arXiv 2203.079502022-03-15math.AP

Navier-Stokes系统旋度算子的性质及其推论

On some properties of the curl operator and their consequences for the Navier-Stokes system

Nicolas Lerner, François Vigneron

摘要：

基于$\operatorname{curl}$算子的对角化及其在有限能量无发散向量场中的伪导数$(-Δ)^{1/2}$的非局部对称性，我们研究了该算子的一些几何性质.在本文中，我们引入了自旋定域的概念，即$(-Δ)^{-1/2}\operatorname{curl}$的本征向量，一般三维不可压缩流的两个自旋定域分量将右手运动与左手运动分开。在观察到Navier-Stokes方程的非线性具有叉积结构，其弱（分布）形式是一个包含涡量、速度和检验函数的行列式之后，我们重新讨论了能量守恒和螺旋度平衡的几何形式。我们表明，在有限时间爆破的情况下，两个自旋确定的组件的流动将同时explose，并以相等的速率，即在3D的奇点是一个冲突的自旋，这是不可能的，在较差的几何形状的2D flow.We调查的本地和非本地的决定因素$$\int_0^T\int_{\mathbb{R}^3}\det(\operatorname{curl} u, u, (-Δ)^θ u)$$和他们的自旋确定的对应物，驱动拟能的作用，更一般地说，是造成流动规律性和奇点或准奇点出现的原因，因此，它们是湍流现象的核心。

英文摘要：

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.

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摘要：

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