# Decay properties of Axially Symmetric D-solutions to the Steady
Incompressible Magnetohydrodynamic Equations

Shangkun Weng School of mathematics and statistics, Wuhan University, Wuhan,
Hubei Province, 430072, People’s Republic of China. Email<EMAIL_ADDRESS>Yan Zhou School of mathematics and statistics, Wuhan University, Wuhan, Hubei
Province, 430072, People’s Republic of China. Email<EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate the decay properties of axially symmetric
solutions to the steady incompressible magnetohydrodynamic equations in
$\mathbb{R}^{3}$ with finite Dirichlet integral. We first derive the decay
rates of general D-solutions to the axisymmetric MHD equations. In the special
case where the magnetic field only has the swirl component ${\bf
h}(r,z)=h_{\theta}(r,z){\bf e_{\theta}}$, we obtain better decay rates. The
last result examines the decay rates along the axis $Oz$ also within the
special class of D-solutions with only swirl magnetic field. The main tool in
this paper is the combination of the scaling argument, the Brezis-Gallouet
inequality and the weighted energy estimate.

Mathematics Subject Classifications 2010: Primary 35Q35; Secondary 76W05.

Key words: Magnetohydrodynamics equations, decay rate, axially symmetric,
scaling argument, weighted energy estimates.

##  1 Introduction

In this paper, we investigate the decay properties of axially symmetric
solutions to the steady Magnetohydrodynamics equations (MHD), which are listed
as follows.

$\left\\{\begin{array}[]{l}\textbf{u}\cdot\nabla\textbf{u}+\nabla
p=\textbf{h}\cdot\nabla\textbf{h}+\triangle\textbf{u}+\textbf{f},\\\
\textbf{u}\cdot\nabla\textbf{h}-\textbf{h}\cdot\nabla\textbf{u}=\triangle\textbf{h}+\rm{curl}\,\textbf{g},\\\
\nabla\cdot\textbf{u}=\nabla\cdot\textbf{h}=0,\end{array}\right.$ (1.1)

with the additional condition at infinity

$\displaystyle\lim\limits_{|x|\to\infty}\textbf{u}(x)=\lim\limits_{|x|\to\infty}\textbf{h}(x)=0,$
(1.2)

and the finite Dirichlet integral

$\int_{\mathbb{R}^{3}}|\nabla\textbf{u}(x)|^{2}+|\nabla{\bf
h}(x)|^{2}dx<+\infty.$ (1.3)

Here $\bf{u}$ is the velocity field, $\bf{h}$ is the magnetic field, $p$ is
the pressure, $\bf{f}$ and $\nabla\times{\bf g}$ are the external forces on
the magnetically charged fluid flows. Set $\bar{\textbf{g}}\coloneqq{\rm
curl\,}\textbf{g}$ and $\bar{\textbf{f}}\coloneqq{\rm curl\,}\textbf{f}$ in
the following. It is well known that Leray [25] constructed the weak solutions
to the stationary Navier-Stokes equations with no-slip boundary conditions and
constant velocity at infinity. Leray’s solution has finite Dirichlet integral,
and is usually refereed as D-solution. One may refer to the reference [12, 13,
14, 15, 20] for different construction methods and solutions in different
kinds of function spaces. One can show that any weak solution to (1.1)
satisfying (1.2) and (1.3) is smooth, by following the argument developed in
[16].

Recently there are several papers working on decay rate estimates for the
axisymmetric D-solutions to the steady Navier-Stokes equations. Our goal here
is to extend the results in [3, 10, 35] to the axisymmetric steady MHD
equations. Note that the weighted energy method developed in [10, 35] needs to
use the special structure of the vorticity equations, such an extension may be
quite nontrivial, although the steady MHD equations has the same scaling as
the steady Navier-Stokes equations, the scaling technique developed in [3]
still works in the MHD setting.

Let us first introduce the cylindrical coordinate

$\displaystyle
r=\sqrt{x^{2}_{1}+x_{2}^{2}},\quad\tan\theta=\frac{x_{2}}{x_{1}},\quad
z=x_{3},$

and the basis vectors $\textbf{e}_{r}$, $\textbf{e}_{\theta}$,
$\textbf{e}_{z}$ are

$\displaystyle{\textbf{e}_{r}}=(\cos\,\theta,\sin\,\theta,0),\quad{\textbf{e}_{\theta}}=(-\sin\,\theta,\cos\,\theta,0),\quad{\textbf{e}_{z}}=(0,0,1).$

For convenience, we write $\textbf{x}^{\prime}=(x_{1},x_{2})$ and $x_{3}=z.$

We call a function $f$ is axially symmetric if it does not depend on $\theta$.
A vector-valued function $\textbf{u}=(u_{r},u_{\theta},u_{z})$ is called
axially symmetric if $u_{r},\,u_{\theta},\,u_{z}$ do not depend on $\theta$.
And we call a vector-valued function $\textbf{u}=(u_{r},u_{\theta},u_{z})$ is
axially symmetric without swirl if $u_{\theta}=0$ while $u_{r}$ and $u_{z}$ do
not depend on $\theta$.

Assume that the vector field $\textbf{u}({\bf
x})=u_{r}(r,z){\textbf{e}_{r}}+u_{\theta}(r,z){\textbf{e}_{\theta}}+u_{z}(r,z){\textbf{e}_{z}}$
and the magnetic field $\textbf{h}({\bf
x})=h_{r}(r,z){\textbf{e}_{r}}+h_{\theta}(r,z){\textbf{e}_{\theta}}+h_{z}(r,z){\textbf{e}_{z}}$
is the axially symmetric solution of (1.1), then we have the following
asymmetric steady MHD equations,

$\left\\{\begin{array}[]{l}(u_{r}\partial_{r}+u_{z}\partial_{z})u_{r}-\frac{u^{2}_{\theta}}{r}+\partial_{r}p=(h_{r}\partial_{r}+h_{z}\partial_{z})h_{r}-\frac{h^{2}_{\theta}}{r}+(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z}-\frac{1}{r^{2}})u_{r}+f_{r},\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})u_{\theta}+\frac{u_{r}u_{\theta}}{r}=(h_{r}\partial_{r}+h_{z}\partial_{z})h_{\theta}+\frac{h_{r}h_{\theta}}{r}+(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z}-\frac{1}{r^{2}})u_{\theta}+f_{\theta},\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})u_{z}+\partial_{z}p=(h_{r}\partial_{r}+h_{z}\partial_{z})h_{z}+(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z})u_{z}+f_{z},\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})h_{r}=(h_{r}\partial_{r}+h_{z}\partial_{z})u_{r}+(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z}-\frac{1}{r^{2}})h_{r}-\partial_{z}g_{\theta},\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})h_{\theta}+\frac{h_{r}u_{\theta}}{r}=(h_{r}\partial_{r}+h_{z}\partial_{z})u_{\theta}+\frac{u_{r}h_{\theta}}{r}+(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z}-\frac{1}{r^{2}})h_{\theta}+(\partial_{z}g_{r}-\partial_{r}g_{z}),\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})h_{z}=(h_{r}\partial_{r}+h_{z}\partial_{z})u_{z}+(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z})h_{z}+\frac{1}{r}\partial_{r}(rg_{\theta}),\\\
\partial_{r}u_{r}+\frac{u_{r}}{r}+\partial_{z}u_{z}=0,\\\
\partial_{r}h_{r}+\frac{h_{r}}{r}+\partial_{z}h_{z}=0.\end{array}\right.$
(1.4)

The vorticity ${\bf\omega^{u}}$ and the current density ${\bf\omega^{h}}$ are
defined as ${\bf\omega^{u}}({\bf x})=\nabla\times{\bf
u(x)}=\omega_{r}^{u}(r,z){\textbf{e}_{r}}+\omega_{\theta}^{u}(r,z){\textbf{e}_{\theta}}+\omega_{z}^{u}(r,z){\textbf{e}_{z}}$
and ${\bf\omega^{h}}(x)=\nabla\times{\bf
h(x)}=\omega_{r}^{h}(r,z){\textbf{e}_{r}}+\omega_{\theta}^{h}(r,z){\bf
e_{\theta}}+\omega_{z}^{h}(r,z){\textbf{e}_{z}}$, where

$\displaystyle\omega_{r}^{u}(r,z)=-\partial_{z}u_{\theta},\;\omega_{\theta}^{u}=\partial_{z}u_{r}-\partial_{r}u_{z},\;\omega_{z}^{u}=\frac{1}{r}\partial_{r}(ru_{\theta}),$
$\displaystyle\omega_{r}^{h}(r,z)=-\partial_{z}h_{\theta},\;\omega_{\theta}^{h}=\partial_{z}h_{r}-\partial_{r}h_{z},\;\omega_{z}^{h}=\frac{1}{r}\partial_{r}(rh_{\theta}).$
(1.5)

