We consider in this paper a hyperbolic quasilinear version of the Navier-Stokes equations in three space dimensions, obtained by using Cattaneo type law instead of a Fourier law. In our earlier work [2], we proved the global existence and uniqueness of solutions for initial data small enough in the space H 4 (R 3) 3 × H 3 (R 3) 3. In this paper, we refine our previous result in [2], we establish the existence under a significantly lower regularity. We first prove the local existence and uniqueness of solution, for initial data in the space H 5 2 +δ (R 3) 3 × H 3 2 +δ (R 3) 3 , δ > 0. Under weaker smallness assumptions on the initial data and the forcing term, we prove the global existence of solutions. Finally, we show that if ε is close to 0, then the solution of the perturbed equation is close to the solution of the classical Navier-Stokes equations.