On the explicit solutions of separation of variables type for the incompressible 2D Euler equations

TL;DR

This paper investigates explicit separation of variables solutions to the 2D Euler equations, identifying all known solutions and introducing three new families, suggesting these may encompass all such solutions.

Contribution

The paper classifies all known separation of variables solutions and introduces three new families, potentially completing the set of such solutions for 2D Euler equations.

Findings

All previously known solutions belong to two families.

Three new families of solutions are introduced.

It is likely these encompass all separation of variables solutions.

Abstract

We study explicit solutions to the 2 dimensional Euler equations in the Lagrangian framework. All known solutions have been of the separation of variables type, where time and space dependence are treated separately. The first such solutions were known already in the 19th century. We show that all the solutions known previously belong to two families of solutions and introduce three new families of solutions. It seems likely that these are all the solutions that are of the separation of variables type.