Global well-posedness and stability of classical solutions to the pressureless Navier-Stokes system in 3D

In this paper, we consider the Cauchy problem of the pressureless Navier-Stokes system which can be derived by taking the high Mach number limit for compressible Navier-Stokes equations. Due to the absence of pressure, the classical perturbation theory developed by Matsumura and Nishida (1980) [34] is not applicable to this model. The major difficulty lies in lack of the uniform boundedness of density. To solve this problem, we employ the spectral analysis and energy method to obtain the $L^2$ time-decay rates of velocity under the additional assumption that the initial velocity is small in the $L^1$ norm. Then combining the time-decay rates of velocity yields the uniform boundedness of density. Furthermore, it is proved that the velocity of pressureless flow decays to the motionless state at an optimal time-decay rate of ${(1+t)}^{-3/4}$ in the $L^2$-norm.