Explicit solutions to the 3D incompressible Euler equations in Lagrangian formulation

TL;DR

This paper presents a novel method for deriving explicit solutions to the 3D incompressible Euler equations using a Lagrangian approach, filling a significant gap in the existing literature of exact solutions.

Contribution

It introduces a general framework for constructing explicit Lagrangian solutions to the 3D Euler equations, expanding the limited set of known exact solutions.

Findings

Multiple solution families with different fluid behaviors identified

A general method for separating variables in the equations developed

Potential for discovering more solutions beyond current findings

Abstract

We introduce many families of explicit solutions to the three dimensional incompressible Euler equations for nonviscous fluid flows using the Lagrangian framework. Almost no exact Lagrangian solutions exist in the literature prior to this study. We search for solutions where the time component and the spatial component are separated, applying the same ideas we used previously in the two dimensional case. We show a general method to derive separate constraint equations for the spatial component and the time component. Using this provides us with a plenty of solution sets exhibiting several different types of fluid behaviour, but since they are computationally heavy to analyze, we have to restrict deeper analysis to the most interesting cases only. It is also possible and perhaps even probable that there exist more solutions of the separation of variables type beyond what we have found.