Equation for self-similar singularity of Euler 3D

TL;DR

This paper derives an integral equation for self-similar solutions of 3D Euler equations, linking the similarity exponent to Kelvin's circulation theorem, and proposes an iterative solution approach feasible with current computational resources.

Contribution

It introduces a novel integral equation framework for self-similar Euler solutions, incorporating circulation conservation and enabling iterative numerical solutions.

Findings

Integral equation for self-similar Euler solutions derived.

Iteration method with nonlinear kernel proposed.

Divergence of energy due to slow decay at large distances noted.

Abstract

The equations for a self-similar solution of an inviscid incompressible fluid are mapped into an integral equation which hopefully can be solved by iteration. It is argued that the exponent of the similarity are ruled by Kelvin's theorem of conservation of circulation. The end result is an iteration with a nonlinear term entering in a kernel given by a 3D integral (in general 3D flow) or 2D (for swirling flows), which seems to be within reach of present day computational power. Because of the slow decay of the similarity solution at large distances the total energy is diverging and recent mathematical results excluding a solution of the self-similar solution of Euler equation do not apply.