Abstract: We consider the Cauchy problem for the three-dimensional pressureless Navier-Stokes/Navier-Stokes system, which consists of the pressureless Navier-Stokes equations for $(n,w)$ coupled with the isentropic compressible Navier-Stokes equations for $(\rho,u)$ through a drag force term $n(w-u)$. We prove the global existence of strong solutions to the coupled system when the initial data are small perturbations of the constant equilibrium state. However, due to the pressureless structure, one can only deal with the density $n$ of the pressureless flow through the transport equation and it is crucial to obtain the exact time-decay rates for the corresponding velocity $w$ of the pressureless flow. To this end, we make use of the spectral analysis, low-high frequency decomposition and time-weighted energy method to deduce the large time behavior of $(w,\rho,u)$ and consequently establish the Lyapunov stability of the density $n$ in Sobolev space.

Key words: pressureless Navier-Stokes/Navier-Stokes system, Cauchy problem, strong solution, global existence