the vector $\vec{u}$ can be represented as:

$$\label{eqn164}
\vec{u}\;=\;\bar{\bar{S}}\;\cdot\;\vec{\tilde{f}}\;+\;\bar{\bar{B}}\cdot\vec{u}^{0}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}\;- \bar{\bar{S}}\cdot(\vec{u}\cdot\nabla)\vec{u}\;+\;\bar{\bar{B}}\cdot\vec{u}^{0}\;,\;\;\;\;\;$$

where $\vec{\tilde{f}}$ is determined by formula $(\ref{eqn14})$.

Here $\;\bar{\bar{S}} \; $ and $\bar{\bar{B}}$ are the matrix integral operators: $$\left( \begin{array}{ccc}
S_{11} & S_{12} & S_{13} \\
S_{21} & S_{22} & S_{23} \\
S_{31} & S_{32} & S_{33}  \end{array} \right)$$ $$\left( \begin{array}{ccc}
B & 0 & 0 \\
0 & B & 0 \\
0 & 0 & B  \end{array} \right)$$

Spaces S, $\stackrel{\longrightarrow}{\textbf{TS}}$
===================================================

$\;\;\cite{GC68}, \cite{RR64}$ $\\$

Let us consider space S of all infinitely differentiable functions $\varphi$(x) defined in N-dimensional space $R^{N}$ (N = 3), such that when $\;\;\mid x \mid \;\rightarrow\; \infty\;\;$ these functions tend to 0, as well as their derivatives of any order, more rapidly than any power of $\frac{1}{\mid x \mid}$.

To define topology in the space S let us introduce countable system of norms

$$\label{eqn200}
\|\varphi\|_{p}\;=\; \mathop{sup_{\;x}}_{\mid k \mid ,\; \mid q \mid \;\leq \;p}\mid x^{k}D^{q} \varphi(x)\mid\;\;\;(p = 0, 1, 2,...)$$

where

$\mid x^{k}D^{q} \varphi(x)\mid\; = \; \mid x_{1}^{k_{1}}\ldots x_{N}^{k_{N}}\frac{\partial^{q_{1}+\cdots + q_{N}}\varphi(x)}{\partial {x_{1}^{q_{1}}}\ldots \partial {x_{N}^{q_{N}}} }\mid$

$k = (k_{1}, \ldots , k_{N}),\;\; q = (q_{1}, \ldots , q_{N}),\;\;x^{k} =  x_{1}^{k_{1}}\ldots x_{N}^{k_{N}} $

$D^{q} = \frac{\partial^{q_{1}+\cdots + q_{N}}}{\partial {x_{1}^{q_{1}}}\ldots \partial {x_{N}^{q_{N}}} },\;\; (k_{1}, \ldots , 