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

TL;DR

This paper develops series solutions for the 3D incompressible Navier-Stokes and Euler equations by transforming them into systems of ordinary differential equations, providing finite and infinite series solutions.

Contribution

It introduces a novel method combining undetermined coefficients and iterative techniques to solve these complex PDEs via series expansions.

Findings

Finite series solutions for Navier-Stokes equations obtained.

Infinite series solutions for Euler equations derived using combinatorial identities.

Transforming PDEs into ODE systems simplifies the solution process.

Abstract

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