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We will prove that solutions of the Allen-Cahn equations that satisfy theequipartition can be transformed into solutions of the Euler equations withconstant pressure. As a consequence, we obtain De Giorgi type results, that is,the level sets of entire solutions are hyperplanes. In addition, we obtain someexamples of smooth entire solutions of the Euler equations in particular cases.For specific type of initial conditions, some of these solutions can beextended to the Navier-Stokes equations. Also, we will determine the structureof solutions of the Allen-Cahn system in two dimensions that satisfy theequipartition. Finally, we apply the Leray projection on the Allen-Cahn systemand provide some explicit entire solutions.

A Relation of the Allen-Cahn equations and the Euler equations and applications of the Equipartition