Then we have

$\displaystyle\left\\{\begin{array}[]{l}(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{r}^{u}-(\omega_{r}^{u}\partial_{r}+\omega_{z}^{u}\partial_{z})u_{r}=(h_{r}\partial_{r}+h_{z}\partial_{z})\omega_{r}^{h}-(\omega_{r}^{h}\partial_{r}+\omega_{z}^{h}\partial_{z})h_{r}\\\
\quad\quad\quad+(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z}-\frac{1}{r^{2}})\omega_{r}^{u}-\partial_{z}f_{\theta},\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{\theta}^{u}-\frac{u_{r}\omega_{\theta}^{u}}{r}+2\frac{u_{\theta}}{r}\omega_{r}^{u}=(h_{r}\partial_{r}+h_{z}\partial_{z})\omega_{\theta}^{h}-\frac{h_{r}\omega_{\theta}^{h}}{r}+2\frac{h_{\theta}}{r}\omega_{r}^{h}\\\
\quad\quad\quad+(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z}-\frac{1}{r^{2}})\omega_{\theta}^{u}+(\partial_{z}f_{r}-\partial_{r}f_{z}),\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{z}^{u}-(\omega_{r}^{u}\partial_{r}+\omega_{z}^{u}\partial_{z})u_{z}=(h_{r}\partial_{r}+h_{z}\partial_{z})\omega_{z}^{h}-(\omega_{r}^{h}\partial_{r}+\omega_{z}^{h}\partial_{z})h_{z}\\\
\quad\quad\quad+(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z})\omega_{z}^{u}+\frac{1}{r}\partial_{r}(rf_{\theta}),\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{r}^{h}-(\omega_{r}^{h}\partial_{r}+\omega_{z}^{h}\partial_{z})u_{r}+\frac{2}{r}\partial_{z}(u_{r}h_{\theta})=(h_{r}\partial_{r}+h_{z}\partial_{z})\omega_{r}^{u}-(\omega_{r}^{u}\partial_{r}+\omega_{z}^{u}\partial_{z})h_{r}\\\
\quad\quad\quad+\frac{2}{r}\partial_{z}(u_{\theta}h_{r})+(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z}-\frac{1}{r^{2}})\omega_{r}^{h}-\partial_{z}(\partial_{z}g_{r}-\partial_{r}g_{z}),\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{\theta}^{h}-\frac{u_{r}}{r}\omega^{h}_{\theta}+2\partial_{z}u_{r}\partial_{r}h_{r}+2\partial_{z}u_{z}\partial_{r}h_{z}=(h_{r}\partial_{r}+h_{z}\partial_{z})\omega_{\theta}^{u}-\frac{h_{r}}{r}\omega_{\theta}^{u}\\\
\quad\quad\quad+2\partial_{z}h_{r}\partial_{r}u_{r}+2\partial_{z}h_{z}\partial_{r}u_{z}+(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z}-\frac{1}{r^{2}})\omega_{\theta}^{h}-\partial_{z}(\partial_{z}g_{\theta})-\partial_{r}(\frac{1}{r}\partial_{r}(rg_{\theta})),\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{z}^{h}+(\partial_{r}u_{r}\partial_{r}+\partial_{r}u_{z}\partial_{z})h_{\theta}+\partial_{r}(\frac{u_{\theta}h_{r}}{r})=(h_{r}\partial_{r}+h_{z}\partial_{z})\omega_{z}^{u}\\\
\quad\quad\quad+(\partial_{r}h_{r}\partial_{r}+\partial_{r}h_{z}\partial_{z})u_{\theta}+\partial_{r}(\frac{u_{r}h_{\theta}}{r})+(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z})\omega_{z}^{h}+\frac{1}{r}\partial_{r}(r(\partial_{z}g_{r}-\partial_{r}g_{z})).\end{array}\right.$
(1.18)

Finn [14] have studied the physically reasonable solutions to the steady
Navier-Stokes with $O(|x|^{-1})$ decay at infinity. Borchers-Miyakawa [1],
Galdi-Simader [17] and Novotny-Padula [30] obtained the existence of such
${\bf u}$ with decay rate $O(|x|^{-1})$ at infinity provided the forcing term
is sufficiently small in different function spaces. Under the assumption that
$|{\bf u}(x)|=O(|x|^{-1})$ as $|x|\to\infty$, Sverak and Tsai [33] had showed
that $|\nabla^{k}{\bf u}(x)|\leq O(|x|^{-k-1})$ for any $k\geq 1$ by employing
a scaling argument and an interior estimate for the Stokes system. In [11],
Deuring and Galdi showed that when these solutions are asymptotically expanded
near infinity, the leading term cannot be the product of a non-zero vector
with the Stokes fundamental solution. One may refer to [29] for a
decomposition of the asymptotic profile at infinity. Korolev and Sverak [23]
showed that Landau’s solution is the right leading term describing the
asymptotic behavior when the solution satisfied some smallness assumptions.

The decay properties of the smooth D-solutions to the steady Navier-Stokes and
MHD equations are closely related to the uniqueness and energy conservation of
the D-solutions. There is a famous open problem which concerns whether ${\bf
u}\equiv 0$ is a unique D-solution to steady Navier-Stokes equations. Galdi
[16] showed that the D-solution must be trivial if ${\bf u}\in
L^{9/2}(\mathbb{R}^{3})$, which was logarithmically improved in [7]. Many
authors have identified different integrability or decay conditions on ${\bf
u}$, which lead to many interesting Liouville type results, one may refer to
[4, 5, 6, 8, 9, 22, 24, 31] for more details. The Liouville theorem in the
case of the axisymmetric three dimensional flows without swirl can be derived
from [21]. Chae [5] explored the maximum principle for the total head
pressure, and proved the triviality of ${\bf u}$ by assuming $\Delta{\bf u}\in
L^{\frac{6}{5}}(\mathbb{R}^{3})$. Seregin [31, 32] applied Caccioppoli type
inequality to show that any smooth solution to the stationary Navier-Stokes
system in $\mathbb{R}^{3}$, belonging to $L^{6}(\mathbb{R}^{3})$ and
$BMO^{-1}$, must be zero. Kozono, Terasawa and Wakasugi [24] proposed a decay
condition on the vorticity to guarantee that ${\bf u}\equiv 0$. For the MHD
system, Liu and Zhang [26] proved the Liouville theorem for the bounded
ancient solution. In [34], it was shown that there are not non-trivial
solutions to MHD equations under the finite Dirichlet integral or the $L^{p}$
integrability of the velocity and magnetic fields. In their proof, the
smallness of the magnetic field plays a vital role.

For the investigation of the decay properties of D-solutions to the exterior
stationary Navier-Stokes equations with large external force, Gilbarg and
Weinberger [18] had made great progress in the two dimensional exterior
domain, showed that the weak solution constructed by Leray [25] was bounded
and converged to a limit $\textbf{u}_{0}$ in a mean square sense, while the
pressure converged pointwise. In [19], they further found that the weak
solution with finite Dirichlet integral may not be bounded, but it must grow
more slowly than $(\ln r)^{1/2}$. By further adapting the ideas in [18, 19] to
the 3D axisymmetric setting, the authors in [10, 35] obtained some decay rates
for smooth axially symmetric solutions to steady Navier-Stokes equations. On
the other hand, it is well-known that if $(\textbf{u},p)$ solves stationary
Navier-Stokes equations, so does $(\textbf{u}_{\lambda},p_{\lambda})$ for all
$\lambda>0$, where
$\textbf{u}_{\lambda}(\textbf{x})=\lambda\textbf{u}(\lambda\textbf{x})$ and
$p_{\lambda}(\textbf{x})=\lambda^{2}p(\lambda\textbf{x})$. According to this,
the authors in [3] utilized the Brezis-Gallouet inequality to improve the
decay rate of the vorticity in [10, 35]. Recently, the decay properties of the
second order derivatives were investigated in [27]. Combining the results in
[3, 10, 35], one has: for $\textbf{x}\in\mathbb{R}^{3}$,

$\displaystyle|\textbf{u}(\textbf{x})|\leq C(\frac{\ln r}{r})^{1/2},\ \ \ \
|\omega_{r}^{u}(\textbf{x})|+|\omega^{u}_{z}(\textbf{x})|\leq C\frac{(\ln
r)^{11/8}}{r^{9/8}},$
$\displaystyle|\omega_{\theta}^{u}(\textbf{x})|+|\nabla\omega_{\theta}^{u}(\textbf{x})|\leq
C\frac{(\ln r)^{3/4}}{r^{5/4}},$ $\displaystyle|\nabla u_{r}|+|\nabla
u_{z}|\leq C\frac{(\ln r)^{7/4}}{r^{5/4}},$

where $C$ is a positive constant.

