Scaling-Invariant Serrin Criterion via One Row of the Strain Tensor for the Navier–Stokes Equations

Abstract

Miller (Arch. Rational Mech. Anal., 2020) posed the question of whether it is possible to prove the Navier–Stokes regularity criterion using only one entry of the strain tensor $S_{ij}$. Although this paper does not fully address this question, we do establish several scaling-invariant Serrin-type criteria based on one row of the strain tensor.

1. Introduction

The Navier–Stokes equation (NSE for short) is one of the fundamental equations of fluid dynamics. The three-dimensional viscous incompressible NSE reads as

$$\left\{\begin{array}{cc}{\partial }_{t}u+u·\nabla u-\Delta u+\nabla \pi =0,\hfill & x\in {\mathbb{R}}^3,t>0\hfill \\ \nabla ·u=0,\hfill & x\in {\mathbb{R}}^3,t>0\hfill \\ {\left.u\right|}_{t=0}=u_0\left(x\right),\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right.$$

(1)

where $u=(u_1,u_2,u_3)$—the velocity field—and $\pi$—the pressure—are the unknowns, and $u_0$ is the initial velocity field. We will denote ${\nabla }_h=({\partial }_{x_1},{\partial }_{x_2})$ as the horizontal gradient operator and ${\Delta }_h={\partial }_{x_1}^2+{\partial }_{x_2}^2$ as the horizontal Laplacian, where $\Delta$ and ∇ are the Laplacian and the gradient operators, respectively. Here, we utilize the following notations:

$$(u·\nabla )v=\sum _{i=1}^3{u}_i{\partial }_{x_i}{v}_k,(k=1,2,3),\nabla ·u=\sum _{i=1}^3{\partial }_{x_i}{u}_i,S=S_{ij}={\displaystyle \frac{1}{2}}\left({\partial }_{x_i}{u}_{x_j}+{\partial }_{x_j}{u}_{x_i}\right)$$

and for the sake of simplicity, we denote the partial derivative with respect to ${\partial }_{x_i}$ as ${\partial }_i$.

In the pioneering work of Leray [1] and Hopf [2], they found that, for any $u_0\in L_{\sigma }^2\left({\mathbb{R}}^3\right)$, one can construct a global weak solution to (1), namely, a function u that, for each $T>0$, is in the class

$$u\in L^{\infty }(0,T;L_{\sigma }^2\left({\mathbb{R}}^3\right))\cap L^2(0,T;H^1\left({\mathbb{R}}^3\right))$$

(2)

and solves (1) in a distributional sense. Here, $L_{\sigma }^2\left({\mathbb{R}}^3\right)$ is the subspace of $L^2\left({\mathbb{R}}^3\right)$ of divergence-free vector functions. In addition, such a solution u satisfies the so-called energy inequality:

$${\parallel u\left(t\right)\parallel }_{L^2}^2+2{\int }_0^t{\parallel \nabla u\left(\tau \right)\parallel }_{L^2}^{2}d\tau \le {\parallel u_0\parallel }_{L^2}^2,\forall t\ge 0.$$

(3)

As we all know, for strong solutions, this inequality passes to the corresponding equality, while Leray–Hopf weak solutions, as these solutions are commonly referred to, must exist globally in time. They are not known to be either regular or unique.

It is known that if a weak solution u of NSE (1) satisfies the Ladyzhenskaya–Prodi–Serrin criteria [3,4,5,6]

$$u\in L^q(0,T;L^p\left({\mathbb{R}}^3\right)),{\displaystyle \frac{2}{q}}+{\displaystyle \frac{3}{p}}=1,p\in [3,\infty ],$$

(4)

then u must be regular in ${\mathbb{R}}^3\times (0,T]$. From a physical point of view, the velocity describes how a fluid is advected, the vorticity describes how it is rotated, and the strain describes how a parcel of fluid is deformed. Therefore, the vorticity and the strain tensor are also important physical quantities to describe fluid behaviors. Beale–Kato–Majda [7] and Beirão da Veiga [8] extended the Ladyzhenskaya–Prodi–Serrin criteria to the vorticity $\omega =\nabla \times u$ and showed that if

$$\omega \in L^q(0,T;L^p\left({\mathbb{R}}^3\right)),{\displaystyle \frac{2}{q}}+{\displaystyle \frac{3}{p}}=2,{\displaystyle \frac{3}{2}}<p<\infty ,$$

(5)

then u is regular. More generally, Ponce in [9] proved that the weak solution u became regular if the strain tensor satisfies

$$S\in L^1(0,T;L^{\infty }\left({\mathbb{R}}^3\right)).$$

(6)

We should mention that the in above cases, ${\displaystyle \frac{2}{q}}+{\displaystyle \frac{3}{p}}=1$ and ${\displaystyle \frac{2}{q}}+{\displaystyle \frac{3}{p}}=2$, the function space $L_t^q{L}_x^p$ is invariant under the following scaling:

$$u(x,t)\mapsto u^{\lambda }(x,t)=\lambda u\left(\lambda x,{\lambda }^{2}t\right),\forall \lambda >0,$$

where $u^{\lambda }$ is still a solution to (1) with initial data $u_0^{\lambda }:=\lambda u_0\left(\lambda x\right)$.

There have also been several scale-critical, component-reduction-type regularity criteria that require control over only specific parts of the solution. These include the regulation of just two components of the vorticity $({\omega }_1,{\omega }_2,0)$ [10,11], the derivative in a single direction ${\partial }_{3}u$ [12,13,14], and only one component of the velocity $u_3$ [15,16,17]. In addition, for the literature related to the regularity criteria of NSE in BMO spces, we refer to [18,19].

