In this paper, we present strong numerical evidences that the incompressible axisymmetric Euler equations with degenerate viscosity coefficients and smooth initial data of finite energy develop a potential finite-time locally self-similar singularity at the origin. An important feature of this potential singularity is that the solution develops a two-scale traveling wave that travels towards the origin. The two-scale feature is characterized by the scaling property that the center of the traveling wave is located at a ring of radius $O((T-t)^{1/2})$ surrounding the symmetry axis while the thickness of the ring collapses at a rate $O(T-t)$. The driving mechanism for this potential singularity is due to an antisymmetric vortex dipole that generates a strong shearing layer in both the radial and axial velocity fields. Without the viscous regularization, the $3$D Euler equations develop a sharp front and some shearing instability in the far field. On the other hand, the Navier-Stokes equations with a constant viscosity coefficient regularize the two-scale solution structure and do not develop a finite-time singularity for the same initial data.
翻译:在本文中,我们展示了强大的数字证据,证明不压缩轴轴偏偏偏偏偏振动方程式与低粘度系数和平滑的有限能源初始数据形成了一种潜在的有限时间局部本地自异原点。这种潜在奇点的一个重要特征是,溶液开发出向原点移动的双尺度旅行波。两个尺度特征的特征是,流动波的中心位于半径((T-t) ⁇ 1/2})一环周围的方圆(O(T-t) ⁇ 1/2}),而环状轴的厚度以美元(T-t)计算,而环状体以美元($-t)的速率崩溃。这一潜在奇点的驱动机制是由于一个反对称的旋涡偏偏偏角在辐射场和轴向原点之间产生强烈的剪切层。没有粘固度,3美元的Euler方程式在远处形成尖锐的前方和某种剪裁的不稳定性。另一方面,导航-Stokeks 方程式的方块方程式则不是以恒定的基度基度结构,以同一的硬度的硬度的基度模型,而将两个基度的基度的基的基度调整了。