As was noticed in [3], the derivation of the decay of the velocity itself
almost had no use of the Navier-Stokes equations, it is straightforward to
gain the same decay rates for the magnetic field. However, the estimates of
the vorticity does depend on the structure of the vorticity equations. For the
steady MHD equations, the equations for the electric current $\nabla\times{\bf
h}$ are more complicate than those of the vorticity, therefore the decay rates
of ${\bf\omega^{u}}$ and ${\bf\omega^{h}}$ obtained here are weaker than those
of Navier-Stokes equations. Our first result is stated as follows.

######  Theorem 1.1.

Support $\textbf{f}\in H^{1}(\mathbb{R}^{3})$ and $\textbf{g}\in
H^{2}(\mathbb{R}^{3})$ be axially symmetric vector field with

$\displaystyle\|(\textbf{f},{\rm
curl\,}\textbf{g})\|_{L^{\frac{3}{2}}(\mathbb{R}^{3})}+\|\textbf{f}\|_{H^{1}(\mathbb{R}^{3})}+\|r^{\frac{1}{2}}(\textbf{f},{\rm
curl\,}\textbf{g})\|_{L^{2}(\mathbb{R}^{3})}+\|\textbf{g}\|_{H^{2}(\mathbb{R}^{3})}\leq
M$ (1.19)

for some constant $M$. Let $(\textbf{u},\textbf{h},p)$ be a smooth axially
symmetric solution to the steady MHD equations (1.1) with (1.2)-(1.3), then
there exist a constant $C(M)>0$, such that

$\displaystyle|\textbf{u}|+|\textbf{h}|\leq C(M)(\frac{\ln
r}{r})^{\frac{1}{2}},$ (1.20)
$\displaystyle|\omega_{\theta}^{u}|+|\omega_{\theta}^{h}|\leq C(M)\frac{\ln
r}{r},$ (1.21)
$\displaystyle|\omega_{r}^{u}|+|\omega_{z}^{u}|+|\omega_{r}^{h}|+|\omega_{z}^{h}|\leq
C(M)\frac{\ln r}{r},$ (1.22)

for large $r$.

Next we consider a special class of axisymmetric D-solutions to the steady MHD
equations with $\textbf{h}(r,z)=h_{\theta}(r,z)\bm{e_{\theta}}$. In this
situation, we obtain better decay results for the vorticity field by employing
the Biot-Savart law as in [3]. However, this argument does not improve the
decay rates of $\omega_{r}^{h}$ and $\omega_{z}^{h}$. Here we resort to the
weighted energy estimates in [35] to get a better decay for $\omega_{r}^{h}$
and $\omega_{z}^{h}$.

######  Theorem 1.2.

Support $\textbf{f}\in H^{2}(\mathbb{R}^{3})$ and $\textbf{g}\in
H^{3}(\mathbb{R}^{3})$ be axially symmetric vector field without swirl, and
satisfying (1.19) and

$\displaystyle\|(\frac{r}{\ln
r})^{\frac{1}{2}}f_{\theta}\|_{L^{2}(\mathbb{R}^{3})}+\|r(f_{r},f_{z})\|_{L^{2}(\mathbb{R}^{3})}+\|r^{2}{\rm
curl\,}\emph{{f}}\|_{L^{2}(\mathbb{R}^{3})}\leq M,$ (1.23)
$\displaystyle\|r^{2}\partial_{r}(\partial_{z}g_{r}-\partial_{r}g_{z})\|_{L^{2}(\mathbb{R}^{3})}\leq
M,$ (1.24)

for some constant $M$. Let $(\textbf{u},\textbf{h},p)$ be a smooth axially
symmetric solution to the steady MHD equations (1.1) with (1.2)-(1.3), where
$\textbf{h}(r,z)=h_{\theta}(r,z)\bm{e_{\theta}}$. Then we have better
estimates as follows:

$\displaystyle|\omega_{\theta}^{u}|\leq C(M)\frac{(\ln r)^{3/4}}{r^{5/4}},$
(1.25) $\displaystyle|\omega_{r}^{u}|+|\omega_{z}^{u}|\leq C(M)\frac{(\ln
r)^{11/8}}{r^{9/8}},$ (1.26) $\displaystyle|\nabla u_{r}|+|\nabla u_{z}|\leq
C(M)\frac{(\ln r)^{7/4}}{r^{5/4}},$ (1.27)
$\displaystyle|\omega^{h}_{r}|+|\omega^{h}_{z}|+|\nabla h_{\theta}|\leq
C(M)r^{-(\frac{35}{32})^{-}},$ (1.28)

where $a^{-}$ denotes any number less than $a$.

The last result examines the decay rate in the $Oz$-direction for the
axisymmetric D-solution of steady MHD equations with the magnetic field having
only the swirl component ${\bf h}(r,z)=h_{\theta}(r,z){\bf e}_{\theta}$. The
quantity $\Pi\coloneqq\frac{h_{\theta}}{r}$ satisfies an elliptic equation,
from which weighted energy estimates are available for the weight
$\rho=\sqrt{r^{2}+z^{2}}$. Furthermore, if the swirl of the velocity
$u_{\theta}\equiv 0$, we can deduce some decay estimates for the velocity as
in steady axisymmetric Navier-Stokes case [35]. However, the decay rates are
worse than the Navier-Stokes case.

######  Theorem 1.3.

Suppose $\textbf{f}\in H^{1}(\mathbb{R}^{3})$ and $\textbf{g}\in
H^{2}(\mathbb{R}^{3})$ be axially symmetric vector field without swirl, and
satisfying (1.19), (1.24) and

$\displaystyle\|\rho\frac{\partial_{z}f_{r}-\partial_{r}f_{z}}{r}\|_{L^{2}(\mathbb{R}^{3})}\leq
M,$ (1.29)
$\displaystyle\|\rho\frac{\partial_{z}g_{r}-\partial_{r}g_{z}}{r}\|_{L^{2}(\mathbb{R}^{3})}\leq
M$ (1.31) $\displaystyle\|(\frac{r}{\ln
r})^{\frac{1}{2}}g_{\theta}\|_{L^{2}(\mathbb{R}^{3})}+\|r^{2}{\rm
curl\,}\emph{{g}}\|_{L^{2}(\mathbb{R}^{2})}\leq M,$

for some constant $M$. Let $(\textbf{u},\textbf{h},p)$ be a smooth axially
symmetric solution to the steady MHD equations (1.1) with (1.2)-(1.3), where
$\textbf{h}(r,z)=h_{\theta}(r,z)\bm{e_{\theta}}$. Then there holds

$\displaystyle|h_{\theta}(r,z)|\leq
C(M)(\rho+1)^{-\frac{31}{168}},\quad\quad\rho=\sqrt{r^{2}+z^{2}}.$

Moreover, if the swirl velocity $u_{\theta}\equiv 0$, then

$\displaystyle|u_{r}(r,z)|+|u_{z}(r,z)|$ $\displaystyle\leq$ $\displaystyle
C(M)(\rho+1)^{-(\frac{13}{179})^{-}},$ (1.32)
$\displaystyle|\omega_{\theta}(r,z)|$ $\displaystyle\leq$ $\displaystyle
C(M)(\rho+1)^{-(\frac{39}{179})^{-}},$ (1.33) $\displaystyle|h_{\theta}(r,z)|$
$\displaystyle\leq$ $\displaystyle C(M)(\rho+1)^{-(\frac{3041}{15036})^{-}}.$
(1.34)

This paper is organized as follows. In section 2 we introduce some preliminary
tools. In Section 3, we prove the decay rates of $({\bf u},{\bf h})$ and
$({\bf\omega^{u}},{\bf\omega^{h}})$ by applying the scaling argument and the
Brezis-Gallouet inequality. In Section 4, we will use the Biot-Savart law and
also weighted energy estimates for the magnetic field to get some improved
decay rate estimates for the special class of solutions with ${\bf
h}(r,z)=h_{\theta}(r,z){\bf e}_{\theta}$. In Section 5, we will investigate
the decay properties of $({\bf u},{\bf h})$ in the $O_{z}$-direction also
within the special class of solutions with ${\bf h}(r,z)=h_{\theta}(r,z){\bf
e}_{\theta}$.

