1 Introduction

In this paper, we consider the Cauchy problem for the incompressible Navier–Stokes equations

$$\begin{aligned} (\partial _t - \varDelta ) u+ (u\cdot \nabla ) u+ \nabla p= & {} 0 \nonumber \\ {{\,\mathrm{div}\,}}u= & {} 0 \end{aligned}$$
(1)

in \(Q_T := \mathbb {R}^3\times (0,T)\) and \(T>0\) with a smooth and rapidly decaying solenoidal initial vector field \(u(x,0)=u_0(x)\) in \(\mathbb {R}^3\). The state variables u and p denote the velocity field of the fluid and its pressure. Leray [1] proved that the Cauchy problem has a unique smooth solution for a short time. He also proved that there exists at least one global weak solution satisfying an energy inequality. Hopf [2] extended the result in the case of bounded domains with a modern concept of weakly differentiable functions. The weak solution u lies in the space

$$\begin{aligned} V^2_\sigma (Q_T) := L^{ \infty }(0,T; L_\sigma ^2(\mathbb {R}^3)) \cap L^2(0,T;W_\sigma ^{1, 2}(\mathbb {R}^3)), \end{aligned}$$
(2)

but uniqueness and regularity of the weak solution are still open problems. The exact concept of weak solutions and notations will be given in the next section.

Since there are plenty of important contributions for the regularity question of the Navier–Stokes equations, we briefly describe a few of them closely related to our results. To guarantee the regularity of weak solutions, one of the most important conditions is the so-called Ladyzhenskaya–Prodi–Serrin [3,4,5] condition, that is,

$$\begin{aligned} u \in L^l(0,T;L^s(\mathbb {R}^3)) \end{aligned}$$
(3)

for some s and l satisfying

$$\begin{aligned} \frac{3}{s} + \frac{2}{l} = 1, \quad 3 < s \le \infty . \end{aligned}$$

Under this condition, the weak solution u to the Cauchy problem (1) is unique and smooth. Later, Escauriaza–Sergin–Šverák [6] proved that the regularity of a weak solution can also be assured by the marginal case,

$$\begin{aligned} u \in L^{\infty }(0,T;L^3(\mathbb {R}^3)). \end{aligned}$$
(4)

However, we do not know yet that kinds of higher integrability hold for weak solutions. By standard embeddings of the solution space (2), any weak solution satisfy the mixed integrability condition with the range of integrability exponents

$$\begin{aligned} \frac{3}{s} + \frac{2}{l} = \frac{3}{2}, \quad 2 \le s \le 6. \end{aligned}$$

There is a considerable gap compared with the Ladyzhenskaya–Prodi–Serrin condition.

To guarantee the local regularity of weak solutions, there are other conditions the so called \(\varepsilon \) regularity conditions. For the Cauchy problem of the Navier–Stokes equations, there is a natural scaling structure

$$\begin{aligned} u(x,t)&\rightarrow \lambda u(\lambda x, \lambda ^2 t), \\ p(x,t)&\rightarrow \lambda ^2 p(\lambda x, \lambda ^2 t). \end{aligned}$$

Many of the local regularity results have been established under the various smallness assumptions on some scaling invariant quantities. We denote by \(\varSigma \) the set of possible singular points for the weak solution u. Utilizing regularity criteria, one can estimate the size of \(\varSigma \) by means of some fractal measures and extract some geometric information of \(\varSigma \). In this direction, Scheffer [7, 8] introduced the concept of suitable weak solutions for the Navier–Stokes equations and then gave partial regularity results. Caffarelli–Kohn–Nirenberg [9] further strengthened Scheffer’s results and gave an improved bound for the Hausdorff dimension of \(\varSigma \). Lin [10] presented a greatly simplified proof. Ladyzhenskaya–Seregin [11] gave more details and considered the case that external forces lie in some Morrey spaces. Choe–Lewis [12] presented an improved estimate of \(\varSigma \) in terms of general Hausdorff measures. Gustafson–Kang–Tsai [13] unified several known regularity criteria. For the case (4), Neustupa [14] investigated the structure of \(\varSigma \) and then Escauriaza–Sergin–Šverák [6] resolved the regularity qestion.

In this paper, we shall present a new regularity criterion for weak solutions to the Cauchy problem (1) satisfying the condition

$$\begin{aligned} u \in L^{\infty }(0,T;L^{3,w}(\mathbb {R}^{3})) \end{aligned}$$
(5)

where \(L^{3,w}(\mathbb {R}^{3})\) denotes the weak Lebesgue space. Because the condition (5) is significantly weaker than the condition (4) encompassing type I singularity, the regularity question under that condition draws many mathematicians’ attention. However, in the authors knowledge, all results were established under the smallness assumption on that scale (5). See, for example, [15,16,17] and the references therein .

We shall use the following notation.

Notation 1

We denote the space ball of radius r and center x by \(B(x,r) := \{y \in \mathbb {R}^3 : |y-x|<r\}\) and the space-time cylinder at \(z = (x,t)\) by

$$\begin{aligned} Q(z,r) := B(x,r) \times (t-r^2, t). \end{aligned}$$

If the center is at the origin, we simply put \(B_r = B(0,r)\) and \(Q_r = Q(0,r)\).

The following theorem is our new regularity criterion.

Theorem 1

For each \(M>0\) there exists a positive number \(\varepsilon (M) < 1/4\) such that if a weak solution \(u\in V_\sigma ^2(Q_T)\) to the Cauchy problem (1) satisfies the conditions

$$\begin{aligned} \mathop {\hbox {ess}\,\hbox {sup}}\limits _{0 \le t \le T} \Vert u\Vert _{L^{3,w}(\mathbb {R}^{3})} \le M \end{aligned}$$
(6)

and for some \(z_0 = (x_0, t_0) \in Q_T\) and \(0 < r \le \sqrt{t_0}\)

$$\begin{aligned} \frac{1}{r^3} {m} \left\{ x\in B(x_0,r) : |u(x, t_0)| > \frac{\varepsilon }{r}\right\} \le \varepsilon , \end{aligned}$$
(7)

where m(E) denotes the Lebesgue measure of the set E, then u is bounded in the space-time cylinder \(Q(z_0,\varepsilon r)\).

As an application of this criterion, we are able to estimate the size of possible singular points at a singular time t, denoted by

$$\begin{aligned} \varSigma (t) = \{x : (x,t) \in \varSigma \}. \end{aligned}$$

We know that the Hausdorff dimension of the possible singular time is at most 1 / 2. Many researchers have been investigating the size of \(\varSigma (t)\) at the singular time t under various conditions on u. Seregin [18] obtained a result on estimating \(\varSigma (t)\) under the slightly weaker condition than (4). A unifying results on the number of singular points under the Ladyzhenskaya–Prodi–Serrin type conditions can be found in [17].

Utilizing Theorem 1, we can obtain the following theorem which shows that the number of possible singular points at any singular time t is at most finite.

