# Decay Properties of Axially Symmetric D-Solutions to the Steady Navier–Stokes Equations

Decay Properties of Axially Symmetric D-Solutions to the Steady Navier–Stokes Equations 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 $$u_{\theta }$$ u θ and $$\nabla \mathbf{u}$$ ∇ u , especially we show that $$|u_{\theta }(r,z)|\le c(\frac{\log r}{r})^{\frac{1}{2}}$$ | u θ ( r , z ) | ≤ c ( log r r ) 1 2 for any smooth axially symmetric D-solutions to the Navier–Stokes equations. These improvement are based on improved weighted estimates of $$\omega _{\theta }$$ ω θ and $$A_p$$ A p weight for singular integral operators, which yields good decay estimates for $$(\nabla u_r, \nabla u_z)$$ ( ∇ u r , ∇ u z ) and $$(\omega _r, \omega _{z})$$ ( ω r , ω z ) , where $${\varvec{\omega }}=\textit{curl }{} \mathbf{u}= \omega _r \mathbf{e}_r + \omega _{\theta } \mathbf{e}_{\theta }+ \omega _z \mathbf{e}_z$$ ω = curl u = ω r e r + ω θ e θ + ω z 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Fluid Mechanics Springer Journals

# Decay Properties of Axially Symmetric D-Solutions to the Steady Navier–Stokes Equations

, Volume 20 (1) – Jan 5, 2017
19 pages

Publisher
Springer International Publishing
Subject
Physics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Classical and Continuum Physics
ISSN
1422-6928
eISSN
1422-6952
D.O.I.
10.1007/s00021-016-0310-5
Publisher site
See Article on Publisher Site

### 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 $$u_{\theta }$$ u θ and $$\nabla \mathbf{u}$$ ∇ u , especially we show that $$|u_{\theta }(r,z)|\le c(\frac{\log r}{r})^{\frac{1}{2}}$$ | u θ ( r , z ) | ≤ c ( log r r ) 1 2 for any smooth axially symmetric D-solutions to the Navier–Stokes equations. These improvement are based on improved weighted estimates of $$\omega _{\theta }$$ ω θ and $$A_p$$ A p weight for singular integral operators, which yields good decay estimates for $$(\nabla u_r, \nabla u_z)$$ ( ∇ u r , ∇ u z ) and $$(\omega _r, \omega _{z})$$ ( ω r , ω z ) , where $${\varvec{\omega }}=\textit{curl }{} \mathbf{u}= \omega _r \mathbf{e}_r + \omega _{\theta } \mathbf{e}_{\theta }+ \omega _z \mathbf{e}_z$$ ω = curl u = ω r e r + ω θ e θ + ω z 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.

### Journal

Journal of Mathematical Fluid MechanicsSpringer Journals

Published: Jan 5, 2017

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