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

Solution of the Cauchy problem for the Navier - Stokes and Euler equations