# Fractal dimension of potential singular points set in the Navier-Stokes
equations under supercritical regularity

Yanqing Wang111 Department of Mathematics and Information Science, Zhengzhou
University of Light Industry, Zhengzhou, Henan 450002, P. R. China Email:
<EMAIL_ADDRESS>and Gang Wu222Corresponding author. School of
Mathematical Sciences, University of Chinese Academy of Sciences, Beijing
100049, P. R. China Email<EMAIL_ADDRESS>
###### Abstract

The main objective of this paper is to answer the questions posed by Robinson
and Sadowski [21, p. 505, Comm. Math. Phys., 2010] 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
$1\leq\frac{2}{q}+\frac{3}{p}\leq\frac{3}{2}$ 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 $1-2s$ 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 $0\leq s\leq\frac{1}{2}$ 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.

MSC(2000): 35B65, 35D30, 76D05
Keywords: Navier-Stokes equations; singular points; Hausdorff dimension; Box
dimension

##  1 Introduction

We consider the three-dimensional incompressible non-stationary Navier-Stokes
equations

$\left\\{\begin{aligned} &u_{t}-\Delta u+u\cdot\nabla
u+\nabla\Pi=0,~{}~{}\text{div}\,u=0~{}~{}\text{in}~{}~{}~{}~{}\mathbb{R}^{3}\times(0,\,T),\\\
&u|_{t=0}=u_{0}(x)~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{on}~{}~{}~{}\mathbb{R}^{3}\times\\{t=0\\}.\end{aligned}\right.$
(1.1)

Here $u$ describes the velocity of the flow and the scalar function $\Pi$
represents the pressure of the fluid. The initial data $u_{0}(x)$ satisfies
divergence free condition.

The full regularity of solutions of the three dimensional Navier-Stokes
equations is not known, the partial regularity theory of suitable weak
solutions of this system starting from Scheffer’s work [23, 24, 25] is well-
known. The famous Caffarelli-Kohn-Nirenberg theorem in [4] is that one
dimensional Hausdorff measure of the potential space-time singular points set
$\mathcal{S}$ of suitable weak solutions to the 3D Navier-Stokes equations is
zero. The critical tool is the following so-called $\epsilon$-regularity
criterion: there is an absolute constant $\epsilon$ such that if

$\limsup_{\varrho\rightarrow 0}\frac{1}{\varrho}\iint_{Q(\varrho)}|\nabla
u|^{2}dxdt\leq\epsilon,$ (1.2)

then $u$ is bounded in a neighborhood of $(0,0)$, where
$Q(\varrho):=B(\varrho)\times(-\varrho^{2},0)$ and $B(\varrho)$ denotes the
ball of center $0$ and radius $\varrho$. To this end, Caffarelli-Kohn-
Nirenberg [4] established $\epsilon$-regularity criterion at one scale below

$\left\|u\right\|_{L^{3}(Q(1))}+\|u\Pi\|_{L^{1}(Q(1))}+\|\Pi\|_{L^{1,5/4}(Q(1))}\leq\epsilon.$
(1.3)

An alternative approach of Caffarelli-Kohn-Nirenberg theorem based on blow-up
argument was duo to Lin, Ladyzhenskaya and Seregin [15, 17], where the
corresponding $\epsilon$-regularity criterion at one scale reads

$\left\|u\right\|_{L^{3}(Q(1))}+\|\Pi\|_{L^{3/2}(Q(1))}\leq\epsilon.$ (1.4)

In what follows, a point $z=(x,t)$ in (1.1) is said to be regular if $u$
belongs to $L^{\infty}$ at a neighborhood of $z$. Otherwise, it is called
singular. The estimation of the size of potential singular points set in 3D
the Navier-Stokes equations can be found in [31, 30, 19, 20, 21, 13, 14, 9,
1].

On the other hand, the integral (Serrin) type conditions based on the
velocity, the gradient of the velocity or the pressure leads to the the full
regularity of Leray-Hopf weak solutions of the 3D Navier-Stokes equations.
Precisely, a weak solution $u$ is smooth on $(0,T]$ if it satisfies one of the
following three conditions

  1. (1)

Serrin [27], Struwe [26], Escauriaza, Seregin and Šverák [7]

$u\in
L^{p}(0,T;L^{q}(\mathbb{R}^{3}))~{}~{}~{}\text{with}~{}~{}~{}~{}2/p+3/q=1,~{}~{}q\geq
3.$ (1.5)

  2. (2)

Beirao da Veiga [2]

$\nabla u\in
L^{p}(0,T;L^{q}(\mathbb{R}^{3}))~{}~{}~{}\text{with}~{}~{}~{}~{}2/p+3/q=2,~{}~{}q>3/2.$
(1.6)

  3. (3)

Berselli and Galdi [3], Zhou [36, 37]

$\Pi\in
L^{p}(0,T;L^{q}(\mathbb{R}^{3}))~{}~{}~{}\text{with}~{}~{}~{}~{}2/p+3/q=2,~{}~{}q>3/2.$
(1.7)

The aforementioned integral (Serrin) type conditions can be seen as the
critical regularity, which is scale invariant under the natural scaling of the
Navier-Stokes equations (1.1). The full regularity means that the set of
$\mathcal{S}$ is empty. The natural (supercritical) regularity $u\in
L^{q}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0,\,T};\,L^{p}(\mathbb{R}^{3}))$
with $\frac{2}{q}+\frac{3}{p}=\frac{3}{2}$ in suitable weak solutions means
that

$\dim_{H}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{S}})\leq
1~{}~{}\text{and}~{}\dim_{B}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{S}})\leq
5/3,$ (1.8)

which can be found in [4, 19] and $\dim_{H}(S)$ and $\dim_{B}(S)$ denote the
Hausdorff dimension and box dimension of a set $S$, respectively. A natural
question is weather the suitable weak solutions satisfying supercritical
regularity $u\in L^{q}(0,\,T;\,L^{p}(\mathbb{R}^{3}))$ with
$1<\frac{2}{q}+\frac{3}{p}<\frac{3}{2}$ lower the fractal dimension in (1.8).
In this direction, Gustafson, Kang and Tsai [9] proved that the Hausdorff
dimension of the potential singular points set $\mathcal{S}$ of a Leray-Hopf
weak solution belonging in $u\in L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for
$1\leq\frac{2}{q}+\frac{3}{p}$ with $\frac{2}{p}+\frac{2}{q}<1$ and
$\frac{3}{p}+\frac{1}{q}<1$ is at most $3-q+\frac{2q}{p},p>q$ or
$2-q+\frac{3q}{p},p\leq q$. Robinson and Sadowski [21] showed that the upper
box dimension of potential singular points set $\mathcal{S}$ of a suitable
weak solution belonging in $u\in L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for
$1\leq\frac{2}{q}+\frac{3}{p}\leq\frac{3}{2}$ with $3<p,q<\infty$ is no
greater than

$\max\\{p,q\\}(\frac{2}{q}+\frac{3}{p}-1).$ (1.9)

In addition, the Hausdorff dimension of potential singular points set
$\mathcal{S}$ of a suitable weak solution belonging in $\nabla u\in
L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for
$2\leq\frac{2}{q}+\frac{3}{p}\leq\frac{5}{2}$ with $2<p\leq q<\infty$ is less
than or equal to

$\max\\{p,q\\}(\frac{2}{q}+\frac{3}{p}-2).$ (1.10)

In [21, Conclusion, Page 9], Robinson and Sadowski mentioned some natural
questions from their results :

  1. (1)

It would be interesting to relax the assumption $q>3$ in (1.9) and obtain the
same bound for any $q\geq 2$;

  2. (2)

similarly in (1.10) one would like to relax the condition $q\geq p$.

