  Solutions of the Navier-Stokes and Euler equations with initial conditions
for 2D and 3D cases were obtained in the form of converging series, by an
analytical iterative method using Fourier and Laplace transforms
\cite{TT10,TT11}. There the solutions are infinitely differentiable functions,
and for several combinations of parameters numerical results are presented.
This article provides a detailed proof of the existence, uniqueness and
smoothness of the solution of the Cauchy problem for the 3D Navier-Stokes
equations with any smooth initial velocity. When the viscosity tends to zero,
this proof applies also to the Euler equations. A priori estimate of this
solution is presented.