Theorem 2

Suppose \(u\in V_\sigma ^2(Q_T)\) is a weak solution to the Cauchy problem (1) and satisfies the condition (6) for some \(M>0\). Then there exist at most finite number N(M) of singular points at any singular time t.

At each singular time t, only a few singular points exist, yet we do not know that blowup points are of type I or not.

2 Preliminaries

Throughout the paper, we shall use the following notation.

Notation 2

We denote \(A \lesssim B\) if there exists a generic positive constant C such that \(|A| \le C|B|\). We denote the average value of f over the set E by

$$\begin{aligned} \langle f\rangle _E := \fint _E f := \frac{1}{{m}(E)} \int _E f \end{aligned}$$
(8)

where m(E) denotes the Lebesgue measure of the set E. We shall use the same notation m for the space sets in \(\mathbb {R}^3\) and the space-time sets \(\mathbb {R}^3 \times (0,T)\) and it will be clearly understood in the contexts.

We now recall the definition of the weak Lebesgue spaces. For a measurable function f on \(\mathbb {R}^3\), its level set with the height h is denoted by

$$\begin{aligned} E(h) = \{x \in \mathbb {R}^3:|f(x)|>h\}. \end{aligned}$$
(9)

The Lebesgue integral can be expressed by the Riemann integral of such level sets. In particular, for \(0<q<\infty \)

$$\begin{aligned} \int |f(x)|^q dx = \int _{0}^{\infty }qh^{q-1} {m}(E(h)) dh. \end{aligned}$$
(10)

Definition 1

The weak Lebesgue space \(L^{q,w}(\mathbb {R}^3)\) is the set of all measurable function such that the quantity

$$\begin{aligned} \Vert f\Vert _{q,w} := \sup _{h>0} \big [h {m}(E(h))^{1/q}\big ] \end{aligned}$$
(11)

is finite.

As the usual convention, two functions are considered the same if they are equal almost everywhere. In fact, \(\Vert f\Vert _{p,w}\) is not a true norm since the triangle inequality fails. But, it is easy to see that for any \(0<r<q\) the following expression

$$\begin{aligned} \Vert f\Vert := \sup _{0<{m}(E)<\infty } {m}(E)^{1/q} \left( \fint _E |f|^r\right) ^{1/r} \end{aligned}$$
(12)

is comparable to \(\Vert f\Vert _{q,w}\), (see [19] for example). Moreover, \(\Vert f\Vert \) satisfies the triangle inequality if \(1 \le r < q\) and hence it plays the role of true norm for \(q>1\). Furthermore, the weak Lebesgue spaces are Banach spaces and coincide with the Lorentz (Marcinkiewicz) spaces \(L^{q,\infty }\).

Remark 3

Using (12) one can easily see that

$$\begin{aligned} L^{q,w}(\mathbb {R}^3) \subset \bigcap _{1 \le r < q} \mathscr {L}_{{{\,\mathrm{loc}\,}}}^{r,3-3r/q}(\mathbb {R}^3). \end{aligned}$$

where \(\mathscr {L}_{{{\,\mathrm{loc}\,}}}^{r,\lambda }(\mathbb {R}^3)\) denotes the local Morrey space.

The next remark shows that there is no nonzero harmonic function in \(L^{3,w}(\mathbb {R}^{3})\). This fact will be used in the proof of our main theorem.

Remark 4

It is easy to see that if \(f \in L^{3,w}(\mathbb {R}^{3})\) is harmonic, then \(f=0\). Indeed, using the mean value property, we have for all \(x_0 \in \mathbb {R}^3\) and \(R>0\)

$$\begin{aligned} |\nabla f(x_0)|&\lesssim \frac{1}{R^4} \int _{B(x_0,R)} |f(x)| dx \\&\lesssim \frac{1}{R^4} \int _{0}^{\infty }{m}(B(x_0,R) \cap E(h)) dh \\&\lesssim \frac{1}{R^4} \left( \int _0^H R^3 dh +\int _H^{\infty } h^{-3} dh\right) \end{aligned}$$

for all \(H>0\). Taking \(H=R^{-1}\) we get \(|\nabla f(x_0)| \lesssim R^{-2}\) for all \(x_0 \in \mathbb {R}^3\) and \(R>0\). Letting \(R \rightarrow \infty \), we conclude that \(\nabla f=0\) and hence f is constant. Since \(f \in L^{3,w}(\mathbb {R}^{3})\), it should be identically zero.

We denote by \(L^q(\mathbb {R}^3)\) and \(W^{k,q}(\mathbb {R}^3)\) the standard Lebesgue and Sobolev spaces, and we omit these standard definitions. We denote by \(\mathscr {D}_\sigma (\mathbb {R}^3)\) the set of all solenoidal vector fields \(\phi \in C_c^{\infty }(\mathbb {R}^3)\). We define \(L_\sigma ^2(\mathbb {R}^3)\) to be the closure of \(\mathscr {D}_\sigma (\mathbb {R}^3)\) in \(L^2(\mathbb {R}^3)\) and \(W_\sigma ^{1,2}(\mathbb {R}^3)\) to be the closure of \(\mathscr {D}_\sigma (\mathbb {R}^3)\) in \(W^{1,2}(\mathbb {R}^3)\).

We now recall the concept of local pressure projection (cf. [20]). Given a bounded \( C^2\)-domain \( G \subset \mathbb {R}^{n}\), \( n\in \mathbb {R}^{n}\), we define the operator

$$\begin{aligned} E^{*}_{G}: W^{-1,\, s}(G) \rightarrow W^{-1,\, s}(G). \end{aligned}$$
(13)

Appealing to the \( L^p\)-theory of the steady Stokes system (cf. [21]), for any \( F\in W^{-1,\, s}(G)\) there exists a unique pair \( (v, p)\in W^{1,\, s}_{ 0, \sigma }(G)\times L^s_0(G)\) which solves in the weak sense the steady Stokes system

$$\begin{aligned} -\varDelta v + \nabla p&= F \quad \text { in } \quad G, \\ {{\,\mathrm{div}\,}}v&= 0 \quad \text { in } \quad G, \\ v&=0 \quad \text { on } \quad \partial G. \end{aligned}$$

Then we set \(E^{*}_G(F):= \nabla p \), where \( \nabla p \) denotes the gradient functional in \( W^{-1,\, s}(G)\) defined by

$$\begin{aligned} \langle \nabla p, \varphi \rangle = \int _{G} p \nabla \cdot \varphi dx, \quad \varphi \in W^{1,\, s'}_0(G). \end{aligned}$$
(14)

Here we have denoted by \( L^s_0(G)\) the space of all \( f\in L^s(G)\) with \( \int _{G} fdx=0 \).

Remark 5

  1. 1.

    The operator \( E^{*}_G\) is bounded from \( W^{-1,\, s}(G)\) into itself with \( E^{*}_G(\nabla p)=\nabla p\) for all \( p\in L^s_{ 0}(G)\). The norm of \( E^{*}_G \) depends only on s and the geometric properties of G, and independent on G, if G is a ball or an annulus, which is due to the scaling properties of the Stokes equation.

