Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 256, Issue 1, Pages 283-309Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2013.09.002
Keywords
Regularity criterions; Anisotropic inequality; Bony paraproduct decomposition; Leray-Hopf weak solution
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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 [GRAPHICS] , or [GRAPHICS] . 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 [GRAPHICS] . 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 [GRAPHICS] , or [GRAPHICS] , which fill the gap of endpoint alpha = infinity. (C) 2013 Elsevier Inc. All rights reserved.
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