##  2 Preliminary

######  Lemma 2.1.

_[2, Theorem 1]_ Let $f\in H^{2}(\Omega)$ where $\Omega\subset\mathbb{R}^{2}$.
Then there exists a constant $C_{\Omega}$, depending only on $\Omega$, such
that

$\|f\|_{L^{\infty}(\Omega)}\leq
C_{\Omega}(1+\|f\|_{H^{1}(\Omega)})\ln^{1/2}\big{(}e+\|\Delta
f\|_{L^{2}(\Omega)}\big{)}.$

######  Lemma 2.2.

_[3, Lemma 3.2]_ Assume that $K(x,y)$ be a Calderon-Zygmund kernel and $f$ is
a smooth axisymmetric function satisfying, for ${\bf x}=({\bf
x^{\prime}},z)\in\mathbb{R}^{3}$

$\displaystyle|f(x)|+|\nabla
f(x)|\leq\frac{\ln^{b}(e+|x^{\prime}|)}{(1+|x^{\prime}|)^{a}}\quad for\quad
0<a<2,\quad b>0.$

Define $Tf(x)\coloneqq\int K(x,y)f(y)dy$. Then there exists a constant $c_{0}$
such that

$\displaystyle|Tf(x)|\leq
c_{0}\frac{\ln^{b+1}(e+|x^{\prime}|)}{(1+|x^{\prime}|)^{a}}.$

######  Lemma 2.3.

_[35, Lemma 2.1]_ Suppose a smooth axially symmetric function $f(x)$ satisfies
the following weighted energy estimates

$\int_{\mathbb{R}^{3}}\left(r^{e_{1}}|f(r,z)|^{2}+r^{e_{2}}|\nabla
f(r,z)|^{2}+r^{e_{3}}|\partial_{z}\nabla f(r,z)|^{2}\right)dx\leq C,$

with nonnegative constants $e_{1}$, $e_{2}$, $e_{3}$. Then for any $r>0$, we
have

$\displaystyle\int_{-\infty}^{\infty}|f(r,z)|^{2}dz\leq
Cr^{-\frac{1}{2}(e_{1}+e_{2})-1},$
$\displaystyle\int_{-\infty}^{\infty}|\partial_{z}f(r,z)|^{2}dz\leq
Cr^{-\frac{1}{2}(e_{3}+e_{2})-1},$ $\displaystyle|f(r,z)|^{2}\leq
Cr^{-\frac{1}{4}(e_{1}+2e_{2}+e_{3})-1},\quad\text{$\forall
z\in\mathbb{R}^{3}$}.$

##  3 Proof of Theorem 1.1

Before starting to prove Theorem 1.1, we want to sketch the proof as follows.
Following [3], the decay rate of $({\textbf{u}},\textbf{h})$ can be shown by a
scaling argument and the Brezis-Gallouet inequality. Next utilizing the
localized energy estimates of $\omega^{u}$ and $\omega^{h}$ and the Brezis-
Gallout inequality, it will be deduced that

$\displaystyle{|\omega_{\theta}^{u}(r,z)|+|\omega_{\theta}^{h}(r,z)|\leq}{C(M)(\frac{\ln
r}{r})^{\frac{1}{2}}\|({\textbf{u}},\textbf{h})\|_{L^{\infty}([\frac{r}{2},\frac{3r}{2}]\times[-r,r])},}$
$\displaystyle{|\omega_{r}^{u}(r,z)|+|\omega_{z}^{u}(r,z)|+|\omega_{r}^{h}(r,z)|+|\omega_{z}^{h}(r,z)|}{\leq}{C(M)(\frac{\ln
r}{r})^{\frac{1}{2}}\|({\textbf{u}},\textbf{h})\|_{L^{\infty}([\frac{r}{2},\frac{3r}{2}]\times[-r,r])}.}$

The details can be found in (3.5) and (3.6). The estimates (1.21) and (1.22)
follow directly from the above two inequalities.

Now we start to prove Theorem 1.1. Assume that (1.19) holds, the standard
existence theory tells us that there exists a weak solution $({\bf u},{\bf
h},p)$ to MHD equation (1.1)-(1.2) with the finite Dirichlet integral (1.3)
(see Chapter X in [16] for more details). Since $({\bf u},{\bf h})\in
L^{6}(\mathbb{R}^{3})$ and $(\nabla{\bf u},\nabla{\bf h})\in
L^{2}(\mathbb{R}^{3})$, then $({\bf u}\cdot\nabla{\bf u},{\bf
h}\cdot\nabla{\bf h},{\bf u}\cdot\nabla{\bf h},{\bf h}\cdot\nabla{\bf u})\in
L^{\frac{3}{2}}(\mathbb{R}^{3})$. According to the $L^{p}$ estimates to the
Stokes system (See Chapter IV.2 in [16]), if $(\textbf{f},\text{curl }{\bf
g})\in L^{\frac{3}{2}}(\mathbb{R}^{3})$, then $(\nabla^{2}{\bf
u},\nabla^{2}{\bf h})\in L^{\frac{3}{2}}(\mathbb{R}^{3})$, which implies that
$(\nabla{\bf u},\nabla{\bf h})\in L^{3}(\mathbb{R}^{3})$ by Sobolev embedding.
Thus $({\bf u}\cdot\nabla{\bf u},{\bf h}\cdot\nabla{\bf h},{\bf
u}\cdot\nabla{\bf h},{\bf h}\cdot\nabla{\bf u})\in L^{2}(\mathbb{R}^{3})$,
yielding that

$\displaystyle\|(\nabla^{2}{\textbf{u}},\nabla^{2}\textbf{h})\|_{L^{2}(\mathbb{R}^{3})}\leq
C(M),$ (3.1)

where the positive constant $C(M)$ depending only on $M$. And there also hold
$\|({\bf u},{\bf h})\|_{W^{1,6}(\mathbb{R}^{3})}+\|({\bf u},{\bf
h})\|_{L^{\infty}(\mathbb{R}^{3})}\leq C(M)$. Simple calculations show that
$({\bf u}\cdot\nabla{\bf u},{\bf h}\cdot\nabla{\bf h},{\bf u}\cdot\nabla{\bf
h},{\bf h}\cdot\nabla{\bf u})\in H^{1}(\mathbb{R}^{3})$. Then the $L^{p}$
theory to the Stokes system yields that

$\displaystyle\|(\nabla^{3}{\textbf{u}},\nabla^{3}\textbf{h})\|_{L^{2}(\mathbb{R}^{3})}\leq
C(M).$ (3.2)

Fix any point ${\bf x_{0}}\in\mathbb{R}^{3}$ such that $|{\bf
x_{0}^{\prime}}|=\lambda$ is large, and consider the scaled solutions
$\tilde{\textbf{u}}(\tilde{\textbf{x}})$,
$\tilde{\textbf{h}}(\tilde{\textbf{x}})$ and the two dimensional domain
$\tilde{D}$,

$\displaystyle\tilde{\textbf{u}}(\tilde{\textbf{x}})=\lambda{\textbf{u}}(\lambda\tilde{\textbf{x}}),\
\ \
\tilde{\textbf{h}}(\tilde{\textbf{x}})=\lambda{\textbf{h}}(\lambda\tilde{\textbf{x}}),$
$\displaystyle\tilde{D}=\\{(\tilde{r},\tilde{z})|1/2\leq\tilde{r}\leq
2,|\tilde{z}|\leq 1\\},$

where ${\textbf{x}}=\lambda\tilde{\textbf{x}}$.

To prove the estimate (1.20), by carefully checking the argument developed in
[3] for the steady axisymmetric Navier-Stokes equations without external
force, only the estimates (3.1) and the finite Dirichlet integral are used in
their proof, the structure of equations play no role, thus the estimate (1.20)
holds.

It remains to derive the decay rates of $\bf{\omega^{u}}$ and
$\bf{\omega^{h}}$. According to previous scaling, we have

$\displaystyle\tilde{\omega}^{u}(\tilde{\textbf{x}})=\lambda^{2}\omega^{u}(\lambda\tilde{\textbf{x}})=\lambda^{2}\omega^{u}({\textbf{x}}),$
$\displaystyle\tilde{\omega}^{h}(\tilde{\textbf{x}})=\lambda^{2}\omega^{h}(\lambda\tilde{\textbf{x}})=\lambda^{2}\omega^{h}({\textbf{x}}),$
$\displaystyle\tilde{\textbf{f}}(\tilde{{\textbf{x}}})=\lambda^{3}\textbf{f}(\lambda\tilde{{\textbf{x}}})=\lambda^{3}\textbf{f}({\textbf{x}}),$
$\displaystyle{\rm
curl\,}_{\tilde{{\textbf{x}}}}\tilde{\textbf{g}}(\tilde{{\textbf{x}}})=\lambda^{3}{\rm
curl\,}_{{\textbf{x}}}\textbf{g}(\lambda\tilde{{\textbf{x}}})=\lambda^{3}{\rm
curl\,}_{{\textbf{x}}}\textbf{g}({\textbf{x}}).$

For simplification of notation, we will drop the $``\sim"$ when computations
take place under the scaled sense. Select the domains

$\displaystyle\mathcal{C}_{1}=\\{(r,\theta,z):\frac{1}{2}<r<\frac{3}{2},0\leq\theta\leq
2\pi,|z|\leq 1\\},$
$\displaystyle\mathcal{C}_{2}=\\{(r,\theta,z):\frac{3}{4}<r<\frac{5}{4},0\leq\theta\leq
2\pi,|z|\leq\frac{1}{2}\\}.$

