Explicit solutions to the Navier-Stokes equation and the Euler equation

TL;DR

This paper derives explicit series solutions for the Navier-Stokes and Euler equations based on initial conditions and forces, aiding in understanding their regularity, though convergence proofs are pending.

Contribution

It provides explicit series solutions for these fundamental fluid dynamics equations, a novel approach that could advance regularity analysis.

Findings

Explicit solutions expressed as series with known coefficients

Potential to facilitate global regularity studies

Convergence of series remains to be proven

Abstract

In this paper, we obtain explicit solutions to the Navier-Stokes equation and the Euler equation. For any initial velocity u0 and the force vector f, exact solutions can be explicitly solved as series, where the coefficients are all known functions determined only by u0 and f. Explicit solutions of this paper can help in global regularity studies in the Navier-Stokes equation and the Euler equation. We admit that the convergence of these series still need to be proved before completely solving the existence of the Navier-Stokes equation and the Euler equation.