  Some known results regarding the Euler and Navier-Stokes equations were
obtained by different authors. Existence and smoothness of solutions for the
Navier-Stokes equations in two dimensions have been known for a long time.
Leray showed that the Navier-Stokes equations in three space dimensions have a
weak solution. Scheffer and Shnirelman obtained weak solution of the Euler
equations with compact support in spacetime. Caffarelli, Kohn and Nirenberg
improved Scheffer's results, and F.-H. Lin simplified the proof of the results
of J. Leray. Many problems and conjectures about behavior of weak solutions of
the Euler and Navier-Stokes equations are described in the books of Bertozzi
and Majda, Constantin or Lemari\'e-Rieusset. Solutions of the Navier-Stokes and
Euler equations with initial conditions (Cauchy problem) for 2D and 3D cases
were obtained in the convergence series form by analytical iterative method
using Fourier and Laplace transforms in paper $\cite{TT10}$. These solutions
were received in a form of infinitely differentiable functions, and that allows
us to analyze all aspects of the problem on a much deeper level and with more
details. Also such smooth solutions satisfy the conditions required in
$\cite{CF06}$ for the problem of Navier-Stokes equations. For several
combinations of problem parameters numerical results were obtained and
presented as graphs $\cite{TT10}$,$\;\cite{TT11}$. This paper describes
detailed proof of convergence of the analitical iterative method for solution
of the Cauchy problem for the 3D Navier - Stokes equations. The convergence is
shown for wide ranges of the problem's parameters. Estimated formula for the
border of convergence area of the iterative process in the space of system
parameters is obtained. Also we have provided justification of the analytical
iterative method solution for Cauchy problem for the 3D Navier-Stokes
equations.
