Fractional Navier–Stokes equations—featuring a fractional Laplacian-provide a ‘bridge’ between the Euler equations (zero diffusion) and the Navier–Stokes equations (full diffusion). The problem of whether an initially smooth flow can spontaneously develop a singularity is a fundamental problem in mathematical physics, open for the full range of models—from Euler to Navier–Stokes. The purpose of this work is to present a hybrid, geometric-analytic regularity criterion for solutions to the 3D fractional Navier–Stokes equations expressed as a balance—in the average sense—between the vorticity direction and the vorticity magnitude, key geometric and analytic descriptors of the flow, respectively.

分数阶Navier–Stokes方程-具有分数拉普拉斯算子，在Euler方程（零扩散）和Navier–Stokes方程（全扩散）之间提供了“桥梁”。最初的平稳流动是否可以自发地产生奇点的问题是数学物理学中的一个基本问题，适用于从欧拉到纳维尔–斯托克斯的所有模型。这项工作的目的是为3D分数Navier–Stokes方程的解法提供一个混合的，几何解析的正则性判据，该解表示为旋涡方向和旋涡强度之间的平均意义上的平衡，关键的几何和解析描述子流的分别。