Recently, the study of the regularity criterion, which involves the positive part of the middle eigenvalue of the strain tensor, has gained popularity. Specifically, it has been proven that the regularity of weak solutions exists within the class of

$${\lambda }_2^{+}\in L^q(0,T;L^p\left({\mathbb{R}}^3\right))\mathrm{with}{\lambda }_2^{+}=\max \left(0,{\lambda }_2\right),{\displaystyle \frac{2}{q}}+{\displaystyle \frac{3}{p}}=2,{\displaystyle \frac{3}{2}}<p\le \infty .$$

(7)

This was first proven by Neustupa and Penel [20,21,22] and independently by Miller [23] using somewhat different methods. Subsequently, Miller also proved regularity criteria that only require control of the velocity, vorticity, or the positive part of the second eigenvalue of the strain matrix in the sum space of two scale critical spaces [24]. In particular, in [23], the author posed the question of whether it is possible to establish the NSE regularity criterion based solely on one entry of the strain tensor $S_{ij}$. This paper does not fully answer this question; however, we will demonstrate several regularity criteria using one row of the strain tensor in certain critical spaces.

First, we will prove the following.

Theorem 1. 

Let $u_0\in H^1$, and suppose that u be a weak solution in $(0,T)$ of NSE (1) if one row of the strain tensor $S=S_{ij}\equiv {\displaystyle \frac{1}{2}}\left({\partial }_i{u}_j+{\partial }_j{u}_i\right)(1\le i,j\le 3)$ satisfies

$$S_{3j}\in L^{{\displaystyle \frac{2}{2-r}}}\left(0,T;{\dot{B}}_{\infty ,\infty }^{-r}\right),r\in (0,1),j=1,2,3.$$

(8)

Then,

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

and thus, $u\in C^{\infty }\left((0,T]\times {\mathbb{R}}^3\right)$.

Remark 1. 

Theorem 1 addresses the case of the strain tensor component, and it partially answers the question raised by [23]. In addition, due to the embedding

$$L^{{\displaystyle \frac{3}{r}}}\left({\mathbb{R}}^3\right)\hookrightarrow {\dot{B}}_{\infty ,\infty }^{-r}\left({\mathbb{R}}^3\right),$$

with $0<r\le {\displaystyle \frac{3}{2}}$, we know that the condition (8) improves and develops the Ladyzhenskaya–Prodi–Serrin criteria.

Next, we will observe that we can also articulate the regularity criterion for one row of the strain tensor $S_{3j}$ in terms of the sum space $L_T^p{L}_x^q+L_T^1{L}_x^{\infty }$.

Theorem 2. 

Let $u_0\in H^1$, and suppose that u be a weak solution in $(0,T)$ of NSE (1) with initial data $u_0$ if one row of the strain tensor $S_{3j}\equiv {\displaystyle \frac{1}{2}}\left({\partial }_3{u}_j+{\partial }_j{u}_3\right)=\widehat{S}+\check{S}(1\le j\le 3)$ satisfies

$${\int }_0^T\parallel \widehat{S}{\left(t\right)\parallel }_{L^p}^{q}dt+{\int }_0^T{\parallel \check{S}\left(t\right)\parallel }_{L^{\infty }}dt<\infty ,{\displaystyle \frac{2}{q}}+{\displaystyle \frac{3}{p}}=2,{\displaystyle \frac{3}{2}}<p<\infty .$$

(9)

Then, u is regular.

Remark 2. 

We will note that this is a significant advance because the regularity criterion in the sum space $L_T^p{L}_x^q+L_T^1{L}_x^{\infty }$ contains within it the whole family of regularity criteria in the spaces $L_T^{p^{\prime }}{L}_x^{q^{\prime }},$ where ${\displaystyle \frac{2}{p^{\prime }}}+{\displaystyle \frac{3}{q^{\prime }}}=2,$ and $q\le q^{\prime }\le +\infty .$ This theorem is a generalization of the Navier–Stokes regularity criteria in sum spaces in [24].

Finally, we will present the regularity criteria for the 3D NSE (1) on the framework of mixed-norm Lorentz spaces. This approach is particularly motivated by the physical interpretation that fluid behavior can vary in different directions. Consequently, understanding the solutions of the NSE in anisotropic functional spaces appears to be a topic of independent interest.

Theorem 3. 

Let $u_0\in H^1$, and suppose that u be a weak solution in $(0,T)$ of NSE (1) with initial data $u_0$. If one row of the strain tensor $S=S_{ij}\equiv {\displaystyle \frac{1}{2}}\left({\partial }_i{u}_j+{\partial }_j{u}_i\right)(1\le i,j\le 3)$ satisfies

$${\int }_0^T{∥{∥{∥S_{3j}∥}_{L_{x_1}^{p,\infty }}∥}_{L_{x_2}^{q,\infty }}∥}_{L_{x_3}^{r,\infty }}^{{\displaystyle \frac{2}{2-\left(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\right)}}}dt<\infty ,$$

(10)

where $2<p,q,r\le \infty$ with $1-\left({\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{r}}\right)\ge 0$, then the solution u is regular on $(0,T]$.

Remark 3. 

Theorem 3 can be considered a Serrin-type criterion that involves only one row of the strain tensor. In contrast, our global regularity criterion pertains to the components of the strain tensor, which scale like $\nabla u$. From this perspective, all of our results are optimal.

Remark 4. 

To the best of our knowledge, Theorem 3 is the first result that proves the regularity criterion by imposing one row of the strain tensor in the mixed-norm Lorentz space.

Remark 5. 

All the spaces mentioned in Theorem 1, Theorem 2, and Theorem 3 are scaling-invariant spaces under the natural 3D Navier–Stokes scaling. Furthermore, it appears that a slightly modified version of the techniques used in the aforementioned theorems can be applied to other incompressible fluid equations, such as the fractional NSE and the micropolar fluid equations.

