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

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 \[ |v(x, t)| \le \frac{C}{r^2} |\ln r|^{1/2}, \]where $r\in (0,1/2)$ 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 \cite{CSTY2} and \cite{KNSS} gave a lower bound. The gap is polynomial order 1 modulo a half log term.

corrected and updated version following referee's suggestions. arXiv admin note: text overlap with arXiv:0811.1609

Cited by in corpus (2)

• Criticality of the Axially Symmetric Navier-Stokes Equations
• Regularity Criterion to the axially symmetric Navier-Stokes Equations