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 \cite{AB}, we proved the global existence and uniqueness of solutions for initial data small enough in the space $ H^4(\R^3)^3 \times H^3(\R^3)^3$. In this paper, we refine our previous result in \cite{AB}, 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^{\frac{5}{2}+\delta}(\R^3)^3 \times H^{\frac{3}{2}+\delta}(\R^3)^3$, $\delta>0$. Under weaker smallness assumptions on the initial data and the forcing term, we prove the global existence of solutions. Finally, we show that if $\eps$ is close to $0$, then the solution of the perturbed equation is close to the solution of the classical Navier-Stokes equations.