论文标题
在超临界规律下，在Navier-Stokes方程中设置的潜在单数点的分形维度
Fractal dimension of potential singular points set in the Navier-Stokes equations under supercritical regularity
论文作者
论文摘要
本文的主要目的是回答Robinson和Sadowski [21，p。 505，通讯。数学。 Phys。，2010] {[RS3]}对于Navier-Stokes方程。首先，我们证明了潜在单数点的上限尺寸设置了$ l^{q}（0，t; l^{p}（\ lakbbbb {r}^{3}^{3}^{3}^{3}）$ for $ 1 \ leq \ leq \ frac \ frac {2} { 3} {p} \ leq \ frac32 $带有$ 2 \ leq q <\ infty $和$ 2 <p <\ p <\ infty $最多是$ \ max \ {p，q \}（\ frac {2} {2} {q} {q} {q}+\ frac {3} {3} {3} {p} {p} {p} {p} {p} {p} {p} - 1）$ ins in in Systems ins in in oin in in oin in in oin in in oin。 Secondly, it is shown that $1-2 s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $ u\in L^{2}(0,T;\dot{H}^{s+1}(\mathbb{R}^{3}))$ for $0\leq s\leq\frac12$ is zero, whose proof relies on Caffarelli-Silvestre的扩展。受贝克·旺（Baker-Wang）最近的工作的启发[1]，这进一步使我们能够讨论如果在某些超临界规律性下速度的梯度梯度，则可以讨论合适弱解决方案的潜在奇异点集的hausdorff尺寸。
The main objective of this paper is to answer the questions posed by Robinson and Sadowski [21, p. 505, Comm. Math. Phys., 2010]{[RS3]} for the Navier-Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal{S}$ of suitable weak solution $u$ belonging in $ L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for $1\leq\frac{2}{q}+\frac{ 3}{p}\leq\frac32$ with $2\leq q<\infty$ and $2<p<\infty$ is at most $\max\{p,q\}(\frac{2}{q}+\frac{ 3}{p}-1)$ in this system. Secondly, it is shown that $1-2 s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $ u\in L^{2}(0,T;\dot{H}^{s+1}(\mathbb{R}^{3}))$ for $0\leq s\leq\frac12$ is zero, whose proof relies on Caffarelli-Silvestre's extension. Inspired by Baker-Wang's recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity under some supercritical regularity.
