Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 83, pp. 1-17.

Title: Existence, uniqueness and smoothness of a solution for 3D Navier-Stokes
        equations with any smooth initial velocity 
        
Authors: Arkadiy Tsionskiy (Stevens Institute of Tech., Hoboken, NJ, USA)
         Mikhail Tsionskiy (Stevens Institute of Tech., Hoboken, NJ, USA)

Abstract:
 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 in [28,29].
 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.

Submitted December 10, 2012. Published April 05, 2013.
Math Subject Classifications: 35Q30, 76D05.
Key Words: 3D Navier-Stokes equations; Fourier transform; 
           Laplace transform; fixed point principle.

An addendum as posted on December 31, 2013. It indicates that some results are incorrect. See the last page of this article.