Consider the Navier-Stokes equation for a one-dimensional and two-dimensional compressible viscous liquid. It is a well-known fact that there is a strong solution locally in time when the initial data is smooth and the initial density is limited down by a positive constant. In this article, under the same hypothesis, I show that the density remains uniformly limited in time from the bottom by a positive constant, and therefore a strong solution exists globally in time. In addition, most existing results are obtained with a positive viscosity factor, but current results are true even if the viscosity factor disappears with density. Finally, I prove that this solution is unique in a class of weak solutions that satisfy the usual entropy inequalities. The point of this work is the new entropy-like inequalities that Bresch and Desjardins introduced into the shallow water system of equations. This discrepancy gives the density additional regularity (assuming such regularity exists first).