  2. 2.

    In case \( F\in L^s(G)\) using the canonical embedding \(L^s(G) \hookrightarrow W^{-1,\, s}(G)\) and the elliptic regularity we get \( E^{*}_G(F)= \nabla p \in L^s(G)\) together with the estimate

    $$\begin{aligned} \Vert \nabla p\Vert _{s, G} \le c \Vert F\Vert _{ s, G}, \end{aligned}$$
    (15)

    where the constant in (15) depends only on s and G. In case G is a ball or an annulus this constant depends only on s (cf. [21] for more details). Accordingly the restriction of \(E^{*}_G\) to the Lebesgue space \( L^s(G)\) defines a projection in \(L^s(G)\). This projection will be denoted still by \(E^{*}_G\).

Next, we introduce the notion of weak solutions and local suitable weak solutions.

Notation 3

We denote by dz the space-time Lebesgue measure dxdt.

Definition 2

We say that u is a Leray–Hopf weak solution to (1) if the velocity field u lies in the space \(V^2_\sigma (Q_T) = L^{ \infty }(0,T; L_\sigma ^2(\mathbb {R}^3)) \cap L^2(0,T;W_\sigma ^{1, 2}(\mathbb {R}^3))\), there exists a distribution p such that (u, p) solves the Navier–Stokes equations in the sense of distributions, and u satisfies the energy inequality for almost all \(s \in (0,T)\)

$$\begin{aligned} \int _{\mathbb {R}^3} |u(s)|^2 dx + 2 \int _0^s \int _{\mathbb {R}^3} |\nabla u|^2 dz \le \int _{\mathbb {R}^3} |u(0)|^2 dx. \end{aligned}$$

We say that u is a local suitable weak solution to (1) if for every ball \( B \subset \mathbb {R}^{3}\) the following local energy inequality the following local energy inequality holds for almost all \( s \in (0,T)\) and for all non negative \(\phi \in C^{\infty }_\mathrm{c}(B\times (0,T))\),

$$\begin{aligned}&\int |v(s)|^2 \phi (s) dx + 2 \int _{0}^{s} \int | \nabla v|^2 \phi dz \nonumber \\&\quad \le \int _{0}^{s} \int |v|^2 (\partial _t + \varDelta ) \phi dz + \int _{0}^{s} \int |v|^2 (v - \nabla p_h) \cdot \nabla \phi dz \nonumber \\&\qquad + 2 \int _{0}^{s} \int (v \otimes v - v \otimes \nabla p_h : \nabla ^2 p_h) \phi dz + 2 \int _{0}^{s}\int (p_{ 1, B}+ p_{ 2, B}) v \cdot \nabla \phi dz,\nonumber \\ \end{aligned}$$
(16)

where \( v= u+ \nabla p_{ h,B}\), and

$$\begin{aligned} \nabla p_{h,B}&= - E^{*}_{ B} (u), \\ \nabla p_{1,B}&= - E^{*}_{ B} (\nabla \cdot (u \otimes u)), \\ \nabla p_{2,B}&= E^{*}_{ B}(\varDelta u). \end{aligned}$$

Remark 6

If a weak solution u is in \(L^{\infty }(0,T;L^{3,w}(\mathbb {R}^{3}))\), then u lies in \(L^4(Q_T)\) by an interpolation. Thus, the function \(|u|^2|\nabla u|\) is integrable on \(Q_T\), which justifies the integration by parts and one can show that u becomes a local suitable weak solution, too.

Using a standard iteration method one can observe that boundedness of a certain scaling invariant quantity essentially implies the boundedness of many of other scaling invariant quantities. The following form of the Caccioppoli-type inequality is convenient in that purpose.

Lemma 7

(Lemma 2.6 in [22]) If u is a suitable weak solution to (1), then for all \(Q(z_0,r) \subset Q_T\)

$$\begin{aligned}&r^{-1} \left( \int _{Q(z_0,r/2)} |u|^{10/3} dz\right) ^{3/5} + r^{-1} \int _{Q(z_0,r/2)} |\nabla u|^2 dz \nonumber \\&\quad \lesssim \left( r^{-5} \int _{t_0-r^2}^{t_0} \left( \int _{B(x_0,r)} |u|^2 dx \right) ^3 dt\right) ^{1/3} + r^{-5} \int _{t_0-r^2}^{t_0} \left( \int _{B(x_0,r)} |u|^2 dx \right) ^3 dt\nonumber \\ \end{aligned}$$
(17)

where the implied constant is absolute.

We end this section by giving the following version of the local regularity criterion. We include its proof at the end of this paper, Appendix A.

Lemma 8

[23] There exists an absolute positive number \(\zeta \) such that if a local suitable weak solution \( u\in V^2_\sigma (Q(z_0,\rho ))\) to the Navier–Stokes equations satisfies the condition

$$\begin{aligned} \rho ^{-2} \int _{Q(z_0,\rho )} |u|^3 dz \le \zeta ^3 \end{aligned}$$
(18)

for some \(z_0 = (x_0, t_0)\in Q_T\) and \( 0 < \rho \le \sqrt{t_0}\), then

$$\begin{aligned} u \in L^{\infty }(Q(z_0,\rho /2)), \end{aligned}$$

and the following estimate holds true

$$\begin{aligned} \Vert u\Vert _{ L^{\infty }(Q(z_0,\rho /2))}\le & {} C \left( \fint _{Q(z_0,\rho )} | u|^3 dz\right) ^{1/3}\nonumber \\&+\, C \mathop {\hbox {ess}\,\hbox {sup}}\limits _{t\in (t_0-\rho ^2,t_0)} \left( \fint _{B(x_0,\rho )} | u(t)|^2 dx\right) ^{1/2} \end{aligned}$$
(19)

where C is an absolute positive constant.

3 Proof of Theorem 1

Due to Lemma 8, it suffices to show that the following lemma holds true.

Lemma 9

For each \(M>0\) there exists a positive number \(\varepsilon (M) < 1/4\) such that if a weak solution \(u\in V_\sigma ^2(Q_T)\) to the Navier–Stokes equations satisfies the condition

$$\begin{aligned} \mathop {\hbox {ess}\,\hbox {sup}}\limits _{0 \le t \le T} \Vert u\Vert _{L^{3,w}(\mathbb {R}^{3})} \le M \end{aligned}$$
(20)

and for some \(z_0 = (x_0, t_0) \in Q_T\) and \(0 < r \le \sqrt{t_0}\)

$$\begin{aligned} r^{-3} {m} \{x\in B_r(x_0) : |u(x, t_0)| > r^{-1} \varepsilon \} \le \varepsilon , \end{aligned}$$
(21)

then there exists \(\rho \in [2\varepsilon r, \sqrt{t_0}]\) such that

$$\begin{aligned} \rho ^{-2} \int _{Q(z_0,\rho )} |u|^3 dz \le \zeta ^3 \end{aligned}$$
(22)

where \(\zeta \) is the same number in Lemma 8.