2. Definitions and Notations

Before proceeding with the proofs of our results, we need to define several spaces. First, we will introduce some notation. Let $\mathcal{S}\left({\mathbb{R}}^3\right)$ denote the Schwartz class of rapidly decreasing functions. Given $f\in \mathcal{S}\left({\mathbb{R}}^3\right)$, its Fourier transform $\mathcal{F}f=\widehat{f}$ is defined as

$$\widehat{f}\left(\xi \right)={\int }_{{\mathbb{R}}^3}f\left(x\right)e^{-ix·\xi }dx.$$

Let $(\chi ,\phi )$ be a pair of smooth functions taking values in the interval $[0,1]$, where $\chi$ is supported in the set $B=\{\xi \in {\mathbb{R}}^3:|\xi |\le {\displaystyle \frac{4}{3}}\}$, and $\phi$ is supported in $\mathcal{C}=\{\xi \in {\mathbb{R}}^3:{\displaystyle \frac{3}{4}}\le |\xi |\le {\displaystyle \frac{8}{3}}\}$ such that

$$\chi \left(\xi \right)+\sum _{j\ge 0}\phi \left(2^{-j}\xi \right)=1,\forall \xi \in {\mathbb{R}}^3,$$

$$\sum _{j\in Z}\phi \left(2^{-j}\xi \right)=1,\forall \xi \in {\mathbb{R}}^3\setminus \left\{0\right\}.$$

Denote $h={\mathcal{F}}^{-1}\phi$ and $\tilde{h}={\mathcal{F}}^{-1}\chi$. We then define the homogeneous dyadic blocks ${\dot{\Delta }}_j$ and the homogeneous low-frequency cut-off operator ${\dot{S}}_j$ as follows:

$${\dot{\Delta }}_{j}u=\phi \left(2^{-j}D\right)u=2^{3j}{\int }_{{\mathbb{R}}^3}h\left(2^{j}y\right)u(x-y)dy,$$

and

$${\dot{S}}_{j}u=\chi \left(2^{-j}D\right)=2^{3j}{\int }_{{\mathbb{R}}^3}\tilde{h}\left(2^{j}y\right)u(x-y)dy.$$

Definition 1 

(Homogeneous Besov spaces [25]). Let

$$S_h^{\prime }=\left\{f\in S^{\prime };\underset{j\to -\infty }{lim}{∥{\dot{S}}_{j}f∥}_{\infty }=0\right\}.$$

The homogeneous Besov space ${\dot{B}}_{p,r}^s$ is defined as

$${\dot{B}}_{p,r}^s=\left\{f\in S_h^{\prime };{\parallel f\parallel }_{{\dot{B}}_{p,r}^s}<\infty \right\}$$

where

$$\left\{\begin{array}{cc}& {\parallel f\parallel }_{{\dot{B}}_{p,r}^s}={\left(\sum _{j\in Z}{2}^{jsr}{∥{\dot{\Delta }}_{j}f∥}_p^r\right)}^{{\displaystyle \frac{1}{r}}},\mathit{if}r<\infty ,\hfill \\ & {\parallel f\parallel }_{{\dot{B}}_{p,r}^s}=\underset{j\in Z}{sup}{2}^{js}{∥{\dot{\Delta }}_{j}f∥}_p,r=\infty .\hfill \end{array}\right.$$

Lemma 1 

([26]). Let $0<r<1$. For $f\in {\dot{B}}_{\infty ,\infty }^{-r}\left({\mathbb{R}}^3\right)$ and $g,h\in H^1\left({\mathbb{R}}^3\right)$, then there exists a constant $C>0$ such that

$${\int }_{{\mathbb{R}}^3}fghdx\le C{\parallel f\parallel }_{{\dot{B}}_{\infty }^{-r},\infty }\left({\parallel g\parallel }_{L^2}{\parallel h\parallel }_{L^2}^{1-r}{\parallel \nabla h\parallel }_{L^2}^r+{\parallel h\parallel }_{L^2}{\parallel g\parallel }_{L^2}^{1-r}{\parallel \nabla g\parallel }_{L^2}^r\right).$$

We will further define the Sobolev spaces.

Definition 2. 

For all $\alpha \in \mathbb{R},$ let

$${\parallel u\parallel }_{H^{\alpha }}^2={\int }_{{\mathbb{R}}^3}{\left(1+{\left|\xi \right|}^2\right)}^{\alpha }{\left|\widehat{u}\left(\xi \right)\right|}^{2}d\xi ,$$

(11)

and let

$$H^{\alpha }\left({\mathbb{R}}^3\right)=\left\{u\in {\mathcal{S}}^{\prime }\left({\mathbb{R}}^3\right):{\parallel u\parallel }_{H^{\alpha }}<+\infty \right\},$$

(12)

where ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^3\right)$ is the space of tempered distributions.

We have defined the space $H^{\alpha }\left({\mathbb{R}}^3\right)$; now, we will define the next space ${\dot{H}}^{\alpha }\left({\mathbb{R}}^3\right)$.

Definition 3. 

For all $\alpha \in \mathbb{R},$ let

$${\parallel u\parallel }_{{\dot{H}}^{\alpha }}^2={\int }_{{\mathbb{R}}^3}{\left|\xi \right|}^{2\alpha }{\left|\widehat{u}\left(\xi \right)\right|}^{2}d\xi ,$$

(13)

and let

$${\dot{H}}^{\alpha }\left({\mathbb{R}}^3\right)=\left\{u\in {\mathcal{S}}^{\prime }\left({\mathbb{R}}^3\right):{\parallel u\parallel }_{{\dot{H}}^{\alpha }}<+\infty \right\}.$$

(14)

Next, we define the sum spaces, which plays an essential role our in Theorem 2.

Definition 4 

(Lebesgue sum spaces). Let X and Y be Banach spaces, and let V be a vector space such that $X,Y\subset V.$ Then,

$$X+Y=\left\{x+y:x\in X,y\in Y\right\}.$$

(15)

Furthermore, $X+Y$ is a Banach space with norm

$${\parallel f\parallel }_{X+Y}=\underset{g+h=f}{inf}{\parallel g\parallel }_X+{\parallel h\parallel }_Y.$$

(16)

We now introduce Lorentz spaces and mixed-norm Lorentz spaces.

Given a measurable function $f:{\mathbb{R}}^n\to \mathbb{R}$ on X, its distribution function $d_f$ defined on $[0,\infty )$ is as follows:

$$d_f\left(\alpha \right)=\mu \left(\right\{x\in X:\left|f\left(x\right)\right|>\alpha \left\}\right).$$

We now define its decreasing rearrangement $f^{*}:[0,\infty )\to [0,\infty ]$ as

$$f^{*}\left(t\right)=inf\left\{\alpha :d_f\left(\alpha \right)\le t\right\},$$

with the convention that $inf\varnothing =\infty$. The point of this definition is that f and $f^{*}$ have the same distribution function,

$$d_{f^{*}}\left(\alpha \right)=d_f\left(\alpha \right),$$

but $f^{*}$ is a positive non-increasing scalar function.

