Abstract
In this paper we establish regularity conditions for the three dimensional incompressible Navier-Stokes equations in terms of one entry of the velocity gradient tensor, say for example, \( \partial _{3}u_{3}\). We show that if \(\partial _{3}u_{3}\) satisfies certain integrable conditions with respect to time and space variables in anisotropic Lebesgue spaces, then a Leray-Hopf weak solution is actually regular. The anisotropic Lebesgue space helps us to almost reach the Prodi-Serrin level 2 in certain special case. Moreover, regularity conditions on non-diagonal element of gradient tensor \(\partial _1 u_3\) are also established, which covers some previous literature.
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Acknowledgements
The authors would like to thank the anonymous referee for his/her very helpful comments on the initial version of this manuscript which make this paper more readable. The first author was partially supported by National Natural Science Foundation of China, under Grant No. 11301394, and China Postdoctoral Science Foundation, under Grant Nos. 2017M620149 and 2018T110387. The third author was supported by the Grant Agency of the Czech republic through Grant 18-09628S and by the Czech Academy of Sciences through RVO: 67985874.
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Guo, Z., Li, Y. & Skalák, Z. Regularity Criteria of The Incompressible Navier-Stokes Equations via Only One Entry of Velocity Gradient. J. Math. Fluid Mech. 21, 35 (2019). https://doi.org/10.1007/s00021-019-0441-6
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DOI: https://doi.org/10.1007/s00021-019-0441-6