论文标题
对3D Navier-Stokes方程的合适解决方案的第二个导数估计
Second derivatives estimate of suitable solutions to the 3D Navier-Stokes equations
论文作者
论文摘要
我们研究了第三维中不可压缩的Navier-Stokes方程的合适弱解的第二个空间衍生物。我们表明，对于任何$ q> \ frac43 $，它是本地$ l ^{\ frac43，q} $，它从当前结果$ l ^{\ frac43，\ infty} $改善。对于平滑溶液，也获得了较高的涡度衍生物的Lorentz空间的类似改进。我们使用爆破技术来获得与缩放尺度兼容的非线性边界。本地研究在涡度方程式上发挥作用，并使用de giorgi迭代。在这项本地研究中，我们可以在没有任何压力的情况下获得任何规律性的涡度。局部到全球步骤使用最近构建的最大函数用于传输方程。
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 $L ^{\frac43, q}$ for any $q > \frac43$, which improves from the current result $L ^{\frac43, \infty}$. 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.
