We study the second spatial derivatives of suitable weak solutions to the incompressible Navier–Stokes equations in dimension three. We show that it is locally \(L ^{\frac{4}{3}, q}\) for any \(q > \frac{4}{3}\), which improves from the current result of \(L ^{\frac{4}{3}, \infty }\). Similar improvements in Lorentz space are also obtained for higher derivatives of the vorticity for smooth solutions. We use a blow-up technique to obtain nonlinear bounds compatible with the scaling. The local study works on the vorticity equation and uses De Giorgi iteration. In this local study, we can obtain any regularity of the vorticity without any a priori knowledge of the pressure. The local-to-global step uses a recently constructed maximal function for transport equations.

我们研究了三维中不可压缩的Navier–Stokes方程的合适弱解的第二空间导数。我们证明对于任何\（q> \ frac {4} {3} \）来说，它都是本地\（L ^ {\ frac {4} {3}，q } \），这比\（当前的结果）有所改善L ^ {\ frac {4} {3}，\ infty} \）。对于光滑解的更高涡度导数，在Lorentz空间中也获得了类似的改进。我们使用爆炸技术来获得与缩放兼容的非线性范围。当地的研究工作在涡度方程上，并使用De Giorgi迭代法。在这项本地研究中，我们无需任何先验压力即可获得涡度的任何规律性。局部到全局步骤将最近构造的最大值函数用于运输方程。