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

TL;DR

This paper establishes a connection between Allen-Cahn equations satisfying equipartition and Euler equations, deriving geometric properties of solutions and providing explicit examples, with implications for Navier-Stokes equations.

Contribution

It demonstrates a transformation from Allen-Cahn solutions to Euler solutions under equipartition and explores the structure of these solutions, including explicit examples and applications.

Findings

Solutions of Allen-Cahn satisfying equipartition can be transformed into Euler solutions.

Level sets of entire solutions are hyperplanes (De Giorgi type results).

Explicit entire solutions of Euler and Navier-Stokes equations are constructed.

Abstract

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