  3. (3)

In order to obtain (1.9) in a bounded domain we would require the analogue of
Lemma 2 (estimates for the pressure when $u\in L^{q}(0,T;L^{p}(\Omega))$.

  4. (4)

An order of magnitude harder is to determine whether any of these partial
regularity results can be proved for general weak solutions, and not only
suitable weak solutions.

In this paper, our first result is the following theorem.

######  Theorem 1.1.

Let $u$ be a suitable weak solution belonging in $u\in
L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for
$1\leq\frac{2}{q}+\frac{3}{p}\leq\frac{3}{2}$ with $2\leq q<\infty$ and
$2<p<\infty$. Then, the upper box dimension of its potential singular points
set $\mathcal{S}$ is at most $\max\\{p,q\\}(\frac{2}{q}+\frac{3}{p}-1)$.

######  Remark 1.1.

Theorem 1.1 answers Robinson and Sadowski’s first question (1).

As observed in [9], the weak solutions in spaces
$L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ with $\frac{2}{p}+\frac{2}{q}<1$ and
$\frac{3}{p}+\frac{1}{q}<1$ are suitable weak solutions. Therefore, towards
the Robinson and Sadowski’s fourth question (4), we have

######  Corollary 1.2.

Let $u$ be a Leray-Hopf weak solution belonging in $u\in
L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for
$1\leq\frac{2}{q}+\frac{3}{p}\leq\frac{3}{2}$ with $\frac{2}{p}+\frac{2}{q}<1$
and $\frac{3}{p}+\frac{1}{q}<1$. Then, the upper box dimension of its
potential singular points set $\mathcal{S}$ is at most
$\max\\{p,q\\}(\frac{2}{q}+\frac{3}{p}-1)$.

With a slight modification of the proof of Theorem 1.1 and using the
$\epsilon$-regularity criterion at one scale without pressure in [34], we can
obtain a parallel result of (1.9) in a bounded domain, which is corresponding
to Robinson and Sadowski’s third issue.

######  Theorem 1.3.

Let $u$ be a suitable weak solution belonging in $u\in
L^{q}(0,T;L^{p}(\Omega))$ for $1\leq\frac{2}{q}+\frac{3}{p}\leq\frac{3}{2}$
with $\frac{5}{2}<q,p<\infty$. Then, the upper box dimension of its potential
singular points set $\mathcal{S}$ is at most
$\max\\{p,q\\}(\frac{2}{q}+\frac{3}{p}-1)$.

Roughly, the following figures summarize the known upper box dimension of its
potential singular points set $\mathcal{S}$ of suitable weak solutions under
supercritical regularity in the Navier-Stokes equations.

$\frac{1}{3}$$\frac{1}{2}$$\frac{3}{4}$$(\frac{1}{9},\frac{1}{3})$$(\frac{5}{18},\frac{1}{3})$$(\frac{1}{3},\frac{1}{4})$$\frac{1}{2}$$\frac{1}{p}$11$\frac{1}{q}$01Fig.
1: Robinson-Sadowski results on
$\mathbb{R}^{3}$$\frac{1}{3}$$\frac{1}{2}$$\frac{3}{4}$$\frac{1}{2}$$\frac{1}{p}$11$\frac{1}{q}$01Fig.
2: Theorem 1.1 on
$\mathbb{R}^{3}$$(\frac{1}{4},\frac{1}{4})$$\frac{1}{3}$$\frac{1}{2}$$\frac{3}{4}$$\frac{1}{2}$$\frac{1}{p}$11$\frac{1}{q}$01Fig.
3: Corollary 1.2 weak
solutions$\frac{1}{2}$$\frac{1}{3}$$\frac{1}{2}$$\frac{3}{4}$$\frac{2}{5}$$(\frac{2}{5},\frac{3}{20})$$(\frac{7}{30},2/5)$$(\frac{1}{15},\frac{2}{5})$$\frac{1}{p}$11$\frac{1}{q}$01Fig.
4: Theorem 1.3 on bounded domain

Next, we study the Robinson and Sadowski’s second issue involving the gradient
of the velocity with additional regularity. It seems that this problem is more
complicated. Very recently, in the other direction, Baker and Wang [1]
estimate the Hausdorff dimension of the singular set for the Navier-Stokes
equations with supercritical assumptions on the pressure. There are two new
ingredients in their proof. The first one is is the high regularity of the
solutions with certain supercritical assumptions on pressure in the Navier-
Stokes equations. The second one is the $\epsilon$-regularity criterion in
terms of quantity $|\nabla u|^{2}|v|^{q-2}$ with $2<q<3$, which usually arises
in the $L^{p}$ type energy estimates of the Navier-Stokes equations. In the
spirit of [1], we consider the $\epsilon$-regularity criterion via quantity
$\Lambda^{s+1}u$ with $s>0$, which usually appears in the $\dot{H}^{s+1}$ type
energy estimates of the Navier-Stokes equations. One naturally invokes the
Caffarelli-Silvestre extension used in [18, 29, 6] to overcome non-local
derivatives. However, since $s>0$, one requires higher order Caffarelli-
Silvestre (Yang) extensions [38]. To this end, we observe that that the
following identity due to [6], for $\alpha=s+1>1$,

$c_{\alpha}\int_{\mathbb{R}^{4}_{+}}y^{3-2\alpha}|\nabla^{\ast}(\nabla
u)^{\ast}|^{2}(x,y,t)\,dx\,dy=\int_{\mathbb{R}^{3}}\left|(-\Delta)^{\frac{\alpha-1}{2}}\nabla
u\right|^{2}(x,t)\,dx=\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{\alpha}{2}}u|^{2}(x,t)\,dx,$

that is,

$\|u\|^{2}_{\dot{H}^{s+1}}=c_{s}\int_{\mathbb{R}^{4}_{+}}y^{1-2s}|\nabla^{\ast}(\nabla
u)^{\ast}|^{2}(x,y,t)\,dx\,dy,$ (1.11)

which helps us to reduce the proof of Theorem 1.4 to show Theorem 1.5 just by
Caffarelli-Silvestre extension rather than higher order (Yang) extension.
Theorem 1.4 can be viewed as the interpolation between the Caffarelli-Kohn-
Nirenberg theorem and Kozono-Taniuchi regular class $L^{2}(0,T;BMO)$, which is
of independent interest.