Definition 5 

(Lorentz spaces). Let $(p,q)\in {[1,\infty ]}^2$ be the Lorentz space $L^{p,q}\left({\mathbb{R}}^3\right)$ that consists of all measurable functions f for which the quantity

$${\parallel f\parallel }_{L^{p,q}}:=\left\{\begin{array}{cc}{\left({\int }_0^{\infty }{\left[t^{{\displaystyle \frac{1}{p}}}{f}^{*}\left(t\right)\right]}^q{\displaystyle \frac{\mathrm{d}t}{t}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill & q<\infty ,\hfill \\ \underset{0<t<\infty }{sup}{t}^{{\displaystyle \frac{1}{p}}}{f}^{*}\left(t\right)\hfill & q=\infty ,\hfill \end{array}\right.$$

is finite.

In order to lead the following definition involving anisotropic Lorentz space, we denote $f=f(x_1,x_2,x_3)$ as a measurable function defined on ${\mathbb{R}}^3$, $f^{*}\left(t\right)=f^{{*}_1,{*}_2,{*}_3}\left(t_1,t_2,t_3\right)$.

Definition 6 

(Mixed-norm Lorentz spaces). Let multi-indexes $p=(p_1,p_2,p_3),q=(q_1,q_2,q_3)$ be such that if $0<p_i<\infty$, then $0<q_i\le \infty$, and if $p_i=\infty$, then $q_i=\infty$ for every $i=1,2,3$. An anisotropic Lorentz space $L^{p_1,q_1}\left({\mathbb{R}}_{x_1};L^{p_2,q_2}\left({\mathbb{R}}_{x_2};L^{p_3,q_3}\left({\mathbb{R}}_{x_3}\right)\right)\right)$ is the set of functions for which the following norm is finite:

$${∥{∥{∥f∥}_{L_{x_1}^{p_1,q_1}}∥}_{L_{x_2}^{p_2,q_2}}∥}_{L_{x_3}^{p_3,q_3}}:={\left({\int }_0^{\infty }{\left({\int }_0^{\infty }{\left({\int }_0^{\infty }{\left[t_1^{{\displaystyle \frac{1}{p_1}}}{t}_2^{{\displaystyle \frac{1}{p_2}}}{t}_3^{{\displaystyle \frac{1}{p_3}}}{f}^{{*}_1,{*}_2,{*}_3}(t_1,t_2,t_3)\right]}^{q_1}{\displaystyle \frac{\mathrm{d}{t}_1}{t_1}}\right)}^{{\displaystyle \frac{q_2}{q_1}}}{\displaystyle \frac{\mathrm{d}{t}_2}{t_2}}\right)}^{{\displaystyle \frac{q_3}{q_2}}}{\displaystyle \frac{\mathrm{d}{t}_3}{t_3}}\right)}^{{\displaystyle \frac{1}{q_3}}}.$$

Lemma 2. 

There exists a positive constant C such that

$${∥{∥{∥f∥}_{L_{x_1}^{{\displaystyle \frac{2p}{p-2}},2}}∥}_{L_{x_2}^{{\displaystyle \frac{2q}{q-2}},2}}∥}_{L_{x_3}^{{\displaystyle \frac{2r}{r-2}},2}}\le C{∥{\partial }_{1}f∥}_{L^2}^{{\displaystyle \frac{1}{p}}}{∥{\partial }_{2}f∥}_{L^2}^{{\displaystyle \frac{1}{q}}}{∥{\partial }_{3}f∥}_{L^2}^{{\displaystyle \frac{1}{r}}}{\parallel f\parallel }_{L^2}^{1-\left({\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{r}}\right)},$$

(17)

for every $f\in C_0^{\infty }\left({\mathbb{R}}^3\right)$, where $2<p,q,r\le \infty ,1-\left({\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{r}}\right)\ge 0$.

Proof. 

Let ${\mathsf{\Lambda }}_1^p$ be the Fourier multiplier defined as

$${\mathcal{F}}_1\left({\mathsf{\Lambda }}_1^{p}f\right)\left({\xi }_1,x_2,x_3\right)={\left|{\xi }_1\right|}^p{\mathcal{F}}_{1}f\left({\xi }_1,x_2,x_3\right)$$

with

$${\mathcal{F}}_{1}f\left({\xi }_1,x_2,x_3\right)={\int }_{\mathbb{R}}{e}^{-i{\xi }_1{x}_1}f\left(x_1,x_2,x_3\right)\mathrm{d}{x}_1,$$

${\mathsf{\Lambda }}_2^p$ and ${\mathsf{\Lambda }}_3^p$ can be defined analogously. Then, through the Sobolev’s embedding theorem, the Minkowski’s inequality and the Hölder’s inequality to obtain

$$\begin{array}{cc}\hfill {∥{∥{∥f∥}_{L_{x_1}^{{\displaystyle \frac{2p}{p-2}},2}}∥}_{L_{x_2}^{{\displaystyle \frac{2q}{q-2}},2}}∥}_{L_{x_3}^{{\displaystyle \frac{2r}{r-2}},2}}& \le C{∥{∥{∥{\mathsf{\Lambda }}_1^{{\displaystyle \frac{1}{p}}}f∥}_{L_{x_1}^2}∥}_{L_{x_2}^{{\displaystyle \frac{2q}{q-2}},2}}∥}_{L_{x_3}^{{\displaystyle \frac{2r}{r-2}},2}}\le {∥{∥{∥{\mathsf{\Lambda }}_1^{{\displaystyle \frac{1}{p}}}f∥}_{L_{x_2}^{{\displaystyle \frac{2q}{q-2}},2}}∥}_{L_{x_1}^2}∥}_{L_{x_3}^{{\displaystyle \frac{2r}{r-2}},2}}\hfill \\ \hfill & \le C{∥{∥{\mathsf{\Lambda }}_2^{{\displaystyle \frac{1}{q}}}{\mathsf{\Lambda }}_1^{{\displaystyle \frac{1}{p}}}f∥}_{L_{x_1,x_2}^2}∥}_{L_{x_3}^{{\displaystyle \frac{2r}{r-2}},2}}\le C{∥{∥{\mathsf{\Lambda }}_2^{{\displaystyle \frac{1}{q}}}{\mathsf{\Lambda }}_1^{{\displaystyle \frac{1}{p}}}f∥}_{L_{x_3}^{{\displaystyle \frac{2r}{r-2}},2}}∥}_{L_{x_1,x_2}^2}\hfill \\ \hfill & \le C{∥{\mathsf{\Lambda }}_3^{{\displaystyle \frac{1}{r}}}{\mathsf{\Lambda }}_2^{{\displaystyle \frac{1}{q}}}{\mathsf{\Lambda }}_1^{{\displaystyle \frac{1}{p}}}f∥}_{L^2}.\hfill \end{array}$$

