  In this paper we study a vanishing pressure process for highly compressible
Navier-Stokes equations as the Mach number tends to infinity. We first prove
the global existence of weak solutions for the pressureless system in the
framework [Li-Xin, arXiv:1504.06826v2], where the weak solutions are
established for compressible Navier-Stokes equations with degenerate viscous
coefficients. Furthermore, a rate of convergence of the density in
$L^{\infty}\left(0,T;L^{2}(\rr)\right)$ is obtained, in case when the velocity
corresponds to the gradient of density at initial time.