######  Theorem 1.4.

Let $u$ be a suitable weak solution belonging in $u\in
L^{2}(0,T;\dot{H}^{s+1}(\mathbb{R}^{3}))$ for $0\leq s\leq\frac{1}{2}$. Then,
$\mathcal{H}^{1-2s}(\mathcal{S})=0$.

######  Theorem 1.5.

Suppose that $u$ is a suitable weak solution to (1.1). Then there exists an
absolute positive constant $\varepsilon_{01}$ such that $(0,0)$ is a regular
point if

$\frac{1}{\mu^{1-2s}}\iint_{Q^{\ast}(\mu)}y^{1-2s}|\nabla^{\ast}(\nabla
u)^{\ast}|^{2}dxdydt\leq\varepsilon_{01}.$

As an application of Theorem 1.4 and the energy estimate of the Navier-Stokes
equtions, we can partially answer the Robinson and Sadowski’s second question.

######  Corollary 1.6.

Let $u$ be a suitable weak solution belonging in $\nabla u\in
L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for
$2\leq\frac{2}{q}+\frac{3}{p}\leq\frac{5}{2}$.

(1) If $\frac{5}{2}-\frac{3}{p}-\frac{5}{2q}\geq
0,1<p<\frac{54+12\sqrt{14}}{25},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1<q\leq
2}$, then
$\mathcal{H}^{\frac{2(\frac{2}{q}+\frac{3}{p}-2)}{1-\frac{1}{q}}}(\mathcal{S})=0$.

(2) If $2-\frac{3}{p}-\frac{1}{q}\geq 0,\frac{3}{2}<p<\frac{12}{7},q\geq 4$,
then $\mathcal{H}^{q(\frac{2}{q}+\frac{3}{p}-2)}(\mathcal{S})=0$.

At present, the Hausdorff dimension of suitable weak solutions with the
gradient of the velocity under supercritical regularity are summarized in the
following figures.

$\frac{1}{2}$$\frac{1}{2}$$\frac{1}{p}$01$\frac{1}{q}$1$\frac{2}{3}$$\frac{5}{6}$$\frac{5}{4}$Fig.
5: First part of Corollary 1.6
$\frac{1}{p}$01$\frac{1}{q}$1$\frac{2}{3}$$\frac{5}{6}$$\frac{5}{4}$$\frac{1}{4}$$\frac{1}{2}$$\frac{1}{2}$Fig.
6: Second part of Corollary
1.6$\frac{1}{p}$$\frac{1}{2}$$\frac{1}{2}$11$\frac{1}{q}$1$\frac{2}{3}$Fig. 7:
Known Hausdorff dimension ofthe gradient of the velocity
$\mathcal{H}^{\frac{2(\frac{2}{q}+\frac{3}{p}-2)}{1-\frac{1}{q}}}(\mathcal{S})=0$$\mathcal{H}^{q(\frac{2}{q}+\frac{3}{p}-2)}(\mathcal{S})=0$

The remainder of this paper is organized as follows. In Section 2, we will
begin with the notations and the definition of fractal dimension including the
Box dimension and Hausdorff dimension. Then we recall the Caffarelli and
Silvestre’s generalized extension for the fractional Laplacian operator and
$\epsilon$-regularity criterion at one scale. Section 3 is devoted to the
proof to Theorem 1 concerning Box dimension. Partial regularity results
involving Hausdorff dimension is proved in Section 4.

##  2  Preliminaries

First, we introduce some notations used in this paper. Throughout this paper,
we denote

$\displaystyle B(x,\mu)=\\{y\in\mathbb{R}^{3}||x-y|\leq\mu\\},~{}~{}$
$\displaystyle B$ $\displaystyle(\mu):=B(0,\mu),$ $\displaystyle
Q(x,t,\mu)=B(x,\,\mu)\times(t-\mu^{2},t),~{}~{}$ $\displaystyle Q$
$\displaystyle(\mu):=Q(0,0,\mu),$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}B^{\ast}(x,\mu)=B(x,\mu)\times(0,\mu),}~{}~{}$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}B^{\ast}}$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\mu):=B^{\ast}(0,\mu),}$
$\displaystyle
Q^{\ast}(x,t,\mu)=B(x,\mu)\times(0,\mu)\times(t-\mu^{2},t),~{}~{}$
$\displaystyle Q^{\ast}$ $\displaystyle(\mu):=Q^{\ast}(0,0,\mu).$

For $p\in[1,\,\infty]$, the notation $L^{p}(0,\,T;X)$ stands for the set of
measurable functions on the interval $(0,\,T)$ with values in $X$ and
$\|f(\cdot,t)\|_{X}$ belonging to $L^{p}(0,\,T)$. For simplicity, we write
$\|f\|_{L^{p,q}(Q(\mu))}:=\|f\|_{L^{p}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\mu^{2}},0;L^{q}(B(\mu)))}$
and $\|f\|_{L^{p}(Q(\mu))}:=\|f\|_{L^{p}L^{p}(Q(\mu))}$. We shall denote by
$\langle f,\,g\rangle$ the $L^{2}$ inner product of $f$ and $g$. The classical
Sobolev norm $\|\cdot\|_{H^{s}}$ is defined as
$\|f\|^{2}_{{H}^{s}}=\int_{\mathbb{R}^{n}}(1+|\xi|)^{2s}|\hat{f}(\xi)|^{2}d\xi$,
$s\in\mathbb{R}$. We denote by $\dot{H}^{s}$ homogenous Sobolev spaces with
the norm
$\|f\|^{2}_{\dot{H}^{s}}=\int_{\mathbb{R}^{n}}|\xi|^{2s}|\hat{f}(\xi)|^{2}d\xi$.
Denote the average of $f$ on the ball $B(\mu)$ by $\overline{f}_{\mu}$.
$\Gamma$ denotes the standard normalized fundamental solution of Laplace
equation in $\mathbb{R}^{3}$. We denote by Div  the divergence operator in
$\mathbb{R}^{4}_{+}$. $|S|$ represents the Lebesgue measure of the set $S$. We
will use the summation convention on repeated indices. $C$ is an absolute
constant which may be different from line to line unless otherwise stated in
this paper.

######  Definition 2.1.

The (upper) box-counting dimension of a set $X$ is usually defined as

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{dim}_{B}(X)}=\limsup_{\epsilon\rightarrow
0}\frac{\log N(X,\,\epsilon)}{-\log\epsilon},$

where $N(X,\,\epsilon)$ is the minimum number of balls of radius $\epsilon$
required to cover $X$.

Let $\beta>0$, $\delta>0$ and $\Omega\times I$ can be covered by the union of
series of parabolic balls $Q(r)$ with radius $r_{j}$ less than $\delta$ for
$j\in\mathbb{N}$. Define

$\mathcal{P}^{\beta}_{\delta}=\inf\Big{\\{}\Sigma r_{j}^{\beta}|\Omega\times
I\subseteq\cup Q(r_{j}),r_{j}<\delta,j\in\mathbb{N}\Big{\\}}$

and $\mathcal{P}^{\beta}=\lim_{\delta\rightarrow
0}\mathcal{P}^{\beta}_{\delta}$. If there is $\beta_{0}$ such that
$\mathcal{P}^{\beta}=\infty$ if $\beta<\beta_{0}$ and $\mathcal{P}^{\beta}=0$
if $\beta>\beta_{0}$, then $\beta_{0}$ is called as the parabolic Hausdorff
dimension and $\mathcal{P}^{\beta}$ is the parabolic Hausdorff measure. The
details of fractal dimension can be found in [8].

