---
abstract: |
    Different authors had received a lot of results regarding the Euler and Navier-Stokes equations. Existence and smoothness of solution for the Navier-Stokes equations in two dimensions have been known for a long time. Leray $\cite{jL34}$ showed that the Navier-Stokes equations in three dimensional space have a weak solution. Scheffer $\cite{vS76}, \cite{vS93}$ and Shnirelman $\cite{aS97}$ obtained weak solution of the Euler equations with compact support in spacetime. Caffarelli, Kohn and Nirenberg $\cite{CKN82}$ improved Scheffer’s results, and F.-H. Lin $\cite{fL98}$ 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 Ladyzhenskaya $\cite{oL69}$, Bertozzi and Majda $\cite{BM02}$, Temam $\cite{RT77}$, Constantin $\cite{pC01}$ or Lemarié-Rieusset $\cite{pL02}$.

    Solutions of the Navier-Stokes and Euler equations with initial conditions (Cauchy problem) for 2D and 3D cases were obtained in the converging series form by analytical iterative method using Fourier and Laplace transforms in paper $\cite{TT10}$. These solutions were received as infinitely differentiable functions. That allowed us to analyze essential aspects of the problem on a much deeper level and with more details. For several combinations of problem parameters numerical results were obtained and presented as 