Global Well-posedness for the Generalized Navier-Stokes System

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 $χ^{-1}$. Then we show that if the $χ^{-1}$ norm of the initial data is smaller than C$ν$ in the GNS system where $ν$ 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 $χ^{-1}$ norm of the initial data is less than $ν.$ Our obtained results cover and improve recent results in \cite{Zhen Lei,wu}.