Next, we focus on Caffarelli and Silvestre’s generalized extension for the
fractional Laplacian operator $(-\Delta)^{\alpha}$ with $0<\alpha<1$ in [5].
The fractional power of Laplacian in $\mathbb{R}^{3}$ can be interpreted as

$(-\Delta)^{\alpha}u=-C_{\alpha}\lim\limits_{y\rightarrow
0_{+}}y^{1-2\alpha}\partial_{y}u^{\ast},$

where $u^{\ast}$ satisfies

$\left\\{\begin{aligned}
&\text{Div\,}(y^{1-2\alpha}\nabla^{\ast}u^{\ast})=0~{}~{}\text{in}~{}~{}\mathbb{R}^{4}_{+},\\\
&u^{\ast}|_{y=0}=u,~{}x\in\mathbb{R}^{3}.\end{aligned}\right.$ (2.1)

As a by-product of the above equation, for any $v|_{y=0}=u$, it holds

$\int_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{R}_{+}^{4}}}y^{1-2s}|\nabla^{\ast}u^{\ast}|^{2}dxdy\leq\int_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(2.2)

Moreover, from Section 3.2 in [5], the definition of the $\dot{H}^{\alpha}$
norm can be written as

$\|u\|^{2}_{\dot{H}^{\alpha}}=\int_{\mathbb{R}^{3}}|\xi|^{2\alpha}|\hat{u}(\xi)|^{2}d\xi=\int_{\mathbb{R}_{+}^{4}}y^{1-2\alpha}|\nabla^{\ast}u^{\ast}|^{2}dxdy.$
(2.3)

We recall the following observation due to [6], for $\alpha>1$,

$c_{\alpha}\int_{\mathbb{R}^{4}_{+}}y^{3-2\alpha}|\nabla^{\ast}(\nabla
u)^{\ast}|^{2}(x,y,t)\,dx\,dy=\int_{\mathbb{R}^{3}}\left|(-\Delta)^{\frac{\alpha-1}{2}}\nabla
u\right|^{2}(x,t)\,dx=\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{\alpha}{2}}u|^{2}(x,t)\,dx\,.$

Hence,

$\|u\|^{2}_{\dot{H}^{s+1}}=c_{s}\int_{\mathbb{R}^{4}_{+}}y^{1-2s}|\nabla^{\ast}(\nabla
u)^{\ast}|^{2}(x,y,t)\,dx\,dy.$ (2.4)

Base on the natural scaling of the Navier-Stokes equations, we set the
following two dimensionless quantities

$\displaystyle E^{\ast}_{\ast}(\nabla^{\ast}(\nabla
u)^{\ast},\mu)=\frac{1}{\mu^{1-2s}}\iint_{Q^{\ast}(\mu)}y^{1-2s}|\nabla^{\ast}(\nabla
u)^{\ast}|^{2}dxdydt,$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{\ast}(\nabla
u,\mu)=\frac{1}{\mu}\iint_{Q(\mu)}|\nabla u|^{2}dxdt}.$

To make our paper more self-contained and more readable, we outline the proof
of Poincaré inequality concerning Caffarelli and Silvestre’s generalized
extension.

######  Lemma 2.1.

Let $u$ and $u^{\ast}$ be defined in (2.1). There exist a constant $C$ such
that

$\displaystyle\|u-\overline{u}_{\mu}\|_{L^{\frac{6}{3-2s}}(B(\mu/2))}\leq
C\Big{(}\int_{B^{\ast}(\mu)}y^{1-2s}|\nabla^{\ast}u^{\ast}|^{2}dxdy\Big{)}^{1/2},$
(2.5) $\displaystyle\|u-\overline{u}_{\mu}\|_{L^{2}(B(\mu/2))}\leq
C\mu^{s}\Big{(}\int_{B^{\ast}(\mu)}y^{1-2s}|\nabla^{\ast}u^{\ast}|^{2}dxdy\Big{)}^{1/2}.$
(2.6)

###### Proof.

Consider the usual cut-off functions

$\eta_{1}(x)=\left\\{\begin{aligned}
&1,\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x\in}B(\hbar\mu),~{}0<\hbar<1,\\\
&0,\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x\in}B^{c}(\mu),\end{aligned}\right.$

and

$\eta_{2}(y)=\left\\{\begin{aligned} &1,\,0\leq y\leq\hbar\mu,\\\
&0,\,y>\mu,\end{aligned}\right.$

satisfying

$0\leq\eta_{1},\,\eta_{2}\leq
1,~{}~{}~{}\text{and}~{}~{}~{}\mu|\partial_{x}\eta_{1}(x)|+\mu|\partial_{y}\eta_{2}(y)|\leq
C.$

By the Young inequality and (2.3) and (2.2),

$\displaystyle\|u\eta_{1}\|^{2}_{\dot{H}^{s}}$
$\displaystyle=\int_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{R}_{+}^{4}}}y^{1-2s}|\nabla^{\ast}(u\eta_{1})^{\ast}|^{2}dxdy$
$\displaystyle\leq
C\int_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{R}_{+}^{4}}}y^{1-2s}|\nabla^{\ast}(u^{\ast}\eta_{2}\eta_{1})|^{2}dxdy$
$\displaystyle\leq
C\mu^{-2}\int_{B^{\ast}(\mu)}y^{1-2s}|u^{\ast}|^{2}+C\int_{B^{\ast}(\mu)}y^{1-2s}|\nabla^{\ast}u^{\ast}|^{2}.$

Thanks to the classical weight Poincaré inequality, we infer that

$\int_{B^{\ast}(\mu)}y^{1-2s}|u^{\ast}-\overline{u^{\ast}}_{B^{\ast}(\mu)}|^{2}\leq
C\mu^{2}\int_{B^{\ast}(\mu)}y^{1-2s}|\nabla^{\ast}u^{\ast}|^{2},$ (2.7)

where
$\overline{u^{\ast}}_{B^{\ast}(\mu)}=\frac{1}{|B^{\ast}(\mu)|}\int_{B^{\ast}(\mu)}y^{1-2s}u^{\ast}dxdy$
and $|B^{\ast}(\mu)|=\int_{B^{\ast}(\mu)}y^{1-2s}dydx.$ The above inequalities
imply

$\Big{(}\int_{B(\hbar\mu)}|u-\overline{u^{\ast}}_{B^{\ast}(\mu)}|^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{6}{3-2s}}}\Big{)}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{3-2s}{3}}}\leq
C\int_{B^{\ast}(\mu)}y^{1-2s}|\nabla^{\ast}u^{\ast}|^{2}.$ (2.8)

It follows from $u^{\ast}=u(x)+\int^{y}_{0}\partial_{z}u^{\ast}dz$ and the
Hölder inequality that

