A regularity criterion for the tridimensional Navier-Stokes equations in term of one velocity component

作者：Zheng Xiaoxin^*

来源：Journal of Differential Equations, 2014, 256(1): 283-309.

DOI：10.1016/j.jde.2013.09.002

• Regularity criterions
• Anisotropic inequality
• Bony paraproduct decomposition
• Leray-Hopf weak solution

In this paper, we investigate the regularity criterion of the tridimensional Navier-Stokes equations via one velocity component. Our strategy is to establish the following version of regularity criterions of Leray-Hopf weak solutions in the framework of anisotropic Lebesgue space
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or
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This allows us to obtain regularity criterion of Leray-Hopf weak solutions via only one element Lambda(gamma)(i) u(j) with gamma is an element of [0, 1] and i, j is an element of {1, 2, 3}, that is
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Here F-1, F-2 and F are the sets of indexes (alpha, beta) which appear in our results and the fractional operator Lambda(i) := root-partial derivative(2)(i). This extends and improves some known regularity criterions of Leray-Hopf weak solutions in term of one velocity component, including the notable works of C. Cao and E.S. Titi [4]. More importantly, by making full use of the Bony paraproduct decomposition, we show that Leray-Hopf weak solutions are smooth on [0, T] if
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or
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which fill the gap of endpoint alpha = infinity.

• 出版日期2014-1-1
• 单位中国工程物理研究院