2d incompressible Euler equations: new explicit solutions

TL;DR

This paper introduces a general method for constructing explicit solutions to the 2D incompressible Euler equations in Lagrangian coordinates, expanding beyond classical harmonic map-based solutions by utilizing matrix Lie groups and their geodesics.

Contribution

A new, more general approach to generating explicit solutions to 2D Euler equations, encompassing all known solutions and revealing broader structures involving Lie groups.

Findings

Classical solutions are special cases of a larger family.

Matrix Lie groups and their geodesics are key to describing solutions.

The method unifies previous solutions and introduces new explicit solutions.

Abstract

There are not too many known explicit solutions to the 2-dimensional incompressible Euler equations in Lagrangian coordinates. Special mention must be made of the well-known ones due Gerstner and Kirchhoff, which were already discovered in the 19th century. These two classical solutions share a common characteristic, namely, the dependence of the coordinates from the initial location is determined by a harmonic map, as recognized by Abrashkin and Yakubovich, who more recently -- in the 1980s -- obtained new explicit solutions with a similar feature. We present a more general method for constructing new explicit solutions in Lagrangian coordinates which contain as special cases all previously known ones. This new approach shows that in fact "harmonic labelings" are special cases of a much larger family. In the classical solutions, the matrix Lie groups were essential in describing the time evolution. We see that also the geodesics in these groups are important.