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\big{|}}\overline{u^{\ast}}_{B^{\ast}(\mu)}-\overline{u}_{\mu}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\big{|}}$
$\displaystyle=\frac{1}{|B^{\ast}(\mu)|}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\big{|}}\int_{B^{\ast}(\mu)}y^{1-2s}\int^{y}_{0}\partial_{z}u^{\ast}dz{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\big{|}}$
$\displaystyle\leq
C\frac{1}{|B^{\ast}(\mu)|}\int_{B(\mu)}\int_{0}^{\mu}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y^{1-2s}\Big{(}\int^{y}_{0}z^{1-2s}|\partial_{z}u^{\ast}|^{2}dz\Big{)}^{1/2}\Big{(}\int^{y}_{0}z^{-(1-2s)}dz\Big{)}^{1/2}dydx}$
$\displaystyle\leq
C\mu^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s-\frac{3}{2}}}\Big{(}\int_{B^{\ast}(\mu)}z^{1-2s}|\nabla^{\ast}u^{\ast}|^{2}dxdz\Big{)}^{1/2}.$
(2.9)

Combining (2.8) with the latter inequality, we deduct that

$\displaystyle\Big{(}\int_{B(\hbar\mu)}|u-\overline{u}_{\mu}|^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{6}{3-2s}}}\Big{)}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{3-2s}{6}}}\leq$
$\displaystyle\Big{(}\int_{B(\hbar\mu)}|u-\overline{u^{\ast}}_{B^{\ast}(\mu)}|^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{6}{3-2s}}}\Big{)}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{3-2s}{6}}}$
$\displaystyle+\Big{(}\int_{B(\hbar\mu)}|\overline{u^{\ast}}_{B^{\ast}(\mu)}-\overline{u}_{\mu}|^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{6}{3-2s}}}\Big{)}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{3-2s}{6}}}$
$\displaystyle\leq$ $\displaystyle
C{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bigg{(}\int_{B^{\ast}(\mu)}y^{1-2s}|\nabla^{\ast}u^{\ast}|^{2}\bigg{)}^{\frac{1}{2}}},$

which means (2.5) and (2.6). ∎

######  Proposition 2.2.

([10]) Let the pair $(u,\Pi)$ be a suitable weak solution to the 3D Navier-
Stokes system (1.1) in $Q(1)$. There exists an absolute positive constant
$\varepsilon$ depending only on $p$ and $q$ such that if the pair $(u,\Pi)$
satisfies

$\|u\|_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{q,p}(Q(1))}}+\|\Pi\|_{L^{1}(Q(1))}<\varepsilon,$
(2.10)

for $1\leq 2/q+3/p<2,1\leq p,\,q\leq\infty$, then $u\in L^{\infty}(Q(1/2))$.

######  Proposition 2.3.

([34]) Let the pair $(u,\Pi)$ be a suitable weak solution to the 3D Navier-
Stokes system (1.1) in $Q(1)$. For any $\delta>0$, there exists an absolute
positive constant $\varepsilon$ such that if $u$ satisfies

$\iint_{Q(1)}|u|^{\frac{5}{2}+\delta}dxdt\leq\varepsilon,$ (2.11)

then, $u\in L^{\infty}(Q(1/16)).$

######  Lemma 2.4.

(Kato-Ponce Commutator and Product Estimates [11]) Let $\alpha>0$,
$p\in(1,\infty)$ and $p_{i}\in(1,\infty)$, $i=1,2,3,4.$ Then there exists a
positive constant $C$ such that

$\|\Lambda^{\alpha}(fg)-f\Lambda^{\alpha}g\|_{L^{p}}\leq C(\|\nabla
f\|_{L^{p_{1}}}\|\Lambda^{\alpha-1}g\|_{L^{p_{2}}}+\|\Lambda^{\alpha}f\|_{L^{p_{3}}}\|g\|_{L^{p_{4}}})$
(2.12)

and

$\|\Lambda^{\alpha}(fg)\|_{L^{p}}\leq
C(\|\Lambda^{\alpha}f\|_{L^{p_{1}}}\|g\|_{L^{p_{2}}}+\|f\|_{L^{p_{3}}}\|\Lambda^{\alpha}g\|_{L^{p_{4}}}),$
(2.13)

where
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{4}}$.

##  3 Box dimension of possible singular points set of suitable weak solutions

This section contains the proof of Theorem 1.1, Corollary 1.2 and Theorem 1.3.
The key point is an application of the $\epsilon$-regularity criterion (2.10)
and (2.11) at one scale.

###### Proof of Theorem 1.1.

We present the proof by contradiction. We suppose that
$\dim_{B}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{S}})>\max\\{p,q\\}(\frac{2}{q}+\frac{3}{p}-1).$
We pick a constant $\alpha$ such that
$\alpha_{0}=\max\\{p,q\\}(\frac{2}{q}+\frac{3}{p}-1)<\alpha<\dim_{B}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{S}})$.
Therefore, using the definition of the box dimension, we know that there
exists a sequence $\delta_{j}\rightarrow 0$ such that
$N(\mathcal{S},\delta_{j})>\delta_{j}^{-\alpha}.$We assume that
$(x_{i},t_{i})_{i=1}^{N(\mathcal{S},\delta_{j})}$ be a collection of
$\delta_{j}$\- separated points in $\mathcal{S}$. By the regularity criterion
in Proposition 2.2, for any $(x_{i},t_{i})\in\mathcal{S}$, we get

$\displaystyle\int_{t_{i}-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{j}^{2}}}^{t_{i}}\Big{[}\Big{(}\int_{B_{i}(\delta_{j})}|u|^{p}dx\Big{)}^{\frac{q}{p}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+}\Big{(}\int_{B_{i}(\delta_{j})}|\Pi|^{p/2}dx\Big{)}^{\frac{q}{p}}\Big{]}dt>\delta_{j}^{(-p+3)\frac{q}{p}+2}\varepsilon_{1},$

where $B_{i}(\mu):=B(x_{i},\mu)$. Thus we have

$\displaystyle\sum^{N(\mathcal{S},\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{j}})}_{i=1}\int_{t_{i}-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{j}^{2}}}^{t_{i}}\Big{[}\Big{(}\int_{B_{i}(\delta_{j})}|u|^{p}dx\Big{)}^{\frac{q}{p}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+}\Big{(}\int_{B_{i}(\delta_{j})}|\Pi|^{p/2}dx\Big{)}^{\frac{q}{p}}\Big{]}dt>N(\mathcal{S},\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{j}})\delta_{j}^{(-p+3)\frac{q}{p}+2}\varepsilon_{1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(3.1)

The pressure equation help us to obtain, for $p>2$, $q\geq 2$,

$\|\Pi\|^{p/2}_{L^{q/2}(0,T;L^{p/2}(\mathbb{R}^{3}))}\leq
C\|u\|^{p}_{L^{q}(0,T;L^{p}(\mathbb{R}^{3}))}.$ (3.2)

For the case $\frac{q}{p}>1$, we know that
$\alpha_{0}=q(\frac{2}{q}+\frac{3}{p}-1)$.