(18)

Combining the Fourier–Plancherel formula and the Hölder’s inequality, we have

$$\begin{array}{cc}\hfill & C{∥{\mathsf{\Lambda }}_3^{{\displaystyle \frac{1}{r}}}{\mathsf{\Lambda }}_2^{{\displaystyle \frac{1}{q}}}{\mathsf{\Lambda }}_1^{{\displaystyle \frac{1}{p}}}f∥}_{L^2}\le C{\left({\int }_{{\mathbb{R}}^3}{\left|{\xi }_1\right|}^{{\displaystyle \frac{2}{p}}}{\left|{\xi }_2\right|}^{{\displaystyle \frac{2}{q}}}{\left|{\xi }_3\right|}^{{\displaystyle \frac{2}{r}}}{\left|\mathcal{F}f\left({\xi }_1,{\xi }_2,{\xi }_3\right)\right|}^2\mathrm{d}{\xi }_1\mathrm{d}{\xi }_2\mathrm{d}{\xi }_3\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & =C{\left({\int }_{{\mathbb{R}}^3}{\left|{\xi }_1\right|}^{{\displaystyle \frac{2}{p}}}{\left|\mathcal{F}f\left(\xi \right)\right|}^{{\displaystyle \frac{2}{p}}}{\left|{\xi }_2\right|}^{{\displaystyle \frac{2}{q}}}{\left|\mathcal{F}f\left(\xi \right)\right|}^{{\displaystyle \frac{2}{q}}}{\left|{\xi }_3\right|}^{{\displaystyle \frac{2}{r}}}{\left|\mathcal{F}f\left(\xi \right)\right|}^{{\displaystyle \frac{2}{r}}}{\left|\mathcal{F}f\left(\xi \right)\right|}^{2-\left({\displaystyle \frac{2}{p}}+{\displaystyle \frac{2}{q}}+{\displaystyle \frac{2}{r}}\right)}\mathrm{d}{\xi }_1\mathrm{d}{\xi }_2\mathrm{d}{\xi }_3\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le {C\parallel \mathcal{F}f\parallel }_{L^2}^{1-{\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{q}}-{\displaystyle \frac{1}{r}}}{\left({\int }_{{\mathbb{R}}^3}{\left|{\xi }_1\right|}^2{\left|\mathcal{F}f\right|}^2\mathrm{d}\xi \right)}^{{\displaystyle \frac{1}{2p}}}{\left({\int }_{{\mathbb{R}}^3}{\left|{\xi }_2\right|}^2{\left|\mathcal{F}f\right|}^2\mathrm{d}\xi \right)}^{{\displaystyle \frac{1}{2q}}}{\left({\int }_{{\mathbb{R}}^3}{\left|{\xi }_3\right|}^2{\left|\mathcal{F}f\right|}^2\mathrm{d}\xi \right)}^{{\displaystyle \frac{1}{2r}}}\hfill \\ \hfill & \le C{∥{\partial }_{1}f∥}_{L^2}^{{\displaystyle \frac{1}{p}}}{∥{\partial }_{2}f∥}_{L^2}^{{\displaystyle \frac{1}{q}}}{∥{\partial }_{3}f∥}_{L^2}^{{\displaystyle \frac{1}{r}}}{\parallel f\parallel }_{L^2}^{1-\left({\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{r}}\right)}.\hfill \end{array}$$

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□

3. Proof of Theorem 1

In this section, we will consider the regularity criteria for one row of strain tensors in critical Besov spaces.

3.1. $\parallel {\nabla }_h{u\parallel }_{L^2}$ Estimates

First, we obtain some estimates of the horizontal gradient. Taking the inner product of NSE (1) with $-{\Delta }_{h}u$ in $L^2\left({\mathbb{R}}^3\right)$, one has

$$\begin{array}{cc}& {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel {\nabla }_h{u\parallel }_{L^2}^2+{\parallel \nabla {\nabla }_{h}u\parallel }_{L^2}^2\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^3}\left[(u·\nabla )u\right]·{\Delta }_{h}udx\hfill \\ \hfill =& -{\int }_{{\mathbb{R}}^3}\sum _{i,j=1}^3\sum _{k=1}^2{\partial }_k{u}_i{\partial }_i{u}_j{\partial }_k{u}_{j}dx\hfill \\ \hfill =& -\left({\int }_{{\mathbb{R}}^3}\sum _{i=1}^3\sum _{j=1}^2\sum _{k=1}^2{\partial }_k{u}_i{\partial }_i{u}_j{\partial }_k{u}_{j}dx+{\int }_{{\mathbb{R}}^3}\sum _{i=1}^3\sum _{k=1}^2{\partial }_k{u}_i{\partial }_i{u}_3{\partial }_k{u}_{3}dx\right)\hfill \\ \hfill =& -({\int }_{{\mathbb{R}}^3}\sum _{i=1}^2\sum _{j=1}^2\sum _{k=1}^2{\partial }_k{u}_i{\partial }_i{u}_j{\partial }_k{u}_{j}dx+{\int }_{{\mathbb{R}}^3}\sum _{j=1}^2\sum _{k=1}^2{\partial }_k{u}_3{\partial }_3{u}_j{\partial }_k{u}_{j}dx\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^3}\sum _{i=1}^2\sum _{k=1}^2{\partial }_k{u}_i{\partial }_i{u}_3{\partial }_k{u}_{3}dx+{\int }_{{\mathbb{R}}^3}\sum _{k=1}^2{\partial }_k{u}_3{\partial }_3{u}_3{\partial }_k{u}_{3}dx)\hfill \\ \hfill =& I\left(t\right)+J\left(t\right)+K\left(t\right)+L\left(t\right).\hfill \end{array}$$

