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