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We utilize undetermined coefficient method and an iterative method toconstruct the series solutions of the 3D Cauchy problem for a class ofincompressible Navier-Stokes and Euler Equations. Then we can turn theNavier-Stokes Equations (Euler Equations) into the Cauchy problem for finitely(infinitely) many ordinary differential equations. We get the finite seriessolution of the Navier-Stokes Equations. By using some combinatorial identitiestechniques, we prove that the sum of the solutions to these ordinarydifferential equations is an infinite series solution of the Euler Equations insome cases.

Series solutions to the 3D cauchy problem for some incompressible Navier-Stokes and Euler Equations