(20)

Next, we will estimate the right-hand side of the above term $I\left(t\right)$, $J\left(t\right)$, $K\left(t\right)$, and $L\left(t\right)$ by using the incompressibility condition and Lemma 1.

$$\begin{array}{cc}\hfill I\left(t\right)=& -{\int }_{{\mathbb{R}}^3}\sum _{i=1}^2\sum _{j=1}^2\sum _{k=1}^2{\partial }_k{u}_i{\partial }_i{u}_j{\partial }_k{u}_{j}dx\hfill \\ \hfill =& -\left({\int }_{{\mathbb{R}}^3}\sum _{i=1}^2\sum _{j=1}^2{\partial }_1{u}_i{\partial }_i{u}_j{\partial }_1{u}_{j}dx+{\int }_{{\mathbb{R}}^3}\sum _{i=1}^2\sum _{j=1}^2{\partial }_2{u}_i{\partial }_i{u}_j{\partial }_2{u}_{j}dx\right)\hfill \\ \hfill =& -({\int }_{{\mathbb{R}}^3}\sum _{i=1}^2{\partial }_1{u}_i{\partial }_i{u}_1{\partial }_1{u}_{1}dx+{\int }_{{\mathbb{R}}^3}\sum _{i=1}^2{\partial }_1{u}_i{\partial }_i{u}_2{\partial }_1{u}_{2}dx\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^3}\sum _{i=1}^2{\partial }_2{u}_i{\partial }_i{u}_1{\partial }_2{u}_{1}dx+{\int }_{{\mathbb{R}}^3}\sum _{i=1}^2{\partial }_2{u}_i{\partial }_i{u}_2{\partial }_2{u}_{2}dx)\hfill \\ \hfill =& -({\int }_{{\mathbb{R}}^3}{\left({\partial }_1{u}_1\right)}^3+{\left({\partial }_2{u}_2\right)}^{3}dx+{\int }_{{\mathbb{R}}^3}({\partial }_1{u}_1+{\partial }_2{u}_2){\left({\partial }_1{u}_2\right)}^{2}dx\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^3}({\partial }_1{u}_1+{\partial }_2{u}_2){\partial }_1{u}_2{\partial }_2{u}_{1}dx+{\int }_{{\mathbb{R}}^3}({\partial }_1{u}_1+{\partial }_2{u}_2){\partial }_1{u}_2{\partial }_2{u}_{1}dx)\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^3}{\partial }_3{u}_3\left({\left({\partial }_1{u}_1\right)}^2-{\partial }_1{u}_1{\partial }_2{u}_2+{\left({\partial }_2{u}_2\right)}^2\right)dx+{\int }_{{\mathbb{R}}^3}{\partial }_3{u}_3{\left({\partial }_1{u}_2\right)}^{2}dx\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^3}{\partial }_3{u}_3{\partial }_1{u}_2{\partial }_2{u}_{1}dx+{\int }_{{\mathbb{R}}^3}{\partial }_3{u}_3{\partial }_1{u}_2{\partial }_2{u}_{1}dx\hfill \\ \hfill \le & C{\int }_{{\mathbb{R}}^3}|S_{33}\left|\right|{\nabla }_h{u|}^{2}dx\hfill \\ \hfill \le & C\parallel S_{33}{\parallel }_{{\dot{B}}_{\infty ,\infty }^{-r}}\parallel {\nabla }_h{u\parallel }_{L^2}^{2-r}{\parallel \nabla {\nabla }_{h}u\parallel }_{L^2}^r\hfill \\ \hfill \le & C\parallel S_{33}{\parallel }_{{\dot{B}}_{\infty ,\infty }^{-r}}^{{\displaystyle \frac{2}{2-r}}}\parallel {\nabla }_h{u\parallel }_{L^2}^2+ϵ{\parallel \nabla {\nabla }_{h}u\parallel }_{L^2}^2.\hfill \end{array}$$

(21)

As to $J\left(t\right)+K\left(t\right)$, we take the same trick as $I\left(t\right)$ to obtain

$$\begin{array}{cc}\hfill J\left(t\right)+K\left(t\right)=& -\left({\int }_{{\mathbb{R}}^3}\sum _{k=1}^2\sum _{j=1}^2{\partial }_k{u}_3{\partial }_3{u}_j{\partial }_k{u}_{j}dx+{\int }_{{\mathbb{R}}^3}\sum _{k=1}^2\sum _{i=1}^2{\partial }_k{u}_i{\partial }_i{u}_3{\partial }_k{u}_{3}dx\right)\hfill \\ \hfill =& -\left({\int }_{{\mathbb{R}}^3}\sum _{k=1}^2{\partial }_k{u}_3{\partial }_k{u}_1({\partial }_3{u}_1+{\partial }_1{u}_3)+\sum _{k=1}^2{\partial }_k{u}_3{\partial }_k{u}_2({\partial }_3{u}_2+{\partial }_2{u}_3)dx\right)\hfill \\ \hfill \le & C{\int }_{{\mathbb{R}}^3}|S_{31}+S_{32}\left|\right|{\nabla }_h{u|}^{2}dx\hfill \\ \hfill \le & C\parallel S_{3j}{\parallel }_{{\dot{B}}_{\infty ,\infty }^{-r}}\parallel {\nabla }_h{u\parallel }_{L^2}^{2-r}{\parallel \nabla {\nabla }_{h}u\parallel }_{L^2}^r\hfill \\ \hfill \le & C\parallel S_{3j}{\parallel }_{{\dot{B}}_{\infty ,\infty }^{-r}}^{{\displaystyle \frac{2}{2-r}}}\parallel {\nabla }_h{u\parallel }_{L^2}^2+ϵ{\parallel \nabla {\nabla }_{h}u\parallel }_{L^2}^2,j=1,2.\hfill \end{array}$$

