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 [16] 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 [3], [2], [9], [13], [11], [15]. 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 [3], [2]. Solutions to this model blow up in finite time.

流体力学 3D 欧拉方程的有限时间爆炸与全局规律性问题是一个主要的开放问题。几年前，Luo 和 Hou [16] 基于广泛的数值模拟提出了一种新的有限时间爆炸方案。该场景是轴对称的，其特点是涡量在位于包含流体的圆柱体边界处的一圈双曲线点附近的涡量快速增长。观察到强烈生长的小边界层发挥了重要作用。已经考虑了该场景的几个简化模型，所有模型都导致有限时间爆炸 [3]、[2]、[9]、[13]、[11]、[15]。在本文中，我们提出了两个模型，这些模型专门设计用于深入了解位于边界处的流动双曲停滞点附近的流体演化。一种模型侧重于分析问题中存在的非线性效应的损耗。该模型的解决方案被证明是全局规则的。与 [3]、[2] 等一维模型相比，第二个模型可以看作是将边界附近的速度场捕获到下一个精度等级的尝试。这个模型的解决方案在有限的时间内爆炸。