Vanishing pressure limit for compressible Navier-Stokes equations with degenerate viscosities

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.