Global Regularity Criterion for the 3D Incompressible Navier–Stokes Equations Involving theVelocity Partial Derivative

where u andp denote the velocity field and the pressure, respectively, and u0(x) is the initial fluid which satisfied ∇ · u0 � 0. ,e existence of weak solutions of N-S equations was proved by Leray [1] and Hopf [2]. However, the existence of 3D global regular solutions is still an open question. Prodi [3] and Serrin [4] first considered the regularity of solutions. ,ey, respectively, proved that the weak solution of 3D N-S equations is regular when the exponents p and q satisfy


Introduction and the Main Result
is paper focuses on the following three-dimensional incompressible Navier-Stokes (N-S) equations: where u and p denote the velocity field and the pressure, respectively, and u 0 (x) is the initial fluid which satisfied ∇ · u 0 � 0. e existence of weak solutions of N-S equations was proved by Leray [1] and Hopf [2]. However, the existence of 3D global regular solutions is still an open question. Prodi [3] and Serrin [4] first considered the regularity of solutions. ey, respectively, proved that the weak solution of 3D N-S equations is regular when the exponents p and q satisfy u ∈ L p 0, T; L q R 3 � L p t L q x , In 1995, Veiga [5] generalized the result to ∇u ∈ L p t L q x , When ∇u, ∇u 3 , u 3 , and the like satisfy a certain integrable condition, the weak solution is regular, and a large number of results are obtained (for details, refer to [6][7][8][9][10][11][12][13][14][15][16]). And Penel and Pokorný [13], Kukavica and Ziane [14], Cao [15], and Zhang [16], respectively, proved that the weak solution is regular on (0, T] when the weak solution satisfies the following conditions: Recently, Zhang, Yuan, and Zhou in [17] proved if or the weak solution is regular on(0, T]. Very recently, LI and Dong in [18] proved if or the weak solution is regular on(0, T]. One of our main tasks is that when the value of the equation is as constant as possible, we expand the range of q to make the value of q as small as possible. Second, when the range of q does not shrink, the equation is tried to enlarge. Inspired by the texts [14,17], we get better results than the above [17] as follows.
then the weak solution is regular on (0, T].
then the weak solution is regular on (0, T].

Remark 1.
Comparing (9) and (7) and (10) and (8), respectively, the value of q is reduced, and within the range of q, the value of the equation is larger than that, so it is better. And when q obtains the minimum value, respectively, each equation takes the critical value of 2, so this condition is the optimal critical value. However, we find that, in the existing results, the maximum value of q is bounded when the equality value can reach 2. e highlight of this article is that the equation value goes to 2, and the q value reaches infinity. When the value of the equation reached 2, it is a difficult problem to prove the regularity of the weak solutions to the 3D incompressible N-S equations by adding the value of q getting to infinity and getting to the current minimum of (3 �� 37 √ /4) − 3. We hope we can overcome this problem in the near future.
We shall give the proof of our main result in the third part. In order to facilitate reading, we will give the necessary preparatory knowledge in the following section.

Preliminaries
roughout this text, C stands for a generic positive constant which may differ in value from one line to another. We use ‖·‖ L p to denote the norm of the Lebesgue space L p (1 ≤ p ≤ ∞) as follows: Definition 1 (see [19]). Assume u 0 ∈ L 2 (R 3 ) and ∇ · u 0 � 0, T > 0. e measurable function u defined on (0, T] × R 3 is called the weak solution of equation (1) if and equation (1) holds in the sense of distributions.
2 Mathematical Problems in Engineering

Proof of Main Results
In this part, we give the proof of main results. In order to prove eorems 1 and 2, we thank the results in [13]. In [13], Penel and Pokorný showed that if then the weak solution of the N-S equations is regular on (0, T]. For arbitrary (15/4) ≤ q ≤ ∞ and (9/4) ≤ q ≤ ∞, we have respectively.

(19)
Hence, they just have to prove that and in both cases, u 3 ∈ L (2s/(s−3)) (0, T; L s (R 3 )), 3 < s ≤ ∞, is established. Taking the inner product for u 3 by |u 3 |u 3 and integrating over R 3 , we have where we use ∇ · u � 0 and the following identities: Proof of eorem 1. We estimate the right side of (21). By using the Hölder inequality, the Young inequality, (14), (16), and (17), we get that Substituting (23) into (21), we have Dividing both sides by ‖u 3 ‖ 2 L 3 and integrating with respect to t imply that We deduce from (9) and (14) that It is available from type (23) that So, by the embedding inequality, we get So, eorem 1 is proved.

□
Proof of eorem 2. We have another estimation for the right side of (21). By using the Hölder inequality, the Young inequality, (14), (15), and (17), we have Inserting (29) into (21), one has Dividing both sides by ‖u 3 ‖ L 3 and integrating with respect to t imply that By (10) and (14), we have e same can be proved: So, eorem 2 is proved.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.