We divide the proof of Lemma 9 into several steps.

Step 1:

We first observe that the condition (20) yields

$$\begin{aligned} u \in C([0,T]; L^2(\mathbb {R}^{3})). \end{aligned}$$

Indeed, (20) implies that for almost all \(0 \le t \le T\) and all \(h>0\)

$$\begin{aligned} h^3 {m}(E_t(h)) \le M^3 \end{aligned}$$
(23)

where \(E_t(h)\) denotes the level set

$$\begin{aligned} E_t(h) := \{x:|u(x,t)|>h\}. \end{aligned}$$

By the Chebyshev inequality we also have

$$\begin{aligned} h^6 {m}(E_t(h)) \le \Vert u(t)\Vert _{L^6}^6. \end{aligned}$$
(24)

Using the two estimates (23) and (24), we obtain that for any \(H>0\)

$$\begin{aligned} \int _{\mathbb {R}^3} |u(x,t)|^4 dx&= 4 \int _{0}^{\infty }h^3 {m}(E_t(h)) dh \\&\lesssim \int _0^H M^3 dh + \int _H^{\infty } h^{-3} \Vert u(t)\Vert _{L^6}^6 dh \\&\lesssim M^3H + \Vert u(t)\Vert _{L^6}^6H^{-2}. \end{aligned}$$

Taking \(H = M^{-1}\Vert u(t)\Vert _{L^6}^2\) we get

$$\begin{aligned} \int _{\mathbb {R}^3} |u(x,t)|^4 dx \lesssim M^2 \Vert u(t)\Vert _{L^6}^2. \end{aligned}$$

Hence \(u \in L^4(Q_T)\) and so \(|u|^2|\nabla u| \in L^1(Q_T)\). This justifies the required integration by parts to be a local suitable weak solution and also implies the global energy equality so that u is in \(C([0,T];L^2(\mathbb {R}^3))\).

Step 2:

We next claim that the condition (20) also yields that for all \(Q(z_0,r) \subset Q_T\)

$$\begin{aligned} r^{-1} \left( \int _{Q(z_0,r/2)} |u|^{10/3} dz\right) ^{3/5} + r^{-1} \int _{Q(z_0,r/2)} |\nabla u|^2 dz \lesssim M^2+M^6. \end{aligned}$$
(25)

Due to the Caccioppoli–type inequality (17), it suffices to estimate

$$\begin{aligned} \int _{B(x_0,r)} |u(x,t)|^2 dx. \end{aligned}$$

Using the estimate (23), we obtain that for almost all \(0 \le t \le T\) and all \(h>0\)

$$\begin{aligned} \int _{B(x_0,r)} |u(x,t)|^2 dx&= 2 \int _{0}^{\infty }h {m} [B(x_0,r) \cap E_t(h)] dh \\&\lesssim \int _0^H h r^3 dh + \int _H^{\infty } h^{-2} M^3 dh \\&\lesssim r^3 H^2 + M^3 H^{-1}. \end{aligned}$$

Taking \(H = Mr^{-1}\) we get for almost all \(0 \le t \le T\)

$$\begin{aligned} \int _{B(x_0,r)} |u(x,t)|^2 dx \lesssim M^2 r. \end{aligned}$$
(26)

Putting this bound into the right side of the inequality (17), we get the estimate (25).

Step 3:

We now prove Lemma 9 by using an indirect argument. Assume the assertion of the lemma is not true, that is, there exist a positive number M, sequences \(\varepsilon _k \in (0, 1/4)\), \(T_k \in (0,\infty )\), \(z_k = (x_k,t_k) \in Q_{T_k}\), \(r_k \in (0,\sqrt{t_k}]\), and a sequence of weak solutions \(u_k \in V^2_\sigma (Q_{ T_k})\) such that \(\varepsilon _k \rightarrow 0\) as \(k \rightarrow \infty \) and for all \(k \in \mathbb {N}\)

$$\begin{aligned} \mathop {\hbox {ess}\,\hbox {sup}}\limits _{0 \le t \le T_k} \Vert u_k\Vert _{L^{3,w}(\mathbb {R}^{3})}\le & {} M, \nonumber \\ r_k^{-3} {m} \{x \in B(x_k,r_k) : |u_k(x,t_k)| > r_k^{ -1} \varepsilon _k\}\le & {} \varepsilon _k, \end{aligned}$$
(27)

and for all \(\rho \in (2\varepsilon _k r_k, \sqrt{t_k}]\)

$$\begin{aligned} \rho ^{-2} \int _{Q(z_k,\rho )} |u_k|^3 dz > \zeta ^3. \end{aligned}$$
(28)

We define for \((y,s) \in \mathbb {R}^3 \times (-1,0)\)

$$\begin{aligned} U_k(y,s)&= r_k u_k(x_k+ r_k y, t_k+ r_k^2 s), \\ P_k(y,s)&= r^2_k p_k(x_k+ r_k y, t_k+ r_k^2 s). \end{aligned}$$

Then \((U_k, P_k)\) is a weak solution to the Navier–Stokes equations in \(\mathbb {R}^{3}\times (-1,0)\) and satisfies

$$\begin{aligned} \mathop {\hbox {ess}\,\hbox {sup}}\limits _{-1 \le s \le 0} \Vert U_k\Vert _{L^{3,w}(\mathbb {R}^{3})} \le M. \end{aligned}$$

Thanks to (25) and (26), we have for all \( k\in \mathbb {N}\), \(z_0 = (x_0,0)\) and \(0 < \rho \le 1\)

$$\begin{aligned} \rho ^{-1} \left( \int _{Q(z_0,\rho /2)} |U_k|^{10/3} dz\right) ^{3/5} + \rho ^{-1} \int _{Q(z_0,\rho /2)} |\nabla U_k|^2 dz \lesssim M^2+M^6 \end{aligned}$$
(29)

and

$$\begin{aligned} \rho ^{-1} \sup _{-\rho ^2 \le s \le 0} \int _{B(x_0,\rho )} |U_k(s)|^2 dx \lesssim M^2. \end{aligned}$$
(30)

Furthermore, from (27) and (28), we also have for all \( k\in \mathbb {N}\)

$$\begin{aligned} {m} \{x \in B_1 : |U_k(x,0)| > \varepsilon _k\} \le \varepsilon _k \end{aligned}$$
(31)

and for all \(\rho \in [\varepsilon _k,1]\)

$$\begin{aligned} \rho ^{-2} \int _{Q(0,\rho )} |U_k|^3 dz > \zeta ^3. \end{aligned}$$
(32)