Now, we can apply the inequality
$\sum^{N(\mathcal{S},\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{j}})}_{i=1}(a_{i})^{\frac{q}{p}}\leq(\sum^{N(\mathcal{S},\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{j}})}_{i=1}a_{i})^{\frac{q}{p}}$
to control the left hand side of (3.1) by
$\|u\|^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q}}_{L^{q}(0,T;L^{p}(\mathbb{R}^{3}))}+\|\Pi\|^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q/2}}_{L^{q/2}(0,T;L^{p/2}(\mathbb{R}^{3}))}$.
This together with (3.2) implies that

$C\geq
N(\mathcal{S},\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{j}})\delta_{j}^{(-p+3)\frac{q}{p}+2}\varepsilon_{1}\geq\delta_{j}^{(-p+3)\frac{q}{p}+2-\alpha}\varepsilon_{1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(3.3)

We immediately get a contradiction as $j\rightarrow\infty.$

For the rest case $\frac{q}{p}\leq 1$, we invoke the inequality
$\sum^{N(\mathcal{S},\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{j}})}_{i=1}(a_{i})^{\frac{q}{p}}\leq
N^{(1-\frac{q}{p})}(\mathcal{S},\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{j}})(\sum^{N(\mathcal{S},\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{j}})}_{i=1}a_{i})^{\frac{q}{p}}$
in the proof.  With a slight modification of  the above proof, we see that
$C\geq
N^{\frac{q}{p}}(\mathcal{S},\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{j}})\delta_{j}^{(-p+3)\frac{q}{p}+2}\varepsilon_{1}$.
This means that

$C\geq
N(\mathcal{S},\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{j}})\delta_{j}^{\frac{p}{q}[(-p+3)\frac{q}{p}+2]}\varepsilon_{1}\geq\delta_{j}^{\frac{p}{q}[(-p+3)\frac{q}{p}+2]-\alpha}\varepsilon_{1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(3.4)

This lead a contradiction as $j\rightarrow\infty$. The proof of this theorem
is achieved. ∎

###### Proof of Corollary 1.2.

It follows from $1\leq\frac{2}{q}+\frac{3}{p}\leq\frac{3}{2}$ with
$\frac{2}{p}+\frac{2}{q}<1$ and $\frac{3}{p}+\frac{1}{q}<1$ that $u\in
L^{4}(0,T;L^{4}(\mathbb{R}^{n}))$. Thanks to the work [28], we observe that
$u$ is a suitable weak solution. Following the path of the above proof, we
complete the proof. ∎

###### Proof of Theorem 1.3.

As the same manner of proof of Theorem 1.1 and replacing the appliction of the
regularity criterion (2.10) by (2.11), the proof of this theorem is completed.
∎

##  4  Hausdorff dimension of possible singular points set of suitable weak
solutions

First, we prove Theorem 1.5. As an application of this theorem, we can achieve
the proof of Corollary 1.6. To this end, we prove the following lemma, which
roughly indicates that the smallness of $\nabla^{\ast}(\nabla u)^{\ast}$
yields the the smallness of $\nabla u$.

######  Lemma 4.1.

For $0<\mu\leq\frac{1}{2}\rho$, there is an absolute constant $C$ independent
of $\mu$ and $\rho$, such that

$E_{\ast}(\nabla
u;\mu)\leq(\frac{\rho}{\mu})E^{\ast}_{\ast}(\nabla^{\ast}(\nabla
u)^{\ast};\rho)+(\frac{\mu}{\rho})^{2}E_{\ast}(\nabla u;\rho).$

###### Proof.

With the help of the triangle inequality, the Hölder inequality and (2.6), we
see that

$\displaystyle\int_{B(\mu)}|u|^{2}dx$ $\displaystyle\leq
C\int_{B(\mu)}|u-\bar{u}_{{\rho}}|^{2}+C\int_{B(\mu)}|\bar{u}_{{\rho}}|^{2}$
(4.1) $\displaystyle\leq
C\Big{(}\int_{B(\frac{\rho}{2})}|u-\bar{u}_{{\rho}}|^{2}\Big{)}+C\frac{\mu^{3}}{\rho^{3}}\Big{(}\int_{B(\rho)}|u|^{2}\Big{)}$
$\displaystyle\leq
C\rho^{2s}\Big{(}\int_{B^{\ast}(\rho)}y^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1-2s}}|\nabla^{\ast}u^{\ast}|^{2}dxdy\Big{)}+C\frac{\mu^{3}}{\rho^{3}}\Big{(}\int_{B(\rho)}|u|^{2}\Big{)},$

that is,

$\int_{B(\mu)}|\nabla u|^{2}dx\leq
C\rho^{2s}\Big{(}\int_{B^{\ast}(\rho)}y^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1-2s}}|\nabla^{\ast}(\nabla
u)^{\ast}|^{2}dxdy\Big{)}+C\frac{\mu^{3}}{\rho^{3}}\Big{(}\int_{B(\rho)}|\nabla
u|^{2}\Big{)}.$

Integrating in time on
$({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\mu^{2}},\,0)$
this inequality, we obtain

$\iint_{Q(\mu)}|\nabla u|^{2}dx\leq
C\rho^{2s}\Big{(}\iint_{Q^{\ast}(\rho)}y^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1-2s}}|\nabla^{\ast}(\nabla
u)^{\ast}|^{2}dxdy\Big{)}+C\frac{\mu^{3}}{\rho^{3}}\Big{(}\iint_{Q(\rho)}|\nabla
u|^{2}\Big{)}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},}$

which leads to

$E_{\ast}(\nabla
u;\mu)\leq(\frac{\rho}{\mu})E^{\ast}_{\ast}(\nabla^{\ast}(\nabla
u)^{\ast};\rho)+(\frac{\mu}{\rho})^{2}E_{\ast}(\nabla u;\rho).$

This achieves the proof of this lemma. ∎

###### Proof of Theorem 1.5.

From Lemma 4.1 and the iteration method as in [4], we know that

$\limsup_{\mu\rightarrow 0}E_{\ast}(\nabla u;\mu)\leq C\limsup_{\mu\rightarrow
0}E^{\ast}_{\ast}(\nabla^{\ast}(\nabla u)^{\ast};\mu).$

The famous $\epsilon$-regularity criterion (1.2) helps us to finish the proof.
∎

###### Proof of Theorem 1.4.

For the case $s=0$, we complete the proof by the Caffarelli-Kohn-Nirenberg
theorem in [4]. For the other borderline case $s=1/2$, by the fact
$\dot{H}^{\frac{3}{2}}(\mathbb{R}^{3})\hookrightarrow BMO$ and the Serrin
class $L^{2}(0,T;BMO)$ due to Kozono and Taniuchi [12], we know there is no
singular point in the weak solutions of the Navier-Stokes equations. Hence, we
achieve the proof of two borderline cases. For the rest cases, from (2.4), we
derive from $u\in L^{2}(0,T;\dot{H}^{s+1}(\mathbb{R}^{3}))$ with
$0<s<\frac{1}{2}$ that

$\int\int_{\mathbb{R}^{4}_{+}}y^{1-2s}|\nabla^{\ast}(\nabla
u)^{\ast}|^{2}(x,y,t)\,dx\,dydt<+\infty.$

At this stage, the Vitali covering lemma used in [4] together with Theorem 1.5
yields that $1-2s$ dimension of potential singular points set of suitable weak
solutions satisfying $u\in L^{2}(0,T;\dot{H}^{s+1}(\mathbb{R}^{3}))$ for
$0<s<\frac{1}{2}$ is zero. The process is standard, hence, we omit the detail
here. In summary, the desired result is derived. ∎

The proof of Corollary 1.6 is a consequence of the following two lemmas.

