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Register a new userA note on optimal decay rates for the axisymmetric D-solutions to the steady Navier-Stokes equations
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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 $|{bf u}(x)|leq frac{C}{|x|+1}$ for axisymmetric flows without swirl. Furthermore, we find a dichotomy for the decay rates of the swirl component $u_{theta}$, that is, either $O(frac{1}{r+1})leq |u_{theta}(r,z)|leq frac{Clog(r+1)}{(r+1)^{1/2}}$ or $|u_{theta}(r,z)|leq frac{C r}{(rho+1)^3}$, 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: $|u_r(r,z)|+ |u_{z}(r,z)|leq frac{C}{rho+1}$. We do not require any small assumptions on the forcing term.
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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 $u_{th}$ and $ a {bf u}$, especially we show that $|u_{th}(r,z)|leq cleft(f{log r}{r}right)^{f 12}$ for any smooth axially symmetric D-solutions to the Navier-Stokes equations. These improvement are based on improved weighted estimates of $om_{th}$, integral representations of ${bf u}$ in terms of $bm{om}=textit{curl }{bf u}$ and $A_p$ weight for singular integral operators, which yields good decay estimates for $( a u_r, a u_z)$ and $(om_r, om_{z})$, where $bm{om}= om_r {bf e}_r + om_{th} {bf e}_{th}+ om_z {bf e}_z$. 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.
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