论文标题
超脱离纳维尔 - 斯托克斯系统中奇异点的盒子计数尺寸上的更加尖锐的界限
Sharper bounds on the box-counting dimension of singularities in the hyperdissipative Navier-Stokes system
论文作者
论文摘要
我们在适当的3D不可压缩的高度脱离Navier-Stokes System的适当弱解决方案中的一组潜在奇异点的盒子计数维度上研究上限 \ begin {equination*} \ partial_t u+（-Δ）^αu+（u \ cdot \ nabla）u+\ nabla p = 0，\ qquad \ qquad \ operatorname {div} u = 0，\ end End {queration {equination {equination*} for $ nin（1,5/4）$。我们的主要观察结果是，在[11]中开发的经典迭代方案，并在[27]中用于改善全拉普拉斯案例的上限，但可以扩展到过度脱落的情况，尽管有非局部性的分数laplacian，否则适当选择的局部数量，这些局部数量是规模不变的。这是通过匹配所需估计值的时间空间尺度的正确顺序来实现的，该估计有效地量化了迭代过程中$（-Δ）^α$。特别是，我们采用了在最近的突破[5]中内置的高丝质框架，其中\ begin {equation*}给出了$α$的一组潜在奇点的盒子计数尺寸的上限。 l（α）= \ frac {15-2α-8α^2} {3} \ quad \ mbox {for} \ Quad 1 <α<\ frac {5} {4} {4}。 \ end {equation*}在本文中，我们将为$α= 1 $设计的迭代方案[27]概括为$ 1 <α<5/4 $，这导致了新建立的bonged \ bent \ begin \ eken {equination {equination {equination*} j（α）= \ frac {36（3-α）（3+2α）（5-4α）} { - 64α^3+272α^2-300α+369} \ Quad \ Quad \ mbox {for} \ Quad 1 <Quad 1 <α<α<\ frac {5} {5} {4} {4} {4} {4}，$ ligation $ list $ live $ live $ live nequily*aquily*aquint actiant <equint actiant <equint actiant pantiant pantiant unient（等级）在[5]中获得。
We study upper bounds on the box-counting dimension of the set of potential singular points in suitable weak solutions to the 3D incompressible hyperdissipative Navier-Stokes system \begin{equation*} \partial_t u + (-Δ)^αu+(u\cdot \nabla)u+\nabla p = 0, \qquad \operatorname{div} u = 0, \end{equation*} for $α\in(1,5/4)$. Our main observation is that a classical iteration scheme developed in [11] and used in [27] to improve upper bounds for the full Laplacian case can be extended to the hyperdissipative case with properly chosen local quantities that are scale-invariant, despite non-locality of fractional Laplacian. This is achieved by matching up the correct orders of the temporal-spatial scales of the required estimates that effectively quantify $(-Δ)^α$ during the iterations. In particular, we adopt the hyperdissipative framework built in the recent breakthrough [5] where the upper bounds on the box-counting dimension of the set of potential singularities in $α$ are given by \begin{equation*} L(α)= \frac{15-2α-8α^2}{3} \quad \mbox{for}\quad 1<α<\frac{5}{4}. \end{equation*} In this paper, we generalize the iteration scheme [27] designed for $α=1$ to the case $1<α<5/4$, which leads to the newly established bound \begin{equation*} J(α)= \frac{36(3-α)(3+2α)(5-4α)}{-64α^3+272α^2-300α+369} \quad \mbox{for} \quad 1<α<\frac{5}{4}, \end{equation*} improving the aforementioned bound $L(α)$ obtained in [5].
