论文标题
Navier-Stokes方程和应用的定量部分规律性
Quantitative partial regularity of the Navier-Stokes equations and applications
论文作者
论文摘要
我们证明了Navier-Stokes方程的Caffarelli-Kohn-Nirenberg部分规则定理的对数改进。关键思想是找到使用Pigonhole原理的耗散能量绝对连续性的定量对应物。基于相同的方法，对于任何合适的弱解决方案，我们在一个空间方向上显示了规律性的间隔，其长度取决于溶液的自然局部能量。然后，在轴向对称情况下，我们给出了后者的两个应用。第一个是用于与小漩涡合适的弱解决方案的局部定量规则标准。第二个是略有改进的单点CKN标准，这意味着所有已知（略微临界）的I型规律性在文献中导致了。
We prove a logarithmic improvement of the Caffarelli-Kohn-Nirenberg partial regularity theorem for the Navier-Stokes equations. The key idea is to find a quantitative counterpart for the absolute continuity of the dissipation energy using the pigeonhole principle. Based on the same method, for any suitable weak solution, we show the existence of intervals of regularity in one spatial direction with length depending exponentially on the natural local energies of the solution. Then, we give two applications of the latter result in the axially symmetric case. The first one is a local quantitative regularity criterion for suitable weak solutions with small swirl. The second one is a slightly improved one-point CKN criterion which implies all known (slightly supercritical) Type I regularity results in the literature.
