A note on optimal decay rates for the axisymmetric D-solutions to the steady Navier-Stokes equations

Abstract

In this paper, we investigate the decay properties of an axisymmetric D-solutions to stationary incompressible Navier-Stokes systems in ${\mathbb{R}}^3$. We obtain the optimal decay rate $|\mathbf{u}(x)|\le \frac{C}{|x|+1}$ for axisymmetric flows without swirl. Furthermore, we find a dichotomy for the decay rates of the swirl component $u_{\theta }$, that is, either $O(\frac{1}{r+1})\le |u_{\theta }(r,z)|\le \frac{C\log (r+1)}{(r+1{)}^{1/2}}$ or $|u_{\theta }(r,z)|\le \frac{Cr}{(\rho +1{)}^3}$, where $\rho =\sqrt{r^2+z^2}$. In the latter case, we can further deduce that the other two components of the velocity field also attain the optimal decay rates: $|u_r(r,z)|+|u_z(r,z)|\le \frac{C}{\rho +1}$. We do not require any small assumptions on the forcing term.

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References in corpus (1)

• Liouville Type Theorem for Stationary Navier-Stokes Equations