(22)

and

$$\begin{array}{cc}\hfill L\left(t\right)\le & C\parallel S_{33}{\parallel }_{{\dot{B}}_{\infty ,\infty }^{-r}}^{{\displaystyle \frac{2}{2-r}}}\parallel {\nabla }_h{u\parallel }_{L^2}^2+ϵ{\parallel \nabla {\nabla }_{h}u\parallel }_{L^2}^2.\hfill \end{array}$$

(23)

Plugging estimates (21)–(23) into (20) give us

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel {\nabla }_h{u\parallel }_{L^2}^2+{\displaystyle \frac{1}{2}}{\parallel {\nabla }_h\nabla u\parallel }_{L^2}^2\le & C\parallel S_{3j}{\parallel }_{{\dot{B}}_{\infty ,\infty }^{-r}}^{{\displaystyle \frac{2}{2-r}}}{\parallel {\nabla }_{h}u\parallel }_{L^2}^2,j=1,2,3.\hfill \end{array}$$

(24)

Gronwall’s inequality implies that

$$\parallel {\nabla }_h{u\parallel }_{L^2}^2+{\int }_0^T{\parallel {\nabla }_h\nabla u\parallel }_{L^2}^{2}d\tau \le C.$$

(25)

3.2. ${\parallel \nabla u\parallel }_{L^2}$ Estimates

Taking the inner product of the Equation (1) with $-\Delta u$ in $L^2$, one obtains

$$\begin{array}{cc}& {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel \nabla u\parallel }_{L^2}^2+{\parallel \Delta u\parallel }_{L^2}^2\hfill \\ \hfill =& \sum _{i,j,k=1}^3{\int }_{{\mathbb{R}}^3}{u}_i{\partial }_i{u}_j{\partial }_{kk}{u}_{j}dx\hfill \\ \hfill =& \sum _{j=1}^3{\int }_{{\mathbb{R}}^3}{u}_3{\partial }_3{u}_j{\Delta }_h{u}_{j}dx+\sum _{i=1}^2\sum _{j=1}^3{\int }_{{\mathbb{R}}^3}{u}_i{\partial }_i{u}_j\Delta u_{j}dx+\sum _{j=1}^3{\int }_{{\mathbb{R}}^3}{u}_3{\partial }_3{u}_j{\partial }_{33}{u}_{j}dx\hfill \\ \hfill =& M_1\left(t\right)+M_2\left(t\right)+M_3\left(t\right).\hfill \end{array}$$

(26)

The proof of the bound (26) has been shown in [17] and concludes that

$$\begin{array}{cc}\hfill |M_i\left(t\right)|& {\displaystyle \le C{\int }_{{\mathbb{R}}^3}|{\nabla }_h{u\left|\right|\nabla u|}^{2}dx}\hfill \\ \hfill & \le C\parallel {\nabla }_h{u\parallel }_{L^2}{\parallel \nabla u\parallel }_{L^4}^2\hfill \\ \hfill & \le C\parallel {\nabla }_h{u\parallel }_{L^2}{\parallel \nabla u\parallel }_{L^2}^{{\displaystyle \frac{1}{2}}}\parallel {\nabla }_h{\nabla u\parallel }_{L^2}{\parallel \Delta u\parallel }_{L^2}^{{\displaystyle \frac{1}{2}}},i=1,2,3.\hfill \end{array}$$

(27)

Gathering the above two estimates, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel \nabla u\parallel }_{L^2}^2+{\parallel \Delta u\parallel }_{L^2}^2& \le C\parallel {\nabla }_h{u\parallel }_{L^2}{\parallel \nabla u\parallel }_{L^2}^{{\displaystyle \frac{1}{2}}}\parallel {\nabla }_h{\nabla u\parallel }_{L^2}{\parallel \Delta u\parallel }_{L^2}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le ϵ\parallel {\nabla }_h{u\parallel }_{L^2}^{{\displaystyle \frac{4}{3}}}{\parallel \Delta u\parallel }_{L^2}^2+C{\parallel \nabla u\parallel }_{L^2}^2.\hfill \end{array}$$

(28)

Hence,

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel \nabla u\parallel }_{L^2}^2+{\displaystyle \frac{1}{2}}{\parallel \Delta u\parallel }_{L^2}^2& \le {C\parallel \nabla u\parallel }_{L^2}^2.\hfill \end{array}$$

(29)

Theorem 1 follows from the Grönwall inequality.

4. The Proof of Theorem 2

In this section, we will prove Theorem 2. Based on the proofs of Theorem 1, it is sufficient to demonstrate that there exists a positive constant K such that

$${\int }_0^T\parallel \widehat{S}{\left(t\right)\parallel }_{L^p}^{q}dt+{\int }_0^T{\parallel \check{S}\left(t\right)\parallel }_{L^{\infty }}dt\le K,\mathrm{with}{\displaystyle \frac{2}{q}}+{\displaystyle \frac{3}{p}}=2\mathrm{and}{\displaystyle \frac{3}{2}}<p<\infty ,$$

and then we derive

$${\nabla }_{h}u\in L^{\infty }(0,T;L^2\left({\mathbb{R}}^3\right))\cap L^2(0,T;H^1\left({\mathbb{R}}^3\right)).$$

