  We present a new hydrodynamic model consisting of the pressureless Euler
equations and the isentropic compressible Navier-Stokes equations where the
coupling of two systems is through the drag force. This coupled system can be
derived, in the hydrodynamic limit, from the particle-fluid equations that are
frequently used to study the medical sprays, aerosols and sedimentation
problems. For the proposed system, we first construct the local-in-time
classical solutions in an appropriate $L^2$ Sobolev space. We also establish
the \emph{a priori} large-time behavior estimate by constructing a Lyapunov
functional measuring the fluctuation of momentum and mass from the averaged
quantities, and using this together with the bootstrapping argument, we obtain
the global classical solution. The large-time behavior estimate asserts that
the velocity functions of the pressureless Euler and the compressible
Navier-Stokes equations are aligned exponentially fast as time tends to
infinity.
