problem for the 3D Navier-Stokes equations for any value of initial fluid velocity.

**Fourth stage** (section 6) - We have used the newly received equivalent integral equation to prove existence and uniqueness of the solution of the Cauchy problem in all time range \[0,$\infty$) based on the Caccioppoli-Banach fixed point principle (L.V. Kantorovich, G.P. Akilov $\cite{KA64},\;$ V.A. Trenogin $\cite{VT80},\;$W. Rudin $\cite{WR73},\;$W.A. Kirk and B. Sims$\; \cite{KS01},\;$A. Granas and J. Dugundji$\; \cite{GD03},\;$J.M. Ayerbe Toledano, T. Dominguez Benavides, G. Lopez Acedo$\;\cite{ADL97}\;$). For this purpose three following theorems were proven in this paper:

Theorem 1: Integral operator of the problem is a contraction operator;

Theorem 2: Existence and uniqueness of the solution of the problem is valid for any t $\in$ \[0,$\infty$);

Theorem 3: Solution of the problem is depending on t continuously.

**Fifth stage** (section 6) - By using a priori estimation of the solution of the Cauchy problem for the 3D Navier-Stokes equations $\; \cite{oL69},\; \cite{LK63}\;$ we have shown that the energy of the whole process has a finite value for any t $\in$ \[0,$\infty$).

The mathematical setup
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The Navier-Stokes equations describe the motion of a fluid in $R^{N}\; ( N = 3 ) $. We look for a viscous 