Using a standard reflexivity argument along with Cantor’s diagonalization principle and passing to a subsequence from (29) we eventually get \(U \in L^{10/3}(-1,0; L^{10/3}_{{{\,\mathrm{loc}\,}}}(\mathbb {R}^{3}))\) with \( \nabla U \in L^{2}(-1,0; L^{2}_{{{\,\mathrm{loc}\,}}}(\mathbb {R}^{3}))\) and \(H\in L^{5/3}(-1,0; L^{5/3}_{{{\,\mathrm{loc}\,}}}(\mathbb {R}^{3}))\) such that for every \( 0<R<\infty \)

$$\begin{aligned} U_k&\rightarrow U \quad \textit{weakly}\ \textit{in} \quad L^{10/3}(B_R \times (-1,0)) \nonumber \\ \nabla U_k&\rightarrow \nabla U \quad \textit{weakly}\ \textit{in} \quad L^{2}(B_R\times (-1,0)) \nonumber \\ U_k \otimes U_k&\rightarrow H \quad \textit{weakly}\ \textit{in} \quad L^{5/3}(B_R\times (-1,0)) \end{aligned}$$
(33)

as \(k \rightarrow \infty \). Hence, U appears to be a distributional solution to

$$\begin{aligned} \partial _t U - \varDelta U + \nabla \cdot H = -\nabla P \quad \text { in}\quad \mathbb {R}^{3}\times (-1,0). \end{aligned}$$
(34)

According to the weakly lower semi-continuity of the norm we get from (29) and (30) along with (33) for all \(0 < \rho \le 1\)

$$\begin{aligned} \rho ^{-1} \left( \int _{Q(z_0,\rho )} |U|^{10/3} dyds\right) ^{3/5} + \rho ^{-1} \int _{Q(z_0,\rho )} | \nabla U|^{2} dy ds \lesssim M^2+ M^6 \end{aligned}$$
(35)

and

$$\begin{aligned} \rho ^{-1} \sup _{-\rho ^2 \le s \le 0} \int _{B(x_0,\rho )} |U(s)|^2 dy \lesssim M^2. \end{aligned}$$
(36)
Step 4:

Let \(s_0 \in [-1,0]\). Since \(u \in C_{ w}^{*}([0,T]; L^{3,w}(\mathbb {R}^{3}))\), we have \(u(\cdot , t_k + r_k^2 s_0) \in L^{3,w}(\mathbb {R}^{3})\) and

$$\begin{aligned} \Vert u(\cdot , t_k + r_k^2 s_0)\Vert _{ L^{3,w}(\mathbb {R}^{3})} = \Vert U_k(\cdot ,s_0)\Vert _{ L^{3,w}(\mathbb {R}^{3})}. \end{aligned}$$

This shows that \(\{U_k(\cdot ,s_0)\}\) is a bounded sequence in \(L^{3,w}(\mathbb {R}^{3})\). On the other hand, the predual of \( L^{3,w}(\mathbb {R}^{3})\) is the Lorentz space \(L^{3/2,1}(\mathbb {R}^{3})\). By means of the Banach–Alaoglu theorem we get a subsequence \( \{ U_{ k_j}(\cdot , s_0)\}\) and a function \( \eta \in L^{3,w}(\mathbb {R}^{3})\) such that

$$\begin{aligned} U_{ k_j}(\cdot , s_0) \rightarrow \eta \quad \textit{weakly}^{*}\ \textit{in} \quad L^{3,w}(\mathbb {R}^{3})\quad \textit{as}\quad j \rightarrow +\infty . \end{aligned}$$
(37)

Thus, from (34) we infer that for all \( \varphi \in C_c^{\infty }(\mathbb {R}^{3}\times (-1,0)) \) with \( \nabla \cdot \varphi =0\)

$$\begin{aligned} \int _{-1}^{s_0} \int _{ \mathbb {R}^{3}} - U \cdot \partial _t \varphi + \nabla U: \nabla \varphi - H : \nabla \varphi dz = - \int _{ \mathbb {R}^{3}} \eta \cdot \varphi (s_0) dx. \end{aligned}$$

In case \( s_0\) is a Lebesgue point of U with respect to time, we argue that for all \( \psi \in C^{\infty }_\mathrm{c, \sigma }(\mathbb {R}^{3})\)

$$\begin{aligned} \int _{\mathbb {R}^{3}} (U(s_0) - \eta ) \cdot \psi dx=0 \end{aligned}$$

which shows that \( U(s_0) - \eta \) is a gradient field. Together with \( \nabla \cdot (U(s_0) - \eta )\) in the sense of distributions we see that \( U(s_0) - \eta \) is harmonic. Recalling that \( U(s_0) - \eta \in L^{3,w}(\mathbb {R}^{3})\), it follows that \( \eta = U(s_0)\) by Remark 4 in Section 2. Consequently, (37) yields

$$\begin{aligned} U_k(\cdot , s_0) \rightarrow U(s_0) \quad \textit{weakly}^{*} \ \textit{in} \quad L^{3,w}(\mathbb {R}^{3})\quad \textit{as} \quad k \rightarrow +\infty . \end{aligned}$$

Furthermore, we get

$$\begin{aligned} \Vert U(s_0)\Vert _{L^{3,w}(\mathbb {R}^{3})} \le \Vert u\Vert _{L^{\infty }(0,T;L^{3,w}(\mathbb {R}^{3}))}. \end{aligned}$$

In particular, \(U \in L^{\infty }(0,T;L^{3,w}(\mathbb {R}^{3}))\).

Step 5:

Next, let \( s_0\in [-1,0]\). Then we may choose \( s_m \in (0,T)\) in the set of Lebesgue points such that \(s_m \rightarrow s_0\) as \( m \rightarrow +\infty \). Then as above we get a subsequence \( \{ s_{ m_j}\}\) and \( \eta \in L^{3,w}(\mathbb {R}^{3})\) such that

$$\begin{aligned} U(\cdot , s_{ m_j}) \rightarrow \eta \quad \textit{weakly}^{*} \ \textit{in}\quad L^{3,w}(\mathbb {R}^{3})\quad \textit{as}\quad j \rightarrow +\infty . \end{aligned}$$

In addition, we easily verify that the following identity holds for every \( \varphi \in C^{\infty }(\mathbb {R}^{3}\times (-1,0)) \) with \({{\,\mathrm{div}\,}}\varphi = 0\)

$$\begin{aligned} \int _{-1}^{s_0} \int _{ \mathbb {R}^{3}} - U \cdot \partial _t \varphi + \nabla U: \nabla \varphi - H : \nabla \varphi dz = - \int _{ \mathbb {R}^{3}} \eta \cdot \varphi (s_0) dx. \end{aligned}$$
(38)

Arguing as above, we see that this limit is unique, and will be denoted by \( U(s_0)\). Note that (38) holds true for \( \eta = U(s_0)\). We now repeat the same argument as above to prove that for all \( s_0 \in [-1,0]\)

$$\begin{aligned} U_k(\cdot ,s_{ 0})&\rightarrow U(s_0) \quad \textit{weakly}^{*} \ \textit{in}\quad L^{3,w}(\mathbb {R}^{3})\quad \textit{as}\quad k\rightarrow +\infty \end{aligned}$$
(39)
$$\begin{aligned} U(\cdot ,s)&\rightarrow U(s_0) \quad \textit{weakly}^{*} \ \textit{in}\quad L^{3,w}(\mathbb {R}^{3})\quad \textit{as}\quad s\rightarrow s_0. \end{aligned}$$
(40)

This leads to \( U\in C_{ w}^{*}([-1,0]; L^{3,w}(\mathbb {R}^{3}))\).

