incompressible fluid filling all of $R^{N}$ here. The Navier-Stokes equations are then given by

$$\label{eqn1}
\frac{\partial u_{k}}{\partial t} \; + \; \sum_{n=1}^{N} u_{n}\frac{\partial u_{k}}{\partial x_{n}}\; =\;\nu\Delta u_{k}\; - \; \frac{\partial p}{\partial x_{k}} \; + \;f_{k}(x,t)\;\;\;\;\;(x\in R^{N},\;\;t\geq 0,\;\;{1\leq k \leq N})$$

$$\label{eqn2}
\emph{div}\,\vec{u}\;=\; \sum_{n=1}^{N} \frac{\partial u_{n}}{\partial x_{n}}\; =\;0\;\;\;\;\;\;\;\;\;\;  (x\in R^{N},t\geq 0)$$

with initial conditions

$$\label{eqn3}
\vec{u}(x,0)\; = \; \vec{u}^{0}(x)\;\;\;\;\;\;\;\;\;\; (x\in R^{N})$$

Here $\vec{u}(x,t)=(u_{k}(x,t)) \in R^{N},\;\; ({1\leq k \leq N})  \;-\; $is an unknown velocity vector $( N = 3 );\; p\,(x,t)\;-\;$ is an unknown pressure; $\vec{u}^{0}(x)\;$ is a given, $C^{\infty}$ divergence-free vector field$;\; f_{k}(x,t)\;$are components of a given, externally applied force $\vec{f}(x,t)$; $\nu$ is a positive coefficient of the viscosity (if $\nu = 0$ then $(\ref{eqn1})$ - $(\ref{eqn3})$ are the Euler equations); and $ \Delta\;=\; \sum_{n=1}^{N} \frac{\partial^{2}}{\partial x_{n}^{2}}\;$ is the Laplacian in the space variables. Equation $(\ref{eqn1})$ is Newton’s law for a fluid element subject. Equation $(\ref{eqn2})$ says that the fluid is incompressible. For physically reasonable solutions, we accept

$$\label{eqn4}
u_{k}(x,t) \rightarrow 0\;\;, \;\;
\frac{\partial u_{k}}{\partial x_{n}} \;\rightarrow\;0\;\; 
\rm {as} 
\;\;\mid x \mid \;\rightarrow\; \infty\;\;\;( {1\leq k \leq N} ,\;\; {1\leq n \leq N}) \;\;\;$$

Hence, we will restrict attention to initial conditions $\vec{u}^{0}$ and force $\vec{f}$ that satisfy

$$\label{eqn5}
\mid\partial_{x}^{\alpha}\vec{u}^{0}(x)\mid\;\leq\;C_{\alpha 