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Solutions of the Navier-Stokes and Euler equations with initial conditionsfor 2D and 3D cases were obtained in the form of converging series, by ananalytical 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 andsmoothness of the solution of the Cauchy problem for the 3D Navier-Stokesequations with any smooth initial velocity. When the viscosity tends to zero,this proof applies also to the Euler equations. A priori estimate of thissolution is presented.

Existence, uniqueness and smoothness of a solution for 3D Navier-Stokes  equations with any smooth initial velocity. A priori estimate of this  solution