Let $\phi(y)$ be a cut-off function satisfying supp
$\phi(y)\subset\mathcal{C}_{1}$ and $\phi(y)=1$ for $y\in\mathcal{C}_{2}$ such
that the gradient of $\phi$ is bounded. Now testing the vorticity equation
(1.18) with $\omega_{r}^{u}\phi^{2}$, $\omega_{\theta}^{u}\phi^{2}$,
$\omega_{z}^{u}\phi^{2}$, $\omega_{r}^{h}\phi^{2}$,
$\omega_{\theta}^{h}\phi^{2}$ and $\omega_{z}^{h}\phi^{2}$ respectively, and
integrating over $\mathcal{C}_{1}$, after direct computations we obtain

$\displaystyle\int_{\mathcal{C}_{1}}\bigg{(}|\nabla(\omega_{\theta}^{u}\phi)|^{2}+\frac{(\omega_{\theta}^{u}\phi)^{2}}{r^{2}}\bigg{)}dy=\int_{\mathcal{C}_{1}}\bigg{(}|\omega_{\theta}^{u}|^{2}|\nabla\phi|^{2}+\frac{1}{2}|\omega_{\theta}^{u}|^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})$
$\displaystyle\quad\quad+\frac{u_{r}}{r}(\omega_{\theta}^{u}\phi)^{2}-\frac{2}{r}u_{\theta}\omega_{r}^{u}\omega_{\theta}^{u}\phi^{2}+\frac{2}{r}h_{\theta}\omega_{r}^{h}\omega_{\theta}^{u}\phi^{2}+\omega_{\theta}^{u}\phi(h_{r}\partial_{r}+h_{z}\partial_{z})(\omega_{\theta}^{h}\phi)-\frac{h_{r}}{r}\omega_{\theta}^{u}\omega_{\theta}^{h}\phi^{2}$
$\displaystyle\quad\quad-\omega_{\theta}^{u}\omega_{\theta}^{h}\phi(h_{r}\partial_{r}+h_{z}\partial_{z})\phi+\phi(f_{z}\partial_{r}-f_{r}\partial_{z})(\omega_{\theta}^{u}\phi)+\omega_{\theta}^{u}\phi(f_{z}\partial_{r}-f_{r}\partial_{z})\phi+f_{z}\frac{\omega_{\theta}^{u}\phi}{r}\phi\bigg{)}dy$
$\displaystyle\leq C(1+\|({\bf u},{\bf
h})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{\theta}^{u},\omega_{r}^{h},\omega_{\theta}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|(f_{r},f_{z})\|_{L^{2}(\mathcal{C}_{1})}^{2}$
$\displaystyle\quad+\frac{1}{8}\|(\nabla(\omega^{u}_{\theta}\phi),\nabla(\omega^{h}_{\theta}\phi))\|^{2}_{L^{2}(\mathcal{C}_{1})},$
$\displaystyle\int_{\mathcal{C}_{1}}\bigg{(}|\nabla(\omega_{\theta}^{h}\phi)|^{2}+\frac{(\omega_{\theta}^{h}\phi)^{2}}{r^{2}}\bigg{)}dy=\int_{\mathcal{C}_{1}}\bigg{(}|\nabla\phi|^{2}|\omega_{\theta}^{h}|^{2}+\frac{1}{2}|\omega_{\theta}^{h}|^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})$
$\displaystyle\quad\quad+2\partial_{z}(\omega_{\theta}^{h}\phi^{2})(u_{z}\partial_{r}h_{z}-h_{r}\partial_{r}u_{r})+2\partial_{r}(\omega_{\theta}^{h}\phi^{2})(h_{r}\partial_{z}u_{r}-u_{z}\partial_{z}h_{z})+\frac{u_{r}}{r}(\omega_{\theta}^{h}\phi)^{2}$
$\displaystyle\quad\quad+\omega_{\theta}^{h}\phi(h_{r}\partial_{r}+h_{z}\partial_{z})(\omega_{\theta}^{u}\phi)-\omega_{\theta}^{u}\omega_{\theta}^{h}\phi(h_{r}\partial_{r}+h_{z}\partial_{z})\phi-\frac{h_{r}}{r}\omega_{\theta}^{u}\phi\cdot\omega_{\theta}^{h}\phi$
$\displaystyle\quad\quad+2\frac{\omega_{\theta}^{h}\phi}{r}(h_{r}\partial_{z}u_{r}-u_{z}\partial_{z}h_{z})\phi+\partial_{z}g_{\theta}(\partial_{z}(\omega_{\theta}^{h}\phi))+\omega_{\theta}^{h}\phi\partial_{z}\phi)$
$\displaystyle\quad\quad+\partial_{r}(rg_{\theta})(\phi\partial_{r}(\omega_{\theta}^{h}\phi)+\omega_{\theta}^{h}\phi\partial_{r}\phi+\frac{\omega_{\theta}^{h}\phi}{r}\phi)\bigg{)}dy$
$\displaystyle\leq
C(1+\|(u_{r},u_{z},h_{r},h_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{\theta}^{u},\omega_{\theta}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1})}^{2}$
$\displaystyle\quad\quad+C\|(u_{r},u_{z},h_{r},h_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})}\|(\nabla
u_{r},\nabla
h_{z})\|^{2}_{L^{2}(\mathcal{C}_{1})}+\frac{1}{8}\|(\nabla(\omega_{\theta}^{u}\phi),\nabla(\omega_{\theta}^{h}\phi))\|^{2}_{L^{2}(\mathcal{C}_{1})}.$

Combining these two estimates, we find

$\displaystyle\|(\nabla\omega_{\theta}^{u},\nabla\omega_{\theta}^{h})\|^{2}_{L^{2}(\mathcal{C}_{2})}\leq
C(1+\|(\textbf{u},\textbf{h})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{\theta}^{u},\omega_{r}^{h},\omega_{\theta}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}$
$\displaystyle\quad\quad+C\|(f_{r},f_{z},\bar{\textbf{g}})\|_{L^{2}(\mathcal{C}_{1})}^{2}+C\|(u_{r},u_{z},h_{r},h_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})}\|(\nabla
u_{r},\nabla h_{z})\|^{2}_{L^{2}(\mathcal{C}_{1})}.$ (3.3)

