Boundary layer models of the Hou-Luo scenario

TL;DR

This paper investigates boundary layer models related to the Hou-Luo scenario for 3D Euler equations, introducing two models that analyze fluid behavior near hyperbolic stagnation points, one showing global regularity and the other finite time blow-up.

Contribution

It proposes two new boundary layer models specifically designed to understand the Hou-Luo blow-up scenario, with one model demonstrating global regularity and the other finite time blow-up.

Findings

One model exhibits global regularity due to nonlinear depletion.

The other model demonstrates finite time blow-up.

Provides insight into boundary layer dynamics near hyperbolic points.

Abstract

Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou \cite{HouLuo14} proposed a new finite time blow up scenario based on extensive numerical simulations. The scenario is axi-symmetric and features fast growth of vorticity near a ring of hyperbolic points of the flow located at the boundary of a cylinder containing the fluid. An important role is played by a small boundary layer where intense growth is observed. Several simplified models of the scenario have been considered, all leading to finite time blow up \cite{CKY15,CHKLVY17,HORY,KT1,HL15,KY1}. In this paper, we propose two models that are designed specifically to gain insight in the evolution of fluid near the hyperbolic stagnation point of the flow located at the boundary. One model focuses on analysis of the depletion of nonlinearity effect present in the problem. Solutions to this model are shown to be globally regular. The second model can be seen as an attempt to capture the velocity field near the boundary to the next order of accuracy compared with the one-dimensional models such as \cite{CKY15,CHKLVY17}. Solutions to this model blow up in finite time.