x_{2}, x_{3}, t)$ and $p(x_{1}, x_{2}, x_{3}, t)$} is the solution of the Cauchy problem $(\ref{eqn1})$ - $(\ref{eqn3})$. From here we see that solving the Cauchy problem $(\ref{eqn1})$ - $(\ref{eqn3})$ is equivalent to finding continuous in t $\in$ \[0,$\infty$) solution of integral equation $(\ref{eqn203})$.\

The Caccioppoli-Banach fixed point principle
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$\cite{KA64},\;$ $\cite{VT80},\;$ $\cite{WR73},\;$$\; \cite{KS01},\;$$\; \cite{GD03},\;$$\;\cite{ADL97}\;$ $\\$

From the problem statement $\cite{CF06}$ we have $\vec{f}\;\equiv\; 0$. Let us rewrite integral equation $(\ref{eqn203})$ with this condition

$$\label{eqn203a}
\vec{u}\;=\;- \bar{\bar{S}}\cdot(\vec{u}\cdot\nabla)\vec{u}\;+\;\bar{\bar{B}}\cdot\vec{u}^{0}$$

Let us divide all parts of the integral equation $(\ref{eqn203a})$ by some constant V, that we will define appropriately below. Then we receive modified integral equation equivalent to equation $(\ref{eqn203a})$:

$$\label{eqn203ab}
\vec{u}_{\mbox{\tiny V}}\;=\;- \bar{\bar{S}_{\mbox{\tiny V}}}\cdot(\vec{u}_{\mbox{\tiny V}}\cdot\nabla_{\mbox{\tiny V}})\vec{u}_{\mbox{\tiny V}}\;+\;\bar{\bar{B}_{\mbox{\tiny V}}}\cdot\vec{u}_{\mbox{\tiny V}}^{0}\; = \bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}\cdot\vec{u}_{\mbox{\tiny V}}$$

here

$$\label{eqn203z}
\vec{u}_{\mbox{\tiny V}}\;=\;\frac{\vec{u}}{V};\;\vec{u}_{\mbox{\tiny V}}^{0}\;=\;\frac{\vec{u}^{0}}{V};\;\nabla_{\mbox{\tiny V}}\;=\;V\cdot\nabla;\;
x_{k\mbox{\tiny V}}\;=\;\frac{x_{k}}{V};\;\gamma_{k\mbox{\tiny V}}\;=\;V\cdot\gamma_{k};\;\nu_{\mbox{\tiny V}}\;=\;\frac{\nu}{V^{2}};\;\;({1\leq k \leq N})$$

Let us choose the constant V as: $$\label{eqn203e}
max \;\{\|{\vec{u}^0}\|_{C^2}, \;\|{\vec{u}^0}\|_{L_2}, \;(\|{\vec{u}}\|_{C_2} + \|{\vec{u}}\|_{L_2})\}<<\;V$$

Then we have that

$$\label{eqn203f}
|{\vec{u}_{\mbox{\tiny V}}}|<\;1\;\;\;,\;\;\;|{\vec{u}^0_{\mbox{\tiny V}}}|<\;1$$

Below in property 4 of Matrix integral operator $\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}$, and in Theorems 1 - 3, where two vector-functions $\vec{u}_{\mbox{\tiny V}}$ and $\vec{u}_{\mbox{\tiny V}}^{'}$ are presented, constant V is selected as maximum between V($\vec{u}$) and V$^{'}$($\vec{u}^{'}$).

We can use the fixed point principle to 