Step 6:

We shall verify the strong convergence of \(U_k\) in \( L^2(B_R\times (-1,0))\). For this purpose, we define the local pressure introduced in [20],

$$\begin{aligned} \nabla P_{h,k,R}&= -E^{*}_{B_R} (U_k), \\ \nabla P_{1,k,R}&= -E^{*}_{B_R} (\nabla \cdot U_k \otimes U_k), \\ \nabla P_{2,k,R}&= E^{*}_{B_R} (\varDelta U_k) \end{aligned}$$

and

$$\begin{aligned} \nabla P_{h,R}&= -E^{*}_{B_R} (U), \\ \nabla P_{1,R}&= -E^{*}_{B_R} (\nabla \cdot H), \\ \nabla P_{2,R}&= E^{*}_{B_R} (\varDelta U ) \end{aligned}$$

(For the definition of \(E^{*}_{B_R}\) see Appendix B of this paper).

Step 7:

We set \( V_k = U_k + \nabla P_{ h,k,R}\), and \( V = U+ \nabla P_{ h,R}\). Then \( V_k\) solves

$$\begin{aligned} \partial _t V_k - \varDelta U_k + \nabla \cdot (U_k \otimes U_k) =-\nabla (P_{ 1,k,R}+ P_{ 2,k,R}) \quad \text { in}\quad B_R\times (-1,0), \end{aligned}$$

while V solves

$$\begin{aligned} \partial _t V - \varDelta U + \nabla \cdot H = -\nabla (P_{ 1,R}+ P_{ 2,R}) \quad \text { in}\quad B_R \times (-1,0). \end{aligned}$$

By using a standard compactness argument due to Lions-Aubin we see that

$$\begin{aligned} V_k \rightarrow V \quad \textit{in}\quad L^2(B_R\times (-1,0)) \quad \textit{as}\quad k \rightarrow +\infty . \end{aligned}$$

By passing to a subsequence we may also assume that

$$\begin{aligned} V_k \rightarrow V \quad a.\,e.\ in \, B_R\times (-1,0)\quad \textit{as}\quad k \rightarrow +\infty . \end{aligned}$$

Arguing as in [23], by the aid of (40), and noting that \( P_{ h, k, R}\) is harmonic, we also find that

$$\begin{aligned} \nabla P_{h,k,R } \rightarrow \nabla P_{ h, R} \quad a.\,e.\ in \, B_R \times (-1,0)\quad \textit{as}\quad k \rightarrow +\infty . \end{aligned}$$

This leads to the a. e. convergence of \( U_k\) which allows to apply Lebesgue’s dominated convergence theorem. Accordingly,

$$\begin{aligned} U_k \rightarrow U \quad \textit{in} \quad L^3(B_R\times (-1,0)) \quad \textit{as}\quad k \rightarrow +\infty . \end{aligned}$$

This also shows that \( H= U \otimes U\) and therefore U solves the Navier-Stokes equations.

Step 8:

In (32) letting \( k \rightarrow +\infty \), we obtain for every \( 0<\rho \le 1\)

$$\begin{aligned} \rho ^{-2} \int _{Q(0,\rho )} |U|^3 dz \ge \zeta ^3. \end{aligned}$$
(41)

It remains to carry out the passage to the limit \( k \rightarrow +\infty \) in (31). Without loss of generality we may assume \(\varepsilon _k \le 2^{ -k}\). Let

$$\begin{aligned} A := \bigcap _{m=1}^{\infty } \bigcup _{k=m}^{\infty } \{x \in B_1 : |U_k(x,0)| > \varepsilon _k\}. \end{aligned}$$

Then according to (31) we have

$$\begin{aligned} \sum _{k=1}^{\infty } {m} \{x \in B_1 : |U_k(x,0)| > \varepsilon _k\} \le \sum _{k=1}^{\infty } \varepsilon _k < \infty . \end{aligned}$$

Hence the Borel–Cantelli lemma yields \({m}(A)=0\). In other words, for each \(x \in B_1 {\setminus } A\), there exists \(m \in \mathbb {N}\) such that for all \(k \ge m\)

$$\begin{aligned} |U_k(x,0)| \le \varepsilon _k. \end{aligned}$$

Accordingly, \(U_k(x,0) \rightarrow 0\) for almost all \(x \in B_1\). In view of (39) we conclude that

$$\begin{aligned} U(0) = 0 \quad \text { on } \quad B_1. \end{aligned}$$
(42)
Step 9:

Next, we set \(\rho _k = 2^{-k}\) and define for \((x,t) \in \mathbb {R}^{3}\times (-1,0)\)

$$\begin{aligned} {\tilde{U}}_k(x,t)&= \rho _k U(\rho _k x, \rho _k^2 t), \\ {\tilde{P}}_k(x,t)&= \rho ^2_k P(\rho _k x,\rho _k^2 t). \end{aligned}$$

Again \(({\tilde{U}}_k, {\tilde{P}}_k)\) is a solution to the Navier-Stokes equation in \(\mathbb {R}^{3}\times (-1,0)\). Observing (35) and (36), we find for all \(z_0=(x_0,0)\)

$$\begin{aligned} \left( \int _{Q(z_0,1)} |{\tilde{U}}_k|^{10/3} dz\right) ^{3/5} + \int _{Q(z_0,1)} |\nabla {\tilde{U}}_k|^{2} dz \lesssim M^2+M^6, \end{aligned}$$
(43)

and

$$\begin{aligned} \sup _{-1 \le t \le 0} \int _{B(x_0,1)} |{\tilde{U}}_k(t)|^2 dx \lesssim M^2. \end{aligned}$$
(44)

On the other hand, (42) and (41) yield \({\tilde{U}} _k(0) = 0\) on \(B_{2^k}\) and

$$\begin{aligned} 16 \int _{Q(0,1/4)} |{\tilde{U}}_k|^3 dz > \zeta ^3. \end{aligned}$$
(45)

Arguing as in Step 3, we get a solution

$$\begin{aligned} {\tilde{U}} \in C_{w}^{*}([-1,0]; L^{3,w}(\mathbb {R}^{3})) \cap L^2(-1,0; W^{1,2}_{{{\,\mathrm{loc}\,}}}(\mathbb {R}^{3})) \end{aligned}$$

to the Navier-Stokes equations. Furthermore, (43), (44), and (45) yield for all \(z_0=(x_0,0)\)

$$\begin{aligned}&\left( \int _{Q(z_0,1)} |{\tilde{U}}|^{10/3} dz\right) ^{3/5} + \int _{Q(z_0,1)} |\nabla {\tilde{U}}|^{2} dz \le C_0,\nonumber \\&\quad \sup _{-1 \le t \le 0} \int _{B(x_0,1)} |{\tilde{U}}(t)|^2 dx \le C_0, \end{aligned}$$
(46)

