  Some known results regarding the Euler and Navier-Stokes equations were
obtained by different authors. Existence and smoothness of the Navier-Stokes
solutions in two dimensions have been known for a long time. Leray
$\cite{jL34}$ showed that the Navier-Stokes equations in three space dimensions
have a weak solution. Scheffer and Shnirelman obtained weak solution of the
Euler equations with compact support in spacetime. Caffarelli-Kohn-Nirenberg
improved Scheffer's results, and F.-H. Lin simplified the proof of the results
of J. Leray. Many problems and conjectures about the behavior of solutions of
the Euler and Navier-Stokes equations are described in the book of Bertozzi and
Majda or Constantin.
  Solutions of the Navier-Stokes and Euler equations with initial conditions
(Cauchy problem) for two and three dimensions are obtained in the convergence
series form by the iterative method using the Fourier and Laplace transforms in
this paper. For several combinations of problem parameters numerical results
were obtained and presented as graphs.
