  The aim of this work is to study the Navier-Stokes-Voigt equations that
govern flows with non-negative density of incompressible fluids with elastic
properties. For the associated non-linear initial-and boundary-value problem,
we prove the global-in-time existence of strong solutions (velocity, density
and pressure). We also establish some other regularity properties of these
solutions and find the conditions that guarantee the uniqueness of velocity and
density. The main novelty of this work is the hypothesis that, in some
subdomain of space, there may be a vacuum at the initial moment, that is, the
possibility of the initial density vanishing in some part of the space domain.
