Further remarks on the Luo-Hou's ansatz for a self-similar solution to the 3D Euler equations

TL;DR

This paper critically examines Luo-Hou's self-similar ansatz for 3D Euler equations, revealing that it leads to trivial or immediate blow-up solutions inconsistent with numerical observations, thus refining previous related findings.

Contribution

It demonstrates that Luo-Hou's ansatz results in over-determined systems with solutions incompatible with numerical data, refining earlier work by Chae and Tsai.

Findings

Luo-Hou's ansatz yields trivial and blow-up solutions

Solutions are inconsistent with numerical observations

Refines previous theoretical results

Abstract

It is shown that the self-similar ansatz proposed by T. Hou and G. Luo to describe a singular solution of the 3D axisymmetric Euler equations leads, without assuming any asymptotic condition on the self-similar profiles, to an over-determined system of partial differential equations that produces two families of solutions: a class of trivial solutions in which the vorticity field is identically zero, and a family of solutions that blow-up immediately, where the vorticity field is governed by a stationary regime. In any case, the analytical properties of these solutions are not consistent with the numerical observations reported by T. Hou and G. Luo. Therefore, this result is a refinement of the previous work published by D. Chae and T.-P. Tsai on this matter, where the authors found the trivial class of solutions under a certain decay condition of the blow-up profiles.