There also holds

$\displaystyle\int_{\mathcal{C}_{1}}\bigg{(}|\nabla(\omega_{r}^{u}\phi)|^{2}+\frac{(\omega_{r}^{u}\phi)^{2}}{r^{2}}\bigg{)}dy=\int_{\mathcal{C}_{1}}\bigg{(}|\nabla\phi|^{2}|\omega_{r}^{u}|^{2}+\frac{1}{2}|\omega_{r}^{u}|^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})$
$\displaystyle\quad\quad-\frac{u_{r}}{r}(\omega_{r}^{u}\phi)^{2}-u_{r}(\partial_{r}(\omega_{r}^{u}\phi)^{2}+\partial_{z}(\omega_{r}^{u}\phi\cdot\omega_{z}^{u}\phi))+\omega_{r}^{u}\phi(h_{r}\partial_{r}+h_{z}\partial_{z})(\omega_{r}^{h}\phi)$
$\displaystyle\quad\quad-\omega_{r}^{u}\omega_{r}^{h}\phi(h_{r}\partial_{r}+h_{z}\partial_{z})\phi+h_{r}\partial_{r}(\omega_{r}^{u}\phi\cdot\omega_{r}^{h}\phi)+h_{r}\partial_{z}(\omega_{r}^{u}\phi\cdot\omega_{z}^{h}\phi)$
$\displaystyle\quad\quad+\frac{h_{r}}{r}\omega_{r}^{u}\phi\cdot\omega_{r}^{h}\phi+f_{\theta}\phi(\partial_{z}(\omega_{r}^{u}\phi)+\omega_{r}^{u}\partial_{z}\phi)\bigg{)}dy$
$\displaystyle\leq
C(1+\|(u_{r},u_{z},h_{r},h_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u},\omega_{r}^{h},\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|f_{\theta}\|_{L^{2}(\mathcal{C}_{1})}^{2}$
$\displaystyle\quad\quad+\frac{1}{8}\|(\nabla(\omega_{r}^{u}\phi),\nabla(\omega_{z}^{u}\phi),\nabla(\omega_{r}^{h}\phi),\nabla(\omega_{z}^{h}\phi)\|^{2}_{L^{2}(\mathcal{C}_{1})},$
$\displaystyle\int_{\mathcal{C}_{1}}|\nabla(\omega_{z}^{u}\phi)|^{2}dy=\int_{\mathcal{C}_{1}}\bigg{(}|\nabla\phi|^{2}|\omega_{z}^{u}|^{2}+\frac{1}{2}|\omega_{z}^{u}|^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})-u_{z}\partial_{r}(\omega_{r}^{u}\phi\cdot\omega_{z}^{u}\phi)$
$\displaystyle\quad\quad-\frac{u_{z}}{r}\omega_{r}^{u}\phi\cdot\omega_{z}^{u}\phi+\omega_{z}^{u}\phi(h_{r}\partial_{r}+h_{z}\partial_{z})(\omega_{z}^{h}\phi)-\omega_{z}^{u}\omega_{z}^{h}\phi(h_{r}\partial_{r}+h_{z}\partial_{z})\phi$
$\displaystyle\quad\quad+h_{z}(\partial_{r}(\omega_{z}^{u}\phi\cdot\omega_{r}^{h}\phi)+\partial_{z}(\omega_{z}^{u}\phi\cdot\omega_{z}^{h}\phi))-u_{z}\partial_{z}(\omega_{z}^{u}\phi)^{2}+\frac{h_{z}}{r}\omega_{z}^{u}\phi\cdot\omega_{r}^{h}\phi$
$\displaystyle\quad\quad-
f_{\theta}\phi(\partial_{r}(\omega_{z}^{u}\phi)+\omega_{z}^{u}\partial_{r}\phi)\bigg{)}dy$
$\displaystyle\leq
C(1+\|(u_{r},u_{z},h_{r},h_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u},\omega_{r}^{h},\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|f_{\theta}\|_{L^{2}(\mathcal{C}_{1})}^{2}$
$\displaystyle\quad\quad+\frac{1}{8}\|(\nabla(\omega_{r}^{u}\phi),\nabla(\omega_{z}^{u}\phi),\nabla(\omega_{r}^{h}\phi),\nabla(\omega_{z}^{h}\phi))\|^{2}_{L^{2}(\mathcal{C}_{1})},$
$\displaystyle\int_{\mathcal{C}_{1}}\bigg{(}|\nabla(\omega_{r}^{h}\phi)|^{2}+\frac{(\omega_{r}^{h}\phi)^{2}}{r^{2}}\bigg{)}dy=\int_{\mathcal{C}_{1}}\bigg{(}|\nabla\phi|^{2}|\omega_{r}^{h}|^{2}+\frac{1}{2}|\omega_{r}^{h}|^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})$
$\displaystyle\quad\quad-
u_{r}\partial_{r}(\omega_{r}^{h}\phi)^{2}-u_{r}\partial_{z}(\omega_{r}^{h}\phi\cdot\omega_{z}^{h}\phi)-\frac{u_{r}}{r}|\omega_{r}^{h}\phi|^{2}+\omega_{r}^{h}\phi(h_{r}\partial_{r}+h_{z}\partial_{z})(\omega_{r}^{u}\phi)$
$\displaystyle\quad\quad-\omega_{r}^{u}\omega_{r}^{h}\phi(h_{r}\partial_{r}+h_{z}\partial_{z})\phi+h_{r}\partial_{r}(\omega_{r}^{u}\phi\cdot\omega_{r}^{h}\phi)+h_{r}\partial_{z}(\omega_{z}^{u}\phi\cdot\omega_{r}^{h}\phi)$
$\displaystyle\quad\quad+\frac{h_{r}}{r}\omega_{r}^{u}\phi\cdot\omega_{r}^{h}\phi-\frac{2}{r}\partial_{z}(u_{r}h_{\theta})\omega_{r}^{h}\phi^{2}+\frac{2}{r}\partial_{z}(u_{\theta}h_{r})\omega_{r}^{h}\phi^{2}$
$\displaystyle\quad\quad+\partial_{z}(\omega_{r}^{h}\phi)(\partial_{z}g_{r}-\partial_{r}g_{z})\bigg{)}dy$
$\displaystyle\leq
C(1+\|(u_{r},u_{z},h_{r},h_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u},\omega_{r}^{h},\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1})}^{2}$
$\displaystyle\quad\quad+C\|(u_{r},u_{\theta},h_{r},h_{\theta})\|^{2}_{L^{\infty}(\mathcal{C}_{1})}\|\nabla(u_{r},u_{\theta},h_{r},h_{\theta})\|^{2}_{L^{2}(\mathcal{C}_{1})}$
$\displaystyle\quad\quad+\frac{1}{8}\|(\nabla(\omega_{r}^{u}\phi),\nabla(\omega_{z}^{u}\phi),\nabla(\omega_{r}^{h}\phi),\nabla(\omega_{z}^{h}\phi))\|^{2}_{L^{2}(\mathcal{C}_{1})}.$
$\displaystyle\int_{\mathcal{C}_{1}}|\nabla(\omega_{z}^{h}\phi)|^{2}dy=\int_{\mathcal{C}_{1}}\bigg{(}|\omega_{z}^{h}|^{2}|\nabla\phi|^{2}+\frac{1}{2}(\omega_{z}^{h})^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})$
$\displaystyle\quad\quad+\partial_{r}(\omega_{z}^{h}\phi^{2})(h_{\theta}\partial_{r}u_{r}-u_{\theta}\partial_{r}h_{r})+\partial_{z}(\omega_{z}^{h}\phi^{2})(h_{\theta}\partial_{r}u_{z}-u_{\theta}\partial_{r}h_{z})$
$\displaystyle\quad\quad+\frac{\omega^{h}_{z}\phi^{2}}{r}(\partial_{r}(u_{r}h_{\theta})-\partial_{r}(u_{\theta}h_{r}))+\omega_{z}^{h}\phi(h_{r}\partial_{r}+h_{z}\partial_{z})(\omega_{z}^{u}\phi)$
$\displaystyle\quad\quad-\omega_{z}^{u}\omega_{z}^{h}\phi(h_{r}\partial_{r}+h_{z}\partial_{z})\phi-\partial_{r}(\omega_{z}^{h}\phi^{2})(\partial_{z}g_{r}-\partial_{r}g_{z})\bigg{)}dy$
$\displaystyle\leq
C(1+\|(u_{r},u_{z},h_{r},h_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{z}^{u},\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+\frac{1}{8}\|(\nabla(\omega_{z}^{u}\phi),\nabla(\omega_{z}^{h}\phi))\|^{2}_{L^{2}(\mathcal{C}_{1})}$
$\displaystyle\quad\quad+C\|(u_{r},u_{\theta},h_{r},h_{\theta})\|^{2}_{L^{\infty}(\mathcal{C}_{1})}\|(\nabla{\textbf{u}},\nabla\textbf{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1})}^{2}.$

Then we obtain

$\displaystyle\|(\nabla\omega_{r}^{u},\nabla\omega_{z}^{u},\nabla\omega_{r}^{h},\nabla\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{2})}\leq
C(1+\|(u_{r},u_{z},h_{r},h_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u},\omega_{r}^{h},\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}$
$\displaystyle\quad\quad+C\|(u_{r},u_{\theta},h_{r},h_{\theta})\|^{2}_{L^{\infty}(\mathcal{C}_{1})}\|(\nabla{\textbf{u}},\nabla\textbf{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|(f_{\theta},\bar{\textbf{g}})\|_{L^{2}(\mathcal{C}_{1})}^{2}.$
(3.4)

Set

$\mathcal{\bar{C}}_{2}\coloneqq\\{(r,z):\frac{3}{4}<r<\frac{5}{4},|z|\leq
1/2\\}.$

Utilizing the Brezis-Gallouet inequality in Lemma 2.1 and the localized energy
estimates (3.3) and (3.4), we can conclude

$\displaystyle\|(\omega_{r}^{u},\omega_{z}^{u},\omega_{r}^{h},\omega_{z}^{h})\|_{L^{\infty}(\mathcal{\bar{C}}_{2})}\leq
C\bigg{(}1+(1+\|(u_{r},u_{z},h_{r},h_{z})\|_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u},\omega_{r}^{h},\omega_{z}^{h})\|_{L^{2}(\mathcal{C}_{1})}$
$\displaystyle\quad\quad+\|(u_{r},u_{\theta},h_{r},h_{\theta})\|_{L^{\infty}(\mathcal{C}_{1})}\|(\nabla{\textbf{u}},\nabla\textbf{h})\|_{L^{2}(\mathcal{C}_{1})}+\|(f_{\theta},\bar{\textbf{g}})\|_{L^{2}(\mathcal{C}_{1})}\bigg{)}$
$\displaystyle\quad\quad\times\ln^{1/2}\bigg{(}e+\|\Delta(\omega_{r}^{u},\omega_{z}^{u},\omega_{r}^{h},\omega_{z}^{h})\|_{L^{2}(\bar{\mathcal{C}}_{2})}\bigg{)},$
$\displaystyle\|(\omega_{\theta}^{u},\omega_{\theta}^{h})\|_{L^{\infty}(\mathcal{\bar{C}}_{2})}\leq
C\bigg{(}1+(1+\|(\textbf{u},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{\theta}^{u},\omega_{r}^{h},\omega_{\theta}^{h})\|_{L^{2}(\mathcal{C}_{1})}+\|(f_{r},f_{z},\bar{\textbf{g}})\|_{L^{2}(\mathcal{C}_{1})}$
$\displaystyle\quad\quad+\|(u_{r},u_{z},h_{r},h_{z})\|_{L^{\infty}(\mathcal{C}_{1})}\|(\nabla
u_{r},\nabla
h_{z})\|_{L^{2}(\mathcal{C}_{1})}\bigg{)}\ln^{1/2}\bigg{(}e+\|(\Delta\omega_{\theta}^{u},\Delta\omega_{\theta}^{h})\|_{L^{2}(\bar{\mathcal{C}}_{2})}\bigg{)}.$

