Regularity criterion for 3D Navier-Stokes equations in terms of the\n direction of the velocity

2007/05/16 by Alexis Vasseur, Vasseur, Alexis
Engineering · Mathematics · #35B65 #76D03 #76D05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.0705.2446

openalex publication_date 2007/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this short note, we give a link between the regularity of the solution u\nto the 3D Navier-Stokes equation, and the behavior of the direction of the\nvelocity u/|u|. It is shown that the control of Div (u/|u|) in a suitable\nL_t^p(L_x^q) norm is enough to ensure global regularity. The result is\nreminiscent of the criterion in terms of the direction of the vorticity,\nintroduced first by Constantin and Fefferman. But in this case the condition is\nnot on the vorticity, but on the velocity itself. The proof, based on very\nstandard methods, relies on a straightforward relation between the divergence\nof the direction of the velocity and the growth of energy along streamlines.\n

Citations

• Sur le mouvement d'un liquide visqueux emplissant l'espace

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