1. Introduction

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

(1.1)\begin{equation} \left\{\begin{array}{@{}ll@{}} 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{array}\right. \end{equation}

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 the 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 [Reference Scheffer23–Reference Scheffer25] is well-known. The famous Caffarelli–Kohn–Nirenberg theorem in [Reference Caffarelli, Kohn and Nirenberg4] is that the one-dimensional parabolic 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

(1.2)\begin{equation} \limsup_{\varrho\rightarrow0}\frac{1}{\varrho}\iint_{Q(\varrho)}|\nabla u|^{2}\,{\rm d}x\,{\rm d}t\leq \epsilon, \end{equation}

then $u$ is bounded in a neighbourhood of $(0,0)$, where $Q(\varrho ):=B(\varrho )\times (-\varrho ^{2},0)$ and $B(\varrho )$ denotes the ball of centre $0$ and radius $\varrho$. To this end, Caffarelli–Kohn–Nirenberg [Reference Caffarelli, Kohn and Nirenberg4] established an $\epsilon$-regularity criterion at one scale

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

An alternative approach of Caffarelli–Kohn–Nirenberg theorem based on blow-up argument was due to Lin, Ladyzhenskaya and Seregin [Reference Ladyzenskaja and Seregin17, Reference Lin18], where the corresponding $\epsilon$-regularity criterion at one scale reads

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

In what follows, a point $z=(x,t)$ in (1.1) is said to be regular if $u$ belongs to $L^{\infty }$ at a neighbourhood of $z$. Otherwise, it is called singular. Estimates of the size of potential singular points set in the 3D Navier–Stokes equations can be found in [Reference Barker and Wang1, Reference Gustafson, Kang and Tsai9, Reference Kukavica14, Reference Kukavica and Pei15, Reference Robinson and Sadowski20–Reference Robinson and Sadowski22, Reference Wang and Zhang30, Reference Wang and Wu31].

On the other hand, the integral (Serrin) type conditions based on the velocity, the gradient of the velocity or the pressure lead to 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 [Reference Serrin27], Struwe [Reference Struwe26], Escauriaza, Seregin and Šverák [Reference Escauriaza, Seregin and Šverák7]

   (1.5)\begin{equation} u\in L^{p} (0,T;L^{q}( \mathbb{R}^{3}))\ \text{with}\ 2/p+3/q=1, \quad q\geq3. \end{equation}

2. (2) Beirao da Veiga [Reference Beirao da Veiga2]

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

3. (3) Berselli and Galdi [Reference Berselli and Galdi3], Zhou [Reference Zhou36, Reference Zhou37]

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

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 $u \in L^{q}(0,\,T;\,L^{p}(\mathbb {R}^{3}))$ with $\frac 2q+\frac 3p=\frac 32$ in suitable weak solutions means that

(1.8)\begin{equation} \dim_{H}(\mathcal{S})\leq 1\ \text{and}\ \dim_{B}({\mathcal{S}})\leq5/3, \end{equation}

which can be found in [Reference Caffarelli, Kohn and Nirenberg4, Reference Robinson and Sadowski20] and $\dim _{H}(S)$ and $\dim _{B}(S)$ denote the Hausdorff dimension and box dimension of a set $S$, respectively. A natural question is whether the suitable weak solutions satisfying supercritical regularity $u \in L^{q}(0,\, T;\,L^{p}(\mathbb {R}^{3}))$ with $1<\frac 2q+\frac 3p<\frac 32$ lower the fractal dimension in (1.8). In this direction, Gustafson, Kang and Tsai [Reference Gustafson, Kang and Tsai9] proved that the Hausdorff dimension of the potential singular points set $\mathcal {S}$ of a Leray–Hopf weak solution belonging to $u\in L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $1\leq \frac {2}{q}+\frac {3}{p}$ with $\frac 2p+\frac 2q<1$ and $\frac 3p+\frac 1q<1$ is at most $3-q+\frac {2q}{p}, p>q$ or $2-q+\frac {3q}{p}, p\leq q$. Robinson and Sadowski [Reference Robinson and Sadowski22] showed that the upper box dimension of potential singular points set $\mathcal {S}$ of a suitable weak solution belonging to $u\in L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $1\leq \frac {2}{q}+\frac { 3}{p} \leq \frac 32$ with $3< p,q<\infty$ is no greater than

(1.9)\begin{equation} \max\{p,q\}\left(\frac{2}{q}+\frac{ 3}{p}-1\right). \end{equation}

In addition, the Hausdorff dimension of potential singular points set $\mathcal {S}$ of a suitable weak solution belonging to $\nabla u\in L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $2\leq \frac {2}{q}+\frac {3}{p} \leq \frac 52$ with $2< p\leq q<\infty$ is less than or equal to

(1.10)\begin{equation} \max\{p,q\}\left(\frac{2}{q}+\frac{ 3}{p}-2\right). \end{equation}

In [Reference Robinson and Sadowski22, 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.