Then scaling back to the domains

$\displaystyle\mathcal{C}_{1,\lambda}$ $\displaystyle=$
$\displaystyle\\{(r,\theta,z):\frac{\lambda}{2}<r<\frac{3\lambda}{2},0\leq\theta\leq
2\pi,|z|\leq\lambda\\},$ $\displaystyle\mathcal{C}_{2,\lambda}$
$\displaystyle=$
$\displaystyle\\{(r,\theta,z):\frac{3\lambda}{4}<r<\frac{5\lambda}{4},0\leq\theta\leq
2\pi,|z|\leq\frac{\lambda}{2}\\},$

we derive

$\displaystyle\lambda^{2}\|(\omega_{r}^{u},\omega_{z}^{u},\omega_{r}^{h},\omega_{z}^{h})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}\leq
C\bigg{(}1+(1+\lambda\|(\textbf{u},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{1,\lambda})})\lambda^{1/2}\|(\omega_{r}^{u},\omega_{z}^{u},\omega_{r}^{h},\omega_{z}^{h})\|_{L^{2}(\mathcal{C}_{1,\lambda})}$
$\displaystyle\quad+\lambda^{3/2}\|(u_{r},u_{\theta},h_{r},h_{\theta})\|_{L^{\infty}(\mathcal{C}_{1,\lambda})}\|(\nabla{\textbf{u}},\nabla\textbf{h})\|_{L^{2}(\mathcal{C}_{1,\lambda})}+\lambda^{3/2}\|(f_{\theta},\bar{\textbf{g}})\|_{L^{2}(\mathcal{C}_{1,\lambda})}\bigg{)}$
$\displaystyle\quad\times\ln^{1/2}\bigg{(}e+\lambda^{5/2}\|\Delta(\omega_{r}^{u},\omega_{z}^{u},\omega_{r}^{h},\omega_{z}^{h})\|_{L^{2}(\mathcal{C}_{2,\lambda})}\bigg{)},$
(3.5)
$\displaystyle\lambda^{2}\|(\omega_{\theta}^{u},\omega_{\theta}^{h})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}\leq
C\bigg{(}1+(1+\lambda\|(\textbf{u},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{1,\lambda})})\lambda^{1/2}\|(\omega_{r}^{u},\omega_{\theta}^{u},\omega_{r}^{h},\omega_{\theta}^{h})\|_{L^{2}(\mathcal{C}_{1,\lambda})}$
$\displaystyle\quad+\lambda^{3/2}\|(u_{z},h_{r})\|_{L^{\infty}(\mathcal{C}_{1,\lambda})}\|(\nabla
u_{r},\nabla
h_{z})\|_{L^{2}(\mathcal{C}_{1,\lambda})}+\lambda^{3/2}\|(f_{r},f_{z},\bar{\textbf{g}})\|_{L^{2}(\mathcal{C}_{1,\lambda})}\bigg{)}$
$\displaystyle\quad\times\ln^{1/2}\bigg{(}e+\lambda^{5/2}\|(\Delta\omega_{\theta}^{u},\Delta\omega_{\theta}^{h})\|_{L^{2}(\mathcal{C}_{2,\lambda})}\bigg{)}.$
(3.6)

By the finite Dirichlet integral assumption (1.3), the estimates (1.20) and
(3.2), we conclude that

$\displaystyle\|(\omega_{\theta}^{u},\omega_{\theta}^{h})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}\leq
C(M)\frac{\ln\lambda}{\lambda},$
$\displaystyle\|(\omega_{r}^{u},\omega_{z}^{u},\omega_{r}^{h},\omega_{z}^{h})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}\leq
C(M)\frac{\ln\lambda}{\lambda},$

where we use the boundedness of
$\|(\nabla^{3}{\textbf{u}},\nabla^{3}\textbf{h})\|_{L^{2}(\mathbb{R}^{3})}$ to
control
$\|(\Delta\omega^{u},\Delta\omega^{h})\|_{L^{2}(\mathcal{C}_{2,\lambda})}$.
These verify (1.21) and (1.22).

##  4 Proof of Theorem 1.2

In this section, we will consider the special class of axisymmetric D-solution
to the steady MHD equations, where the magnetic field has only the swirl
component $\textbf{h}(r,z)=h_{\theta}(r,z)\bf{e_{\theta}}$. In this case,
${\bf\omega^{h}}(r,z)=\nabla\times\textbf{h(x)}=\omega_{r}^{h}(r,z){\bf
e_{r}}+\omega_{z}^{h}(r,z){\bf e_{z}}$, the equations (1.18) reduce to

$\left\\{\begin{array}[]{l}(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{r}^{u}-(\omega_{r}^{u}\partial_{r}+\omega_{z}^{u}\partial_{z})u_{r}=(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z}-\frac{1}{r^{2}})\omega_{r}^{u}-\partial_{z}f_{\theta},\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{\theta}^{u}-\frac{u_{r}\omega_{\theta}^{u}}{r}+\frac{1}{r}\partial_{z}(h_{\theta}^{2}-u_{\theta}^{2})=(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z}-\frac{1}{r^{2}})\omega_{\theta}^{u}+(\partial_{z}f_{r}-\partial_{r}f_{z}),\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{z}^{u}-(\omega_{r}^{u}\partial_{r}+\omega_{z}^{u}\partial_{z})u_{z}=(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z})\omega_{z}^{u}+\frac{1}{r}\partial_{r}(rf_{\theta}),\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{r}^{h}-(\omega_{r}^{h}\partial_{r}+\omega_{z}^{h}\partial_{z})u_{r}+\frac{2}{r}\partial_{z}(u_{r}h_{\theta})=(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z}-\frac{1}{r^{2}})\omega_{r}^{h}\\\
\quad\quad-\partial_{z}(\partial_{z}g_{r}-\partial_{r}g_{z}),\\\
(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{z}^{h}-(\omega_{r}^{h}\partial_{r}+\omega_{z}^{h}\partial_{z})u_{z}-\frac{2}{r}\partial_{r}(u_{r}h_{\theta})=(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z})\omega_{z}^{h}\\\
\quad\quad+\frac{1}{r}\partial_{r}(r(\partial_{z}g_{r}-\partial_{r}g_{z})).\end{array}\right.$
(4.1)

Comparing with (1.18), (4.1) has a simpler form, from which we derive better
localized energy estimates and it turns out that we can obtain better decay
rates.

The proof of theorem 1.2 will be divided into the following steps. Firstly,
the decay rate of $\omega_{\theta}^{u}$ and a weaker decay rates of
$(\omega_{r}^{u},\omega_{z}^{u})$ will be deduced by using the simpler
structure of the equations in (4.1). Secondly, note that
$\nabla(u_{r}{\textbf{e}}_{r}+u_{z}{\textbf{e}}_{z})=\mathbb{C}(\omega_{\theta}^{u}{\textbf{e}}_{\theta})+\mathbb{K}\ast(\omega_{\theta}^{u}{\textbf{e}}_{\theta})$,
where $\mathbb{C}(\omega_{\theta}^{u}{\textbf{e}}_{\theta})$ is a matrix whose
entries are components of $\omega_{\theta}^{u}{\textbf{e}}_{\theta}$ and
$\mathbb{K}$ is a Calderon-Zymund kernel, then Lemma 2.2 and the estimate
(1.25) will be used to improve the decay rates of
$(\omega_{r}^{u},\omega_{z}^{u})$:

$\displaystyle|\nabla u_{r}|+|\nabla u_{z}|\leq C(M)r^{-5/4}(\ln r)^{7/4},$

and

$\displaystyle|\omega_{r}^{u}(r,z)|+|\omega_{z}^{u}(r,z)|\leq C(M)r^{-9/8}(\ln
r)^{11/8},$ $\displaystyle|\omega_{r}^{h}(r,z)|+|\omega_{z}^{h}(r,z)|\leq
C(M)r^{-1}(\ln r)^{11/8},$

for large $r$. Thirdly, we utilize the weighted energy estimates to obtain
better decay rates of $\nabla h_{\theta}$:

$\displaystyle|\omega^{h}_{r}|+|\omega^{h}_{z}|+|\nabla h_{\theta}|\leq
C(M)r^{-(\frac{35}{32})^{-}}.$

Step 1: Proof of (1.25) for $\omega_{\theta}^{u}$ and a weaker decay of
$(\omega_{r}^{u},\omega_{z}^{u})$.

