论文标题
在卷曲操作员的某些属性及其对Navier-Stokes系统的后果
On some properties of the curl operator and their consequences for the Navier-Stokes system
论文作者
论文摘要
我们根据其对角度化及其作为伪衍生$（δ）^{1/2} $的非本地对称对称性的表达，研究了$ \ operatotorname {curl} $ operator的一些几何特性。在这种情况下，我们介绍了自旋定义字段的概念，即$（ - δ）^{ - 1/2} \ operatotorName {curl} $的特征值。这两个旋转 - definite组件的两个旋转 - definite组件是一般3D不可压缩的流量，从左撇子右手手动的一台不可交接的右手运动。在观察到Navier-Stokes的非线性具有跨产品的结构及其弱（分布）形式的结构是涉及涡度，速度和测试功能的决定因素，我们重新审视了能量的保护和以地面方式的平衡。我们表明，在有限的时间爆破的情况下，流动的两个自旋排定成分将同时且速率同样散发。 3D中的奇异性是旋转冲突的结果，这在2D流的几何形状中是不可能的。驱动腹部的同类产品，并且更普遍地是造成流动的规律性和奇异性或准清点的出现的原因。因此，它们是湍流现象的核心。
We investigate some geometric properties of the $\operatorname{curl}$ operator, based on its diagonalizationand its expression as a non-local symmetry of the pseudo-derivative $(-Δ)^{1/2}$ among divergence-free vector fieldswith finite energy. In this context, we introduce the notion of spin-definite fields, i.e. eigenvectorsof $(-Δ)^{-1/2}\operatorname{curl}$.The two spin-definite components of a general 3D incompressible flow untangle the right-handed motion from the left-handed one. Having observed that the non-linearity of Navier-Stokes has the structure of a cross-productand its weak (distributional) form is a determinant that involves the vorticity, the velocity and a test function,we revisit the conservation of energy and the balance of helicity in a geometrical fashion. We show that in the caseof a finite-time blow-up, both spin-definite components of the flow will explose simultaneously and with equal rates,i.e. singularities in 3D are the result of a conflict of spin, which is impossible in the poorer geometry of 2D flows.We investigate the role of the local and non-local determinants $$\int_0^T\int_{\mathbb{R}^3}\det(\operatorname{curl} u, u, (-Δ)^θ u)$$ and their spin-definite counterparts, which drive the enstrophy and, more generally, are responsible forthe regularity of the flow and the emergence of singularities or quasi-singularities.As such, they are at the core of turbulence phenomena.
