  In this paper we investigate well-posedness of the Cauchy problem of the
three dimensional generalized Navier-Stokes system. We first establish local
well-posedness of the GNS system for any initial data in the Fourier-Herz space
$\chi^{-1}$. Then we show that if the $\chi^{-1}$ norm of the initial data is
smaller than C$\nu$ in the GNS system where $\nu$ is the viscosity coefficient,
the corresponding solution exists globally in time. Moreover, we prove global
well-posedness of the Navier-Stokes system without norm restrictions on the
corresponding solutions provided the $\chi^{-1}$ norm of the initial data is
less than $\nu.$ Our obtained results cover and improve recent results in
\cite{Zhen Lei,wu}.