\({\tilde{U}} (0) = 0\) in \(\mathbb {R}^{3}\), and

$$\begin{aligned} 16 \int _{Q(0,1/4)} |{\tilde{U}}|^3 dyds \ge \zeta ^3. \end{aligned}$$
(47)
Step 10:

By the Fubini theorem, we have

$$\begin{aligned}&{m} \{(x,t) \in \mathbb {R}^{3} \times (-1,0) : |{\tilde{U}}(x,t)| \ge 2^{-5} \zeta \} \\&\quad = \int _{-1}^{0} {m} \{x \in \mathbb {R}^{3} : |{\tilde{U}}(x,t)| \ge 2^{-5} \zeta \} dt \\&\quad \le 2^{15} \zeta ^{ -3} \Vert {\tilde{U}}\Vert _{L^{\infty }(-1,0;L^{3,w}(\mathbb {R}^{3}))}^3\\&\quad < \infty . \end{aligned}$$

Hence, for each \(\eta >0\) there is a radius \(R=R(\eta )>0\) such that

$$\begin{aligned} {m} \{(x,t) \in (\mathbb {R}^{3} {\setminus } B_R) \times (-1,0) : |{\tilde{U}}(x,t)| \ge 2^{-5} \zeta \} \le \eta . \end{aligned}$$

Choose

$$\begin{aligned} \eta = 2^{-10} C_0^{-15} \zeta ^{30}, \end{aligned}$$
(48)

where \(C_0\) is the constant in (46). Then for any \(x_0 \in \mathbb {R}^3 {\setminus } B(0,R+1)\), we obtain, by Hölder’s inequality, (46), and (48), that

$$\begin{aligned} \int _{Q(z_0,1)} |{\tilde{U}}|^3 dz&\le (2^{-5} \zeta )^3 {m}(Q(z_0,1)) +\int _{Q(z_0,1) \cap \{|{\tilde{U}}| \ge 2^{-5} \zeta \}} |{\tilde{U}}|^3 dz\\&\le \frac{\zeta ^3}{2} +\eta ^{1/10} \left( \int _{Q(z_0,1)} |{\tilde{U}}|^{10/3} dz\right) ^{9/10}\\&\le \frac{\zeta ^3}{2} + 2^{-1} C_0^{-3/2} \zeta ^{3} C_0^{3/2}\\&\le \zeta ^3. \end{aligned}$$

Thus, appealing to Lemma 8, and making use of (46), we get for all \(x_0 \in \mathbb {R}^{3} {\setminus } B(0,R+1)\)

$$\begin{aligned} \Vert {\tilde{U}}\Vert _{L^{\infty }(Q(z_0,1/2))} \le C\Vert {\tilde{U}}\Vert _{L^3(Q(z_0,1))} + C \Vert {\tilde{U}}\Vert _{L^{\infty }(-1,0;L^2(B(x_0, 1)))} \le C (\zeta + C_0). \end{aligned}$$
(49)

This shows that \({\tilde{U}}\) is bounded in \(\mathbb {R}^{3} {\setminus } B(0,R+1) \times (-1/4,0)\).

Step 11:

Using a standard bootstrapping argument, we obtain the higher regularity

$$\begin{aligned} \nabla {\tilde{U}} \in L^{\infty } \left( \mathbb {R}^{3} {\setminus } B(0,R+2) \times (-1/4,0)\right) . \end{aligned}$$

Taking the curl operator to the Navier–Stokes equations, we see that \({\tilde{\varOmega }} := \nabla \times {\tilde{U}}\) solves the heat equation

$$\begin{aligned} \partial _t {\tilde{\varOmega }} - \varDelta {\tilde{\varOmega } } = {\tilde{\varOmega } } \cdot \nabla {\tilde{U}} - {\tilde{U}} \cdot \nabla {\tilde{\varOmega }} \end{aligned}$$

in \(\mathbb {R}^{3}\times (-1,0)\). Hence

$$\begin{aligned} \left| \partial _t {\tilde{\varOmega }} - \varDelta {\tilde{\varOmega }}\right| \le \Vert \nabla {\tilde{U}}\Vert _{\infty } |{\tilde{\varOmega }}| + \Vert {\tilde{U}}\Vert _{\infty } |\nabla {{\tilde{\varOmega }}}| \end{aligned}$$

in \(\mathbb {R}^{3} {\setminus } B(0,R+2) \times (-1/16,0)\). Verifying that \({\tilde{\varOmega } } (0)=0\), we are in a position to apply the backward uniqueness of [6] to conclude that \({\tilde{\varOmega } } \equiv 0\) in \(\mathbb {R}^{3} {\setminus } B(0,R+2) \times (-1/16,0)\). By the spatial analyticity of \({\tilde{U}}\) we get the spatial analyticity of \({\tilde{\varOmega } } \) which shows that \( {\tilde{\varOmega } } \equiv 0 \) in \( \mathbb {R}^{3} \times (-1/16,0)\). Recalling that \({{\,\mathrm{div}\,}}{\tilde{U}} = 0\) it follows that \( {\tilde{U}} \) is harmonic in \( \mathbb {R}^{3}\times (-1/16,0)\), and thus \( {\tilde{U}} \) must be identically zero in \( \mathbb {R}^{3}\times (-1/16,0)\). However this contradicts to (47). Therefore the assertion of Lemma 9 must be true.

This completes the proof of Lemma 9. By combining Lemma 8 we obtain Theorem 1. \(\square \)

4 Proof of Theorem 2

We divide the proof of Theorem 2 into a few steps.

Step 1:

Let \(C(x_0,r)\) denote the closed cube of a side-length r and the center \(x_0\). We may replace the condition (7) in Theorem 1 by using cubes, that is,

$$\begin{aligned} r^{-3} {m} \{x\in C(x_0,r) : |u(x, t_0)| > r^{-1} \varepsilon \} \le \varepsilon . \end{aligned}$$
(50)

Then the conclusion also be changed with \(u\in L^{\infty } (\widetilde{Q}(z_0,\varepsilon r))\) where

$$\begin{aligned} \widetilde{Q}(z_0,\varepsilon r) := C(x_0,r) \times (t_0-r^2,t_0). \end{aligned}$$

In fact, \(\varepsilon \) should be changed by a multiplication of some constant which depends only on the volume ratio of the ball of a radius r and the cube of a side-length r. For convenience we just use the same letter \(\varepsilon \).

