Stability of Blow Up for a 1D model of Axisymmetric 3D Euler Equation

TL;DR

This paper proves finite time blow up for a refined 1D model of the axisymmetric 3D Euler equations, demonstrating the robustness of the blow-up mechanism and advancing understanding of potential singularities in fluid dynamics.

Contribution

It establishes finite time blow up for a more accurate 1D model of the 3D Euler equations, including lower order terms, and shows the stability of the blow-up mechanism across a broader class of models.

Findings

Finite time blow up is proven for the refined model.

The blow-up mechanism is shown to be robust across model variations.

The results provide insights into the hyperbolic blow-up scenario in 3D Euler equations.

Abstract

The question of the global regularity vs finite time blow up in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbolic boundary blow up scenario for the 3D Euler equation proposed by Hou and Luo based on extensive numerical simulations. These models generalize the 1D Hou-Luo model suggested in Hou & Luo's paper, for which finite time blow up has been established in the paper by Kyudong Choi, Thomas Y. Hou, Alexander Kiselev, Guo Luo, Vladimir Sverak, Yao Yao. The main new aspects of this work are twofold. First, we establish finite time blow up for a model that is a closer approximation of the three dimensional case than the original Hou-Luo model, in the sense that it contains relevant lower order terms in the Biot-Savart law that have been discarded in the paper by Hou & Luo's and the paper by Choi et al.. Secondly, we show that the blow up mechanism is quite robust, by considering a broader family of models with the same main term as in the Hou-Luo model. Such blow up stability result may be useful in further work on understanding the 3D hyperbolic blow up scenario.