Hyperbolic Navier-Stokes equations in three space dimensions

Abstract

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 H4(R3)3 ? H3(R3)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 H5 2 +?(R3)3 ?H32 +?(R3)3, ? > 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 ? is close to 0, then the solution of the perturbed equation is close to the solution of the classical Navier-Stokes equations.