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

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.