  In two and three space dimensions, and under suitable assumptions on the
initial data, we show global existence for a damped wave equation which
approaches, in some sense, the Navier-Stokes problem. The proofs are based on a
refined energy method. In this paper, we improve the results in two papers by
Y. Brenier, R. Natalini and M. Puel and by M. Paicu and G. Raugel. We relax the
regularity of the initial data of the former, even though we still use energy
methods as a principal tool. Regarding the second paper, the improvement
consists in the simplicity of the proofs since we do not use any Strichartz
estimate and in requiring less regularity for the convergence to the
Navier-Stokes problem. Indeed, the convergence result we obtain is near-optimal
regularity.
