A priori bound on the velocity in axially symmetric Navier-Stokes equations

2013/09/25 by Zhen Lei, Lei, Zhen, Esteban A. Navas +3
Mathematics · #35B07 #35Q30 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1309.6625

openalex publication_date 2013/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let v be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. Under suitable conditions for initial values, we prove the following a priori bound |v(x, t)| ≤ (C)/(r²) |ln r|^1/2,where r ∈ (0, 1/2) is the distance from x to the z axis, and C is a constant depending only on the initial value. This provides a pointwise upper bound (worst case scenario) for possible singularities while the recent papers \citeCSTY2 and \citeKNSS gave a lower bound. The gap is polynomial order 1 modulo a half log term.

Citations

• Liouville theorems for the Navier–Stokes equations and applications

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