Sharper bounds on the box-counting dimension of singularities in the hyperdissipative Navier-Stokes system

Abstract

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 + (-\Delta)^{\alpha}u+(u\cdot \nabla)u+\nabla p = 0, \qquad \operatorname{div} u = 0, \end{equation*} for $\alpha\in(1,5/4)$. 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 $(-\Delta)^{\alpha}$ 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 $\alpha$ are given by \begin{equation*} L(\alpha)= \frac{15-2\alpha-8\alpha^2}{3} \quad \mbox{for}\quad 1<\alpha<\frac{5}{4}. \end{equation*} In this paper, we generalize the iteration scheme [27] designed for $\alpha=1$ to the case $1<\alpha<5/4$, which leads to the newly established bound \begin{equation*} J(\alpha)= \frac{36(3-\alpha)(3+2\alpha)(5-4\alpha)}{-64\alpha^3+272\alpha^2-300\alpha+369} \quad \mbox{for} \quad 1<\alpha<\frac{5}{4}, \end{equation*} improving the aforementioned bound $L(\alpha)$ obtained in [5].