Decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations

Abstract

We investigate the decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations. The achievements of this paper are two folds. One is improved decay rates of $u_{þ}$ and $\na {\bf u}$, especially we show that $|u_þ(r,z)|\leq c\left(\f{\log r}{r}\right)^{\f 12}$ for any smooth axially symmetric D-solutions to the Navier-Stokes equations. These improvement are based on improved weighted estimates of $\om_þ$, integral representations of $\mathbf{u}$ in terms of $\bm{\om}=\textit{curl }{\bf u}$ and $A_p$ weight for singular integral operators, which yields good decay estimates for $(\na u_r, \na u_z)$ and $(\om_r, \om_{z})$, where $\bm{\om}= \om_r {\bf e}_r + \om_þ {\bf e}_þ+ \om_z {\bf e}_z$. Another is the first decay rate estimates in the $Oz$-direction for smooth axially symmetric flows without swirl. We do not need any small assumptions on the forcing term.

We improved the decay rates. 22 pages

Cited by in corpus (2)

• Decay and vanishing of some axially symmetric D-solutions of the Navier-Stokes equations
• A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang