Strong solutions for the Navier–Stokes–Voigt equations with non-negative density

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 nonlinear 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.