Recall that in Theorem 1, the ${\nabla }_{h}u$ satisfies

$$\begin{array}{cc}& {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel {\nabla }_h{u\parallel }_{L^2}^2+{\parallel {\nabla }_h\nabla u\parallel }_{L^2}^2\hfill \\ \hfill \le & C{\int }_{{\mathbb{R}}^3}|S_{3j}\left|\right|{\nabla }_h{u|}^{2}dx(j=1,2,3)\hfill \\ \hfill =& C{\int }_{{\mathbb{R}}^3}{\left|{\nabla }_{h}u\right|}^2(\widehat{S}+\check{S})dx\hfill \\ \hfill \le & C\parallel \widehat{S}{\parallel }_{L^p}{∥{\nabla }_{h}u∥}_{L^{{\displaystyle \frac{2p}{p-1}}}}^2+C\parallel \breve{S}{\parallel }_{L^{\infty }}\parallel {∥{\nabla }_{h}u∥}_{L^2}^2\hfill \\ \hfill \le & C\parallel \widehat{S}{\parallel }_{L^p}{∥{\nabla }_{h}u∥}_{L^2}^{2-{\displaystyle \frac{3}{p}}}{∥\nabla {\nabla }_{h}u∥}_{L^2}^{{\displaystyle \frac{3}{p}}}+C\parallel \check{S}{\parallel }_{L^{\infty }}\parallel {∥{\nabla }_{h}u∥}_{L^2}^2\hfill \\ \hfill \le & C\parallel \widehat{S}{\parallel }_{L^p}^q{∥{\nabla }_{h}u∥}_{L^2}^2+C\parallel \check{S}{\parallel }_{L^{\infty }}\parallel {∥{\nabla }_{h}u∥}_{L^2}^2+ϵ{\parallel \nabla {\nabla }_{h}u\parallel }_{L^2}^2,\hfill \end{array}$$

(30)

where $q={\displaystyle \frac{2p}{2p-3}},{\displaystyle \frac{3}{2}}<p<\infty$. Therefore, we can conclude that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel {\nabla }_h{u\parallel }_{L^2}^2+{\parallel {\nabla }_h\nabla u\parallel }_{L^2}^2\le C\left(\parallel \widehat{S}{\parallel }_{L^p}^q+{\parallel \check{S}\parallel }_{L^{\infty }}\right){∥{\nabla }_{h}u∥}_{L^2}^2.\end{array}$$

(31)

Applying Grönwall’s inequality, this implies that

$${\nabla }_{h}u\in L^{\infty }(0,T;L^2\left({\mathbb{R}}^3\right))\cap L^2(0,T;H^1\left({\mathbb{R}}^3\right)),$$

so this completes the proof of Theorem 2.

5. Proof of Theorem 3

The proof of Theorem 3 is not significantly different from the previous one. We begin with Equation (30), applying Hölder’s inequality and Young’s inequality, and combining these with Lemma 2 yields that

$$\begin{array}{cc}& {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel {\nabla }_h{u\parallel }_{L^2}^2+{\parallel {\nabla }_h\nabla u\parallel }_{L^2}^2\hfill \\ \hfill \le & C{\int }_{{\mathbb{R}}^3}|S_{3j}\left|\right|{\nabla }_h{u|}^{2}dx(j=1,2,3)\hfill \\ \hfill \le & C{∥{∥{∥S_{3j}∥}_{L_{x_1}^{p,\infty }}∥}_{L_{x_2}^{q,\infty }}∥}_{L_{x_3}^{r,\infty }}{∥{∥{∥{\nabla }_{h}u∥}_{L_{x_1}^{{\displaystyle \frac{2p}{p-2}},2}}∥}_{L_{x_2}^{{\displaystyle \frac{2q}{q-2}},2}}∥}_{L_{x_3}^{{\displaystyle \frac{2r}{r-2}},2}}{∥{\nabla }_{h}u∥}_{L^2}\hfill \\ \hfill \le & C{∥{∥{∥S_{3j}∥}_{L_{x_1}^{p,\infty }}∥}_{L_{x_2}^{q,\infty }}∥}_{L_{x_3}^{r,\infty }}{∥{\partial }_1{\nabla }_{h}u∥}_{L^2}^{{\displaystyle \frac{1}{p}}}{∥{\partial }_2{\nabla }_{h}u∥}_{L^2}^{{\displaystyle \frac{1}{q}}}{∥{\partial }_3{\nabla }_{h}u∥}_{L^2}^{{\displaystyle \frac{1}{r}}}\parallel {\nabla }_h{u\parallel }_{L^2}^{1-\left({\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{r}}\right)}{\parallel {\nabla }_{h}u\parallel }_{L^2}\hfill \\ \hfill \le & C{∥{∥{∥S_{3j}∥}_{L_{x_1}^{p,\infty }}∥}_{L_{x_2}^{q,\infty }}∥}_{L_{x_3}^{r,\infty }}{∥\nabla {\nabla }_{h}u∥}_{L^2}^{{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{r}}}{\parallel {\nabla }_{h}u\parallel }_{L^2}^{2-\left({\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{r}}\right)}\hfill \\ \hfill \le & C{∥{∥{∥S_{3j}∥}_{L_{x_1}^{p,\infty }}∥}_{L_{x_2}^{q,\infty }}∥}_{L_{x_3}^{r,\infty }}^{{\displaystyle \frac{2}{2-\left(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\right)}}}\parallel {\nabla }_h{u\parallel }_{L^2}^2+ϵ{\parallel \nabla {\nabla }_{h}u\parallel }_{L^2}^2.\hfill \end{array}$$

(32)

Next, integrating in time interval $(0,T]$ and applying Gronwall’s lemma gives

$$\begin{array}{c}\hfill \parallel {\nabla }_h{u\left(t\right)\parallel }_{L^2}^2+{\int }_0^T\nabla {\nabla }_{h}u\left(t\right)dt\le {\parallel {\nabla }_h{u}_0\parallel }_{L^2}^{2}expC{\int }_0^T{∥{∥{∥S_{3j}∥}_{L_{x_1}^{p,\infty }}∥}_{L_{x_2}^{q,\infty }}∥}_{L_{x_3}^{r,\infty }}^{{\displaystyle \frac{2}{2-\left(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\right)}}}dt.\end{array}$$

(33)

The same arguments as in the conclusion of the previous theorem lead to the desired result. Theorem 3 is proved.

Discussion and Conclusions

As is well known in fluid mechanics, velocity describes the advection of a fluid, vorticity characterizes its rotation, and strain quantifies the deformation of a fluid parcel. Consequently, the strain tensor is a crucial physical quantity for understanding fluid behavior. From a mathematical perspective, our results partially address the questions raised by Miller in [23]. Furthermore, our findings suggest that the strain tensor S plays a less significant role than the individual components of the strain tensor in the regularity theory of solutions to the incompressible Navier–Stokes equations. In a certain sense, our results represent a preliminary step toward resolving Miller’s open problem.