######  Lemma 4.2.

Let $\nabla u\in L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for
$2\leq\frac{2}{q}+\frac{3}{p}\leq\frac{5}{2}$ with
$\frac{5}{2}-\frac{3}{p}-\frac{5}{2q}\geq
0,2<p<\frac{54+12\sqrt{14}}{25},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1<q\leq
2}$. Then

$u\in L^{\infty}(0,T;L^{2}(\mathbb{R}^{3}))\cap
L^{2}(0,T;\dot{H}^{1+s}(\mathbb{R}^{3})),$

where
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0\leq}s=\frac{\frac{5}{2}-\frac{3}{p}-\frac{5}{2q}}{1-\frac{1}{q}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\leq\frac{1}{2}}$.

###### Proof.

The incompressible condition allow us to get

$\langle
u\cdot\nabla{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Lambda^{s}u},\Lambda^{s}u\rangle=0.$
(4.2)

Multiplying the Navier-Stokes equations with $\Lambda^{2s}u$, using the
divergence free condition and (4.2), we know that

$\frac{1}{2}\frac{d}{dt}\|\Lambda^{s}u\|^{2}_{L^{2}(\mathbb{R}^{3})}+\|\Lambda^{s+1}u\|^{2}_{L^{2}(\mathbb{R}^{3})}=-\langle\Lambda^{s}(u\cdot\nabla
u)-u\cdot\nabla(\Lambda^{s}u),\Lambda^{s}u\rangle.$

The Hölder inequality guarantees that

$|\langle\Lambda^{s}(u\cdot\nabla
u)-u\cdot\nabla(\Lambda^{s}u),\Lambda^{s}u\rangle|\leq\|\Lambda^{s}(u\cdot\nabla
u)-u\cdot\nabla(\Lambda^{s}u)\|_{L^{2}(\mathbb{R}^{3})}\|\Lambda^{s}u\|_{L^{2}(\mathbb{R}^{3})}.$

By means of Kato-Ponce commutator inequality (2.12), we infer that

$\|\Lambda^{s}(u\cdot\nabla
u)-u\cdot\nabla(\Lambda^{s}u)\|_{L^{2}(\mathbb{R}^{3})}\leq C\|\nabla
u\|_{L^{p}(\mathbb{R}^{3})}\|\Lambda^{s}u\|_{L^{\frac{2p}{p-2}}(\mathbb{R}^{3})},{p>2}.$

Consequently, we arrive at

$\frac{1}{2}\frac{d}{dt}\|\Lambda^{s}u\|^{2}_{L^{2}(\mathbb{R}^{3})}+\|\Lambda^{s+1}u\|^{2}_{L^{2}(\mathbb{R}^{3})}\leq
C\|\nabla
u\|_{L^{p}(\mathbb{R}^{3})}\|\Lambda^{s}u\|_{L^{\frac{2p}{p-2}}(\mathbb{R}^{3})}\|\Lambda^{s}u\|_{L^{2}(\mathbb{R}^{3})}.$
(4.3)

We conclude by the fractional Gagliardo-Nirenberg inequality (see e.g. [32,
33] and references therein) and the Sobolev inequality that,

$\displaystyle\|\Lambda^{s}u\|_{L^{\frac{2p}{p-2}}(\mathbb{R}^{3})}\leq
C\|\nabla
u\|^{\frac{\frac{3}{p}}{\frac{5}{2}-s-\frac{3}{p}}}_{L^{p}(\mathbb{R}^{3})}\|u\|^{\frac{\frac{5}{2}-s-\frac{6}{p}}{\frac{5}{2}-s-\frac{3}{p}}}_{L^{\frac{3}{\frac{3}{2}-s}}(\mathbb{R}^{3})}\leq
C\|\nabla
u\|^{\frac{\frac{3}{p}}{\frac{5}{2}-s-\frac{3}{p}}}_{L^{p}(\mathbb{R}^{3})}\|\Lambda^{s}u\|^{\frac{\frac{5}{2}-s-\frac{6}{p}}{\frac{5}{2}-s-\frac{3}{p}}}_{L^{2}(\mathbb{R}^{3})},$
(4.4)

where we require

$0\leq\frac{\frac{3}{p}}{\frac{5}{2}-s-\frac{3}{p}}\leq
1,\frac{5}{2}-s-\frac{3}{p}>0\quad\text{and}\quad
s\leq\frac{\frac{3}{p}}{\frac{5}{2}-s-\frac{3}{p}}.$

Indeed, in the light the definition of $s$ and $1<q\leq 2$, we observe that
$\frac{\frac{3}{p}}{\frac{5}{2}-s-\frac{3}{p}}\leq 1$. In addition, taking
advantage of the the definition of $s$ again, we know that
$\frac{5}{2}-s-\frac{3}{p}>0$. Some straightforward computations yields that
$\frac{9-\sqrt{56}}{6}<\frac{1}{p}<\frac{9+\sqrt{56}}{6}$ guarantees that
$s\leq\frac{\frac{3}{p}}{\frac{5}{2}-s-\frac{3}{p}}$.

Inserting (4.4) into (4.3), we have

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\Lambda^{s}u\|^{2}_{L^{2}(\mathbb{R}^{3})}+\|\Lambda^{s+1}u\|^{2}_{L^{2}(\mathbb{R}^{3})}\leq$
$\displaystyle C\|\nabla
u\|^{\frac{\frac{3}{p}}{\frac{5}{2}-s-\frac{3}{p}}+1}_{L^{p}(\mathbb{R}^{3})}\|\Lambda^{s}u\|^{\frac{\frac{5}{2}-s-\frac{6}{p}}{\frac{5}{2}-s-\frac{3}{p}}+1}_{L^{2}(\mathbb{R}^{3})}$
(4.5) $\displaystyle\leq$ $\displaystyle C\|\nabla
u\|^{q}_{L^{p}(\mathbb{R}^{3})}\|\Lambda^{s}u\|^{\frac{\frac{5}{2}-s-\frac{6}{p}}{\frac{5}{2}-s-\frac{3}{p}}+1}_{L^{2}(\mathbb{R}^{3})}.$

Thanks to $\frac{\frac{5}{2}-s-\frac{6}{p}}{\frac{5}{2}-s-\frac{3}{p}}\leq 1$,
we derive from (4.5) and $\nabla u\in L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for
$2\leq\frac{2}{q}+\frac{3}{p}\leq\frac{5}{2}$ that $u\in
L^{2}(0,T;H^{1+s}(\mathbb{R}^{3}))$. ∎

######  Lemma 4.3.

Let $u$ be a suitable weak solution belonging in $\nabla u\in
L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for
$2\leq\frac{2}{q}+\frac{3}{p}\leq\frac{5}{2}$ with
$2-\frac{3}{p}-\frac{1}{q}\geq 0,\frac{3}{2}<p<\frac{12}{7},q\geq 4$. Then

$u\in L^{\infty}(0,T;L^{2}(\mathbb{R}^{3}))\cap
L^{2}(0,T;\dot{H}^{1+s}(\mathbb{R}^{3})),$

where
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0\leq}s=\frac{2-\frac{3}{p}-\frac{1}{q}}{\frac{2}{q}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\leq\frac{1}{2}}$.

