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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. In particular, we do not use any Strichartz estimate. Nevertheless, we lose the natural Sobolev-Strichartz regularity for the convergence to Navier-Stokes equations but only $\delta$-regularity on the initial data for the global existence results.