  In this paper, we study the global Cauchy problem for a two-phase fluid model
consisting of the pressureless Euler equations and the incompressible
Navier-Stokes equations where the coupling of two equations is through the drag
force. We establish the global-in-time existence and uniqueness of classical
solutions for that system when the initial data are sufficiently small and
regular. Main difficulties arise in the absence of pressure in the Euler
equations. In order to resolve it, we properly combine the large-time behavior
of classical solutions and the bootstrapping argument to construct the
global-in-time unique classical solutions.

Analysis of PDEs