Quasi-periodic non-stationary solutions of 3D Euler equations for incompressible flow

TL;DR

This paper derives a novel class of non-stationary solutions for 3D Euler equations in incompressible flow, revealing conditions under which these solutions relate to elliptical integrals and quasi-periodic behavior.

Contribution

It introduces a new method to obtain non-stationary solutions of 3D Euler equations by separating spatial and temporal parts, involving Riccati equations and elliptical integrals.

Findings

Solutions involve Riccati-type equations with no general solution

Conditions lead to solutions expressed via elliptical integrals

Solutions exhibit quasi-periodic, non-stationary behavior

Abstract

A novel derivation of non-stationary solutions of 3D Euler equations for incompressible inviscid flow is considered here. Such a solution is the product of 2 separated parts: - one consisting of the spatial component and the other being related to the time dependent part. Spatial part of a solution could be determined if we substitute such a solution to the equations of motion (equation of momentum) with the requirement of scale-similarity in regard to the proper component of spatial velocity. So, the time-dependent part of equations of momentum should depend on the time-parameter only. The main result, which should be outlined, is that the governing (time-dependent) ODE-system consist of 2 Riccati-type equations in regard to each other, which has no solution in general case. But we obtain conditions when each component of time-dependent part is proved to be determined by the proper elliptical integral in regard to the time-parameter t, which is a generalization of the class of inverse periodic functions.