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 $$|v(x, t)| \le \frac{C |\ln r|^{1/2}}{r^2}, \qquad 0 < r \le 1/2,$$ where r 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 (Chiun-Chuan et al., Commun PDE 34(1–3):203–232, 2009; Koch et al., Acta Math 203(1):83–105, 2009) gave a lower bound. The gap is polynomial order 1 modulo a half log term.