graphs $\cite{TT10}$,$\;\cite{TT11}$.

    This paper describes detailed proof of existence and uniqueness of the solution of the Cauchy problem for the 3D Navier- Stokes equations with any smooth initial velocity. This solution satisfies the conditions required in $\cite{CF06}$ for the problem of Navier-Stokes equations. When viscosity tends to zero this proof is correct for the Euler equations also.
author:
- 'A. Tsionskiy, M. Tsionskiy [^1]'
title: |
    Existence, uniqueness and smoothness of solution for\
    3D Navier-Stokes equations with any smooth initial velocity.
---

Introduction
============

 

The solution of the Cauchy problem for the 3D Navier-Stokes equations described in this paper is based on the form of differential equations in the statement of the problem and also conditions for the initial velocity and applied force $\cite{CF06}$. It grows from classic definition of function and classic methods of analysis.

In the problem statement for Navier-Stokes equations an initial velocity is infinitely differentiable function decreasing rapidly to zero in infinity. Applied force is identically zero.

Solution of the problem will be presented by the following stages:

**First stage** (sections 2, 3) - we have moved non-linear parts of equations to the right sides. Then we have solved the system of linear partial differential equations 