###### Proof.

In view of the standard energy estimate, the integration by parts and the
incompressible condition, we have

$\frac{1}{2}\frac{d}{dt}\|\Lambda^{s}u\|^{2}_{L^{2}(\mathbb{R}^{3})}+\|\Lambda^{s+1}u\|^{2}_{L^{2}(\mathbb{R}^{3})}=\langle\Lambda^{s}(u\otimes
u),\Lambda^{s+1}u\rangle.$

It follows from the Hölder inequality that

$|\langle\Lambda^{s}(u\otimes
u),\Lambda^{s+1}u\rangle|\leq\|\Lambda^{s}(u\otimes
u)\|_{L^{2}(\mathbb{R}^{3})}\|\Lambda^{s+1}u\|_{L^{2}(\mathbb{R}^{3})}.$

We deduce from the Kato-Ponce product estimates (2.13) and the Sobolev
embedding that

$\|\Lambda^{s}(u\otimes u)\|_{L^{2}(\mathbb{R}^{3})}\leq
C\|\Lambda^{s}u\|_{L^{\frac{6p}{5p-6}}(\mathbb{R}^{3})}\|u\|_{L^{\frac{3p}{3-p}}(\mathbb{R}^{3})}\leq
C\|\Lambda^{s}u\|_{L^{\frac{6p}{5p-6}}(\mathbb{R}^{3})}\|\nabla
u\|_{L^{p}(\mathbb{R}^{3})},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\;\frac{6}{5}<p<3.}$

Combining the above estimates together, we observe that

$\frac{1}{2}\frac{d}{dt}\|\Lambda^{s}u\|^{2}_{L^{2}(\mathbb{R}^{3})}+\|\Lambda^{s+1}u\|^{2}_{L^{2}(\mathbb{R}^{3})}\leq
C\|\Lambda^{s}u\|_{L^{\frac{6p}{5p-6}}(\mathbb{R}^{3})}\|\nabla
u\|_{L^{p}(\mathbb{R}^{3})}\|\Lambda^{s+1}u\|_{L^{2}(\mathbb{R}^{3})}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(4.6)

According to the fractional Gagliardo-Nirenberg inequality and the Sobolev
inequality, we discover that

$\|\Lambda^{s}u\|_{L^{\frac{6p}{5p-6}}(\mathbb{R}^{3})}\leq
C\|u\|_{L^{\frac{3p}{3-p}}(\mathbb{R}^{3})}^{\frac{2-\frac{3}{p}}{s-\frac{3}{2}+\frac{3}{p}}}\|\Lambda^{s+1}u\|_{L^{2}(\mathbb{R}^{3})}^{\frac{s-\frac{7}{2}+\frac{6}{p}}{s-\frac{3}{2}+\frac{3}{p}}}\leq
C\|\nabla
u\|_{L^{p}(\mathbb{R}^{3})}^{\frac{2-\frac{3}{p}}{s-\frac{3}{2}+\frac{3}{p}}}\|\Lambda^{s+1}u\|_{L^{2}(\mathbb{R}^{3})}^{\frac{s-\frac{7}{2}+\frac{6}{p}}{s-\frac{3}{2}+\frac{3}{p}}},$
(4.7)

where we need
$p\geq\frac{3}{2},0\leq\frac{s-\frac{7}{2}+\frac{6}{p}}{s-\frac{3}{2}+\frac{3}{p}}\leq
1$, $s-\frac{3}{2}+\frac{3}{p}>0$ and
$\frac{s}{s+1}<\frac{2-\frac{3}{p}}{s-\frac{3}{2}+\frac{3}{p}}$.
On one hand, we can examine
$\frac{2-\frac{3}{p}}{s-\frac{3}{2}+\frac{3}{p}}\leq 1$ via $q\geq 4$ and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}3>p\geq\frac{3}{2}}$.
On the other hand, direct calculation ensures that $q>2,p>\frac{3}{2}$ yields
that $s-\frac{3}{2}+\frac{3}{p}>0$. Moreover, $p<\frac{12}{7}$ means
$\frac{s}{s+1}<\frac{2-\frac{3}{p}}{s-\frac{3}{2}+\frac{3}{p}}.$

Inserting (4.7) into (4.6), we find

$\frac{1}{2}\frac{d}{dt}\|\Lambda^{s}u\|^{2}_{L^{2}(\mathbb{R}^{3})}+\|\Lambda^{s+1}u\|^{2}_{L^{2}(\mathbb{R}^{3})}\leq
C\|\nabla
u\|^{\frac{s+\frac{1}{2}}{s-\frac{3}{2}+\frac{3}{p}}}_{L^{p}(\mathbb{R}^{3})}\|\Lambda^{s+1}u\|^{\frac{s-\frac{7}{2}+\frac{6}{p}}{s-\frac{3}{2}+\frac{3}{p}}+1}_{L^{2}(\mathbb{R}^{3})},$

which implies that

$\|\Lambda^{s}u\|^{2}_{L^{\infty}(0,T;L^{2}(\mathbb{R}^{3}))}+\|\Lambda^{s+1}u\|^{2}_{L^{2}(0,T;L^{2}(\mathbb{R}^{3}))}\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{0}+}C\|\nabla
u\|^{\frac{s+\frac{1}{2}}{s-\frac{3}{2}+\frac{3}{p}}}_{L^{\frac{2s+1}{2-\frac{3}{p}}}(0,T;L^{p}(\mathbb{R}^{3}))}\|\Lambda^{s+1}u\|^{\frac{s-\frac{7}{2}+\frac{6}{p}}{s-\frac{3}{2}+\frac{3}{p}}+1}_{L^{2}(0,T;L^{2}(\mathbb{R}^{3}))}.$

The proof of this lemma is completed. ∎

###### Proof of Corollary 1.6.

Combining Lemma 4.2, Lemma 4.3 and Theorem 1.4, we immediately finish the
proof of Corollary 1.6. ∎

## Acknowledgement

The authors would like to express their sincere gratitude to Dr. Wei Wei at
Northwest University, for the discussion involving the inequality (4.4) and
(4.7). The research of Wang was partially supported by the National Natural
Science Foundation of China (No. 11971446 and No. 11601492). The research of
Wu was partially supported by the National Natural Science Foundation of China
under grant No. 11771423.

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