Research Article | Open Access
TianLi Li, Wen Wang, "Global Regularity Criterion for the 3D Incompressible Navier–Stokes Equations Involving the Velocity Partial Derivative", Mathematical Problems in Engineering, vol. 2021, Article ID 5568180, 5 pages, 2021. https://doi.org/10.1155/2021/5568180
Global Regularity Criterion for the 3D Incompressible Navier–Stokes Equations Involving the Velocity Partial Derivative
In this paper, we study the regularity of the weak solutions for the incompressible 3D Navier–Stokes equations with the partial derivative of the velocity. By the embedded technology, we prove that the weak solution is regular on (0, T] if with , or .
1. Introduction and the Main Result
This paper focuses on the following three-dimensional incompressible Navier–Stokes (N-S) equations:where denote the velocity field and the pressure, respectively, and is the initial fluid which satisfied .
The existence of weak solutions of N-S equations was proved by Leray  and Hopf . However, the existence of 3D global regular solutions is still an open question. Prodi  and Serrin  first considered the regularity of solutions. They, respectively, proved that the weak solution of 3D N-S equations is regular when the exponents and satisfy
In 1995, Veiga  generalized the result to
When , 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 [6–16]). And Penel and Pokorný , Kukavica and Ziane , Cao , and Zhang , respectively, proved that the weak solution is regular on when the weak solution satisfies the following conditions:
Recently, Zhang, Yuan, and Zhou in  proved iforthe weak solution is regular on.
Very recently, LI and Dong in  proved iforthe weak solution is regular on.
One of our main tasks is that when the value of the equation is as constant as possible, we expand the range of to make the value of as small as possible. Second, when the range of does not shrink, the equation is tried to enlarge. Inspired by the texts [14, 17], we get better results than the above  as follows.
Theorem 1. Assume and . Let be a weak solution of N-S equations (1) on which satisfies the initial value of . Ifthen the weak solution is regular on .
Theorem 2. Assume and . Let be a weak solution of N-S equations (1) on which satisfies the initial value of . Ifthen the weak solution is regular on .
Remark 1. Comparing (9) and (7) and (10) and (8), respectively, the value of is reduced, and within the range of , the value of the equation is larger than that, so it is better. And when 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 is bounded when the equality value can reach 2. The highlight of this article is that the equation value goes to 2, and the 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 getting to infinity and getting to the current minimum of . 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.
Throughout this text, C stands for a generic positive constant which may differ in value from one line to another. We use to denote the norm of the Lebesgue space as follows:
Definition 1. (see ). Assume and . The measurable function defined on is called the weak solution of equation (1) if(1).(2) and satisfy and equation (1) holds in the sense of distributions. For and satisfy where .(3)The strong energy inequality, that is,
Lemma 1. (see ).
Lemma 2. (Sobolev embedding inequality).
3. Proof of Main Results
In this part, we give the proof of main results. In order to prove Theorems 1 and 2, we thank the results in . In , Penel and Pokorný showed that ifthen the weak solution of the N-S equations is regular on .
For arbitrary and , we have
Hence, they just have to prove thatand in both cases, , is established.
Taking the inner product for by and integrating over , we havewhere we use and the following identities:
Proof of Theorem 1. We estimate the right side of (21). By using the Hölder inequality, the Young inequality, (14), (16), and (17), we get thatSubstituting (23) into (21), we haveDividing both sides by and integrating with respect to imply thatWe deduce from (9) and (14) thatIt is available from type (23) thatSo, by the embedding inequality, we getSo, Theorem 1 is proved.
Proof of Theorem 2. We have another estimation for the right side of (21). By using the Hölder inequality, the Young inequality, (14), (15), and (17), we haveInserting (29) into (21), one hasDividing both sides by and integrating with respect to imply thatBy (10) and (14), we haveThe same can be proved:So, Theorem 2 is proved.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Li was supported by the Anhui Provincial Natural Science Fund Project (KJ2018B0002). Wang was supported by the NSF of Anhui Province (1908085QA04).
- L. Sur, “Le mouvement dun liquide visqueux emplissant lespace,” Acta Mathematica, vol. 63, pp. 193–248, 1934, (In French).
- E. Hopf, “Uber die Anfangswertaufgabe Fur die hydrodynamischen Grundgleichungen,” Mathematische Nachrichten, vol. 4, pp. 213–231, 1951, (In German).
- G. Prodi, “Un teorema di unicità per le equazioni di Navier-Stokes,” Annali di Matematica Pura ed Applicata, vol. 48, no. 1, pp. 173–182, 1959.
- J. Serrin, “The initial value problems for the Navier-Stokes,” in Nonlinear Problems, R. E. Langer, Ed., 1963.
- B.-D. Veiga, “A new regularity class for the Navier-Stokes equations,” Chinese Annals of Mathematics, Series B, vol. 407, pp. 1–6, 1995.
- M. Pokorn, “On the result of He concerning the smoothness of solutions to the Navier-Stokes equations,” Electronic Journal of Differential Equations, vol. 11, p. 1, 2003.
- Y. Zhou, “A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component,” Methods and Applications of Analysis, vol. 9, no. 4, pp. 563–578, 2002.
- Y. Zhou, “A new regularity criterion for weak solutions to the Navier-Stokes equations,” Journal de Mathématiques Pures et Appliquées, vol. 84, no. 11, pp. 1496–1514.
- C. Cao and E. S. Titi, “Regularity criteria for the three-dimensional Navier-Stokes equations,” Indiana University Mathematics Journal, vol. 57, no. 6, pp. 2643–2662, 2008.
- B.-Q. Dong and Z.-M. Chen, “Regularity criterion of weak solutions to the 3D Navier-Stokes equations via two velocity components,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 1–10, 2008.
- J. Wolf, “A regularity criterion of Serrin-type for the Navier-Stokes equations involving the gradient of one velocity component,” Analysis, vol. 5, pp. 259–292, 2015.
- Z. Guo, M. Caggio, and Z. Skalák, “Regularity criteria for the Navier-Stokes equations based on one component of velocity,” Nonlinear Analysis: Real World Applications, vol. 35, pp. 379–396, 2017.
- P. Penel and M. Pokorný, “Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity,” Applications of Mathematics, vol. 49, no. 5, pp. 483–493, 2004.
- I. Kukavica and M. Ziane, “Navier-Stokes equations with regularity in one direction,” Journal of Mathematical Physics, p. 48, 2007.
- C. Cao, “Sufficient conditions for the regularity to the 3D Navier-Stokes equations,” Discrete & Continuous Dynamical Systems - A, vol. 26, no. 4, pp. 1141–1151, 2010.
- Z. Zhang, “An improved regularity criterion for the Navier-Stokes equations in terms of one directional derivative of the velocity field,” Bulletin of Mathematical Sciences, vol. 8, no. 1, pp. 33–47, 2018.
- Z. Zhang, W. Yuan, and Y. Zhou, “Some remarks on the Navier-Stokes equations with regularity in one direction,” Applications of Mathematics, vol. 64, no. 3, pp. 301–308, 2019.
- T.-L. LI and B.-Q. Dong, “Global regularity criterion for the 3D incompressible Navier-Stokes equations,” College Mathematics, vol. 36, pp. 1–6, 2020, (in Chinese).
- P.-G. Lemarie-Rieussset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall, London, 2002.
Copyright © 2021 TianLi Li and Wen Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.