On a regularity criterion for the Navier–Stokes equations involving gradient of one velocity component

We improve the regularity criterion for the incompressible Navier–Stokes equations in the full three-dimensional space involving the gradient of one velocity component. The method is based on recent results of Cao and Titi [see “Regularity criteria for the three dimensional Navier–Stokes equations,”...

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Bibliographic Details
Published in:Journal of mathematical physics 2009-12, Vol.50 (12), p.123514-123514-11
Main Authors: Zhou, Yong, Pokorný, Milan
Format: Article
Language:English
Subjects:
Differential equations
Exact sciences and technology
Mathematical methods in physics
Mathematics
Navier-Stokes equations
Physics
Sciences and techniques of general use
Velocity
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Summary:We improve the regularity criterion for the incompressible Navier–Stokes equations in the full three-dimensional space involving the gradient of one velocity component. The method is based on recent results of Cao and Titi [see “Regularity criteria for the three dimensional Navier–Stokes equations,” Indiana Univ. Math. J. 57, 2643 (2008)] and Kukavica and Ziane [see “Navier-Stokes equations with regularity in one direction,” J. Math. Phys. 48, 065203 (2007)]. In particular, for s ∊ [ 2 , 3 ] , we get that the solution is regular if ∇ u 3 ∊ L t ( 0 , T ; L s ( R 3 ) ) , 2 / t + 3 / s ≤ 23 12 .
ISSN:0022-2488
1089-7658
DOI:10.1063/1.3268589