Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity - Mathematics > Analysis of PDEsReport as inadecuate




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Abstract: In this short note, we give a link between the regularity of the solution $u$to the 3D Navier-Stokes equation, and the behavior of the direction of thevelocity $u-|u|$. It is shown that the control of $\Div (u-|u|)$ in a suitable$L t^p(L x^q)$ norm is enough to ensure global regularity. The result isreminiscent of the criterion in terms of the direction of the vorticity,introduced first by Constantin and Fefferman. But in this case the condition isnot on the vorticity, but on the velocity itself. The proof, based on verystandard methods, relies on a straightforward relation between the divergenceof the direction of the velocity and the growth of energy along streamlines.



Author: Alexis Vasseur

Source: https://arxiv.org/







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