Step 2:

We shall proceed with an algorithm based on a dyadic decomposition argument. We say that two cubes E and \(E'\) meet if \(E \cap E'\) has nonempty interior. Let \(C = [0,1]^3\) denote the unit cube in \(\mathbb {R}^3\). We define for \(k=0,1,2,\dots \) the following covers

$$\begin{aligned} \mathscr {C}_k := \{2^{-k} (\varepsilon j + C) : j \in \mathbb {Z}^3\}, \end{aligned}$$

which has finite overlapping property. Indeed, each fixed cube in \(\mathscr {C}_k\) can meet \(\varepsilon ^{-3}\) number of cubes in \(\mathscr {C}_k\). We pick a sub-family

$$\begin{aligned} F_0 := \{E \in \mathscr {C}_0 : {m} \{x \in E : |u(x,t_0)|>\varepsilon \} > \varepsilon \}. \end{aligned}$$
(51)

If \(F_0\) has no element, then we have \({m} \{x \in E : |u(x,t_0)|>\varepsilon \} \le \varepsilon \) for all \(E \in \mathscr {C}_0\). Hence we conclude that there is no singularity at all at the moment \(t_0\) due to Theorem 1.

Next, we claim that \(F_0\) has at most a finite number of members, which is bounded by a number depending only on M and \(\varepsilon \). Suppose that \(E_1, E_2, \dots , E_N \in F_0\) don’t meet each other. Then for \(j=1,2,\dots ,N\)

$$\begin{aligned} \varepsilon < {m} \{x \in E_j : |u(x,t_0)|>\varepsilon \}. \end{aligned}$$

Summing both sides for \(j=1,2,\dots ,N\) yields

$$\begin{aligned} N\varepsilon < {m} \left\{ x \in \bigcup _{j=1}^N E_j : |u(x,t_0)| > \varepsilon \right\} \le \varepsilon ^{-3} M^3. \end{aligned}$$

The last inequality follows from the fact \(\Vert u(t_0)\Vert _{L^{3,w}(\mathbb {R}^{3})} \le M\). This implies that the number of maximal disjoint cubes in \(F_0\) is finite and hence \(F_0\) has at most finite members. If we denote by \(N_0^d\) the number of maximal disjoint cubes in \(F_0\), then we should have

$$\begin{aligned} N_0^d \le \varepsilon ^{-4} M^3. \end{aligned}$$

Let \(N_0\) denote the number of cubes in \(F_0\). Then, from the finite overlapping property of \(\mathscr {C}_0\), we have

$$\begin{aligned} \varepsilon ^3 N_0 \le N_0^d \le N_0. \end{aligned}$$

Hence

$$\begin{aligned} N_0 \le \varepsilon ^{-3} N_0^d \le \varepsilon ^{-3} (\varepsilon ^{-4} M^3) = \varepsilon ^{-7} M^3. \end{aligned}$$
(52)

We define \(G_0\) to be the union of \(F_0\) and the cubes \(E \in \mathscr {C}_0\) which meet some element of \(F_0\). Theorem 1 implies that if \((x,t_0) \notin \bigcup _{E \in G_0} E\), then \((x,t_0)\) is a regular piont, that is, possible singularities can only occur in some element of \(G_0\).

Step 3:

We now inductively construct two families of cubes \(\{F_k\}\) and \(\{G_k\}\). For \(k\ge 1\) we define \(F_k\) to be the family of cubes \(E \in \mathscr {C}_k\) satisfying \(E \subset E'\) for some \(E' \in G_{k-1}\) and

$$\begin{aligned} {m} \{E : |u(x,t_0)|>2^k\varepsilon \} > 2^{-3k}\varepsilon . \end{aligned}$$
(53)

Let \(N_k\) denote the number of cubes in \(F_k\) and let \(N_k^d\) denote the number of maximal disjoint cubes in \(F_k\). By the same reasoning \(N_k\) and \(N_k^d\) are finite numbers and have the same bounds. Indeed, since each fixed cube in \(F_k\) can meet at most \(\varepsilon ^{-3}\) number of cubes in \(F_k\), we have

$$\begin{aligned} \varepsilon ^3 N_k \le N_k^d \le N_k. \end{aligned}$$

By the same way in the previous step, we obtain

$$\begin{aligned} N_k^d 2^{-3k} \varepsilon \le {m} \{x \in \mathbb {R}^3 : |u(x,t_0)| > 2^k\varepsilon \} \le (2^k\varepsilon )^{-3} M^3. \end{aligned}$$

Therefore,

$$\begin{aligned} N_k \le \varepsilon ^{-3} N_k^d \le \varepsilon ^{-3} (\varepsilon ^{-4} M^3) = \varepsilon ^{-7} M^3. \end{aligned}$$
(54)

We define \(G_k\) to be the union of \(F_k\) and the cubes \(E \in \mathscr {C}_k\) which meet some element of \(F_k\). Theorem 1 implies that if \((x,t_0) \notin \bigcup _{E \in G_k} E\), then \((x,t_0)\) is a regular piont, that is, the possible singularities can only occur in the elements of \(G_k\).

Step 4:

Finally, we construct nested sequences \(\{E_k\}\) of closed cubes satisfying \(E_k \in G_k\). Fix an element \(E_k\) in \(G_k\). If \(E \in G_{k+1}\), then \(E \subset E_k\) or E does not meet \(E_k\) by the dyadic construction. If there is no \(E \in G_{k+1}\) which meet \(E_k\), then each interior point of \(E_k\) is a regular point. In this case, we stop to choose next elements. Otherwise, there is an element \(E \in G_{k+1}\) such that \(E \subset E_k\). Then we pick E and name it as \(E_{k+1}\). The cardinality of each set \(G_k\) is bounded by \(\varepsilon ^{-7} M^3 + \varepsilon ^{-3}\) from (54) and the finite overlapping property. The number of such choices is also always bounded by \(\varepsilon ^{-7} M^3 + \varepsilon ^{-3}\). After the construction, we only have at most \(\varepsilon ^{-7} M^3 + \varepsilon ^{-3}\) number of sequences \(\{E_k\}\). If the sequence \(\{E_k\}\) is finite, then each interior point of \(E_k\) is regular. If the sequence \(\{E_k\}\) is infinite, then

$$\begin{aligned} E_{k+1} \subset E_k, \quad {{\,\mathrm{diam}\,}}E_{k+1} \le \frac{1}{2} {{\,\mathrm{diam}\,}}E_k. \end{aligned}$$

where \({{\,\mathrm{diam}\,}}E\) denote the diameter of the set E. Since \({{\,\mathrm{diam}\,}}E_k\) goes to 0 as \(k \rightarrow \infty \),

$$\begin{aligned} \bigcap _{k=1}^{\infty } E_k = (x,t_0) \end{aligned}$$

for some \(x \in \mathbb {R}^3\). This point might be a singular point. Therefore, the number of such possible singularities is at most

$$\begin{aligned} N(M) := \varepsilon ^{-7} M^3 + \varepsilon ^{-3} \end{aligned}$$
(55)

at the time \(t_0\). We note that \(\varepsilon \) actually depends only on M, and hence the number of possible singularities is bounded by the uniform norm of the weak Lebesgue space.

This completes the proof of Theorem 2. \(\square \)