As observed in [Reference Gustafson, Kang and Tsai9], the weak solutions in spaces $L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ with $\frac 2p+\frac 2q<1$ and $\frac 3p+\frac 1q<1$ are suitable weak solutions. Therefore, towards the Robinson and Sadowski's fourth question (4), we have

With a slight modification of the proof of theorem 1.1 and using the $\epsilon$-regularity criterion at one scale without pressure in [Reference Wang, Wu and Zhou33], we can obtain a parallel result of (1.9) in a bounded domain, which is corresponding to Robinson and Sadowski's third issue.

Figure 1. Robinson–Sadowski results on $\mathbb {R}^{3}$.

Figure 2. Theorem 1.1 on $\mathbb {R}^{3}$.

Figure 3. Corollary 1.3 weak solutions.

Figure 4. Theorem 1.4 on bounded domain.

Roughly, the following figures (Figures 1–4) 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.

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, Barker and Wang [Reference Barker and Wang1] 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 the higher integrability 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 $|\nabla u|^{2}|u|^{q-2}$ with $2< q<3$, which usually arises in the $L^{p}$ type energy estimates of the Navier–Stokes equations. In the spirit of [Reference Barker and Wang1], 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 [Reference Colombo, De Lellis and Massaccesi6, Reference Ren, Wang and Wu19, Reference Tang and Yu29] to overcome non-local derivatives. However, since $s>0$, one requires higher-order Caffarelli–Silvestre (Yang) extensions [Reference Yang35]. To this end, we observe that the following identity due to [Reference Colombo, De Lellis and Massaccesi6], for $\alpha =s+1>1$,

\begin{align*} c_\alpha\int_{\mathbb{R}^4_+} y^{3-2\alpha} |\nabla^{{\ast}} (\nabla u)^{{\ast}}|^2 (x,y,t)\,{\rm d}x\,{\rm d}y & = \int_{\mathbb{R}^3} \left|(-\Delta)^{\frac{\alpha-1}{2}} \nabla u\right|^2 (x,t)\,{\rm d}x\\ & = \int_{\mathbb{R}^3} |(-\Delta)^\frac{\alpha}{2} u|^2 (x,t)\,{\rm d}x, \end{align*}

that is,

(1.11)\begin{equation} \|u\|^{2} _{\dot{H}^{s+1}}= c_s\int_{\mathbb{R}^4_+} y^{1-2s} |\nabla^{{\ast}} (\nabla u)^{{\ast}}|^2 (x,y,t)\,{\rm d}x\,{\rm d}y, \end{equation}

which helps us to reduce the proof of theorem 1.5 to show theorem 1.6 just by Caffarelli–Silvestre extension rather than higher-order (Yang) extension. Theorem 1.5 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.

Proposition 1.6 Suppose that $u$ is a suitable weak solution to (1.1). Then there exists an absolute positive constant $\epsilon _{01}$ such that $(0,0)$ is a regular point if

(1.12)\begin{equation} \limsup\limits_{\mu\rightarrow0}\frac{1}{\mu^{1-2s}}\iint_{Q^{{\ast}}(\mu)}y^{1-2s}|\nabla^{{\ast}} (\nabla u)^{{\ast}}|^2\,{\rm d}x\,{\rm d}y\,{\rm d}t \leq\epsilon_{01}. \end{equation}

Figure 5. First part of corollary 1.7.

Figure 6. Second part of corollary 1.7.

Figure 7. Known Hausdorff dimension of the gradient of the velocity.

As an application of theorem 1.5 and the energy estimate of the Navier–Stokes equations, we can partially answer the Robinson and Sadowski's second question.

Currently, the Hausdorff dimension of suitable weak solutions with the gradient of the velocity under supercritical regularity are summarized in the following figures (Figures 5–7).

The remainder of this paper is organized as follows. In §2, we 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 of theorem 1.1 concerning Box dimension. Partial regularity results involving Hausdorff dimension is proved in §4.