the
pressureless Euler equations and incompressible Navier-Stokes equations coupled
through the drag force. To overcome the difficulties arising from the absence
of the pressure term in the Euler equations, we establish the time decay
estimates of the high-order derivative of the velocity to obtain uniform
estimates of the fluid density. The upper bound decay rates are obtained by
designing a new functional and the lower bound decay rates are achieved by
selecting specific initial data. Moreover, the upper bound decay rates are the
same order as the lower one. Therefore, the time decay rates are optimal. When
the fluid density in the pressureless Euler flow vanishes, the system is
reduced into an incompressible Navier-Stokes flow. In this case, our works
coincide with the classical results by Schonbek \cite{M.S3} [JAMS,1991], which
can be regarded as a generalization from a single fluid model to the two-phase
fluid one.