Same as in section 3, consider the scaled solution
$\tilde{\textbf{u}}(\tilde{\textbf{x}})$,
$\tilde{\textbf{h}}(\tilde{\textbf{x}})$,
$\tilde{\omega}(\tilde{\textbf{x}})$, and
$\tilde{\textbf{f}}(\tilde{\textbf{x}})$, ${\rm
curl\,}_{\tilde{\textbf{x}}}\tilde{\textbf{g}}(\tilde{\textbf{x}})$, where
$\tilde{\textbf{x}}=\lambda{\textbf{x}}$. Drop the ”$\sim$” for simplification
of notation when computations take place under the scaled sense. Taking the
same cut-off function $\phi(y)$ as previous, and testing the vorticity
equation (4.1) with $\omega_{r}^{u}\phi^{2}$, $\omega_{\theta}^{u}\phi^{2}$,
$\omega_{z}^{u}\phi^{2}$, $\omega_{r}^{h}\phi^{2}$ and
$\omega_{z}^{h}\phi^{2}$ respectively, we deduce that

$\displaystyle\int_{\mathcal{C}_{1}}\bigg{(}|\nabla(\omega_{\theta}^{u}\phi)|^{2}+\frac{(\omega_{\theta}^{u}\phi)^{2}}{r^{2}}\bigg{)}dy=\int_{\mathcal{C}_{1}}\bigg{(}|\omega_{\theta}^{u}|^{2}|\nabla\phi|^{2}+\frac{1}{2}|\omega_{\theta}^{u}|^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})$
$\displaystyle\quad\quad+\frac{u_{r}}{r}(\omega_{\theta}^{u})^{2}\phi^{2}-2\frac{\omega_{\theta}^{u}\phi^{2}}{r}(-\omega_{r}^{h}h_{\theta}+\omega_{r}^{u}u_{\theta})+f_{z}\frac{\omega_{\theta}^{u}\phi}{r}\phi$
$\displaystyle\quad\quad+\phi(f_{z}\partial_{r}-f_{r}\partial_{z})(\omega_{\theta}^{u}\phi)+\omega_{\theta}^{u}\phi(f_{z}\partial_{r}-f_{r}\partial_{z})\phi\bigg{)}dy$
$\displaystyle\leq
C(1+\|(u_{r},u_{\theta},u_{z},h_{\theta})\|_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{\theta}^{u},\omega_{r}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|(f_{r},f_{z})\|_{L^{2}(\mathcal{C}_{1})}^{2}$
$\displaystyle\quad\quad+\frac{1}{8}\|\nabla(\omega_{\theta}^{u}\phi)\|^{2}_{L^{2}(\mathcal{C}_{1})},$
$\displaystyle\int_{\mathcal{C}_{1}}\bigg{(}|\nabla(\omega_{r}^{u}\phi)|^{2}+\frac{(\omega_{r}^{u}\phi)^{2}}{r^{2}}\bigg{)}dy=\int_{\mathcal{C}_{1}}\bigg{(}(\omega_{r}^{u})^{2}|\nabla\phi|^{2}+\frac{1}{2}|\omega_{r}^{u}|^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})$
$\displaystyle\quad-
u_{r}(\partial_{r}((\omega_{r}^{u}\phi)^{2})+\partial_{z}(\omega_{r}^{u}\phi\cdot\omega_{z}^{u}\phi))-\frac{(\omega_{r}^{u}\phi)^{2}}{r}u_{r}+f_{\theta}\phi(\partial_{z}(\omega_{r}^{u}\phi)+\omega_{r}^{u}\partial_{z}\phi)\bigg{)}dy$
$\displaystyle=\int_{\mathcal{C}_{1}}\bigg{(}(\omega_{r}^{u})^{2}|\nabla\phi|^{2}+\frac{1}{2}(\omega_{r}^{u})^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})+(|\omega_{r}^{u}|^{2}\phi^{2}\partial_{r}+\omega_{r}^{u}\phi\cdot\omega_{z}^{u}\phi\partial_{z})u_{r}$
$\displaystyle\quad\quad+f_{\theta}\phi(\partial_{z}(\omega_{r}^{u}\phi)+\omega_{r}^{u}\partial_{z}\phi)\bigg{)}dy$
$\displaystyle\leq\begin{cases}C(1+\|(u_{r},u_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|f_{\theta}\|_{L^{2}(\mathcal{C}_{1})}^{2}\\\
\quad\quad+\frac{1}{8}\|(\nabla(\omega_{r}^{u}\phi),\nabla(\omega_{z}^{u}\phi)\|^{2}_{L^{2}(\mathcal{C}_{1})},\\\
C(1+\|(u_{r},u_{z})\|_{L^{\infty}(\mathcal{C}_{1})}+\|\nabla
u_{r}\|_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|f_{\theta}\|_{L^{2}(\mathcal{C}_{1})}^{2}\\\
\quad\quad+\frac{1}{8}\|\nabla(\omega_{r}^{u}\phi)\|^{2}_{L^{2}(\mathcal{C}_{1})},\end{cases}$
$\displaystyle\int_{\mathcal{C}_{1}}|\nabla(\omega_{z}^{u}\phi)|^{2}dy=\int_{\mathcal{C}_{1}}\bigg{(}(\omega_{z}^{u})^{2}|\nabla\phi|^{2}+\frac{1}{2}(\omega_{z}^{u})^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})$
$\displaystyle\quad\quad-
u_{z}\partial_{r}(\omega_{r}^{u}\phi\cdot\omega_{z}^{u}\phi)-u_{z}\partial_{z}(\omega_{z}^{u}\phi)^{2}-\frac{u_{z}}{r}\omega_{r}^{u}\omega_{z}^{u}\phi^{2}-f_{\theta}\phi(\partial_{r}(\omega_{z}^{u}\phi)+\omega_{z}^{u}\partial_{r}\phi)\bigg{)}dy$
$\displaystyle=\int_{\mathcal{C}_{1}}\bigg{(}(\omega_{z}^{u})^{2}|\nabla\phi|^{2}+\frac{1}{2}(\omega_{z}^{u})^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})+(\omega_{z}^{u})^{2}\phi^{2}\partial_{z}u_{z}$
$\displaystyle\quad\quad+\omega_{r}^{u}\omega_{z}^{u}\phi^{2}\partial_{r}u_{z}-f_{\theta}\phi(\partial_{r}(\omega_{z}^{u}\phi)+\omega_{z}^{u}\partial_{r}\phi)\bigg{)}dy$
$\displaystyle\leq\begin{cases}C(1+\|(u_{r},u_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|f_{\theta}\|_{L^{2}(\mathcal{C}_{1})}^{2}\\\
\quad\quad+\frac{1}{8}\|(\nabla(\omega_{r}^{u}\phi),\nabla(\omega_{z}^{u}\phi))\|^{2}_{L^{2}(\mathcal{C}_{1})}\\\
C(1+\|(u_{r},u_{z})\|_{L^{\infty}(\mathcal{C}_{1})}+\|\nabla
u_{z}\|_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|f_{\theta}\|_{L^{2}(\mathcal{C}_{1})}^{2}\\\
\quad\quad+\frac{1}{8}\|\nabla(\omega_{z}^{u}\phi)\|^{2}_{L^{2}(\mathcal{C}_{1})},\end{cases}$

$\displaystyle\int_{\mathcal{C}_{1}}\bigg{(}|\nabla(\omega_{r}^{h}\phi)|^{2}+\frac{(\omega_{r}^{h}\phi)^{2}}{r^{2}}\bigg{)}dy=\int_{\mathcal{C}_{1}}\bigg{(}|\omega_{r}^{h}|^{2}|\nabla\phi|^{2}+\frac{1}{2}(\omega_{r}^{h})^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})$
$\displaystyle\quad\quad-
u_{r}\partial_{r}(\omega_{r}^{h}\phi)^{2}-u_{r}\partial_{z}(\omega_{r}^{h}\phi\cdot\omega_{z}^{h}\phi)-\frac{u_{r}}{r}(\omega_{r}^{h}\phi)^{2}+2\frac{u_{r}h_{\theta}}{r}\partial_{z}(\omega_{r}^{h}\phi^{2})$
$\displaystyle\quad\quad+\partial_{z}(\omega_{r}^{h}\phi)(\partial_{z}g_{r}-\partial_{r}g_{z})\bigg{)}dy$
$\displaystyle=\int_{\mathcal{C}_{1}}\bigg{(}|\omega_{r}^{h}|^{2}|\nabla\phi|^{2}+\frac{1}{2}|\omega_{r}^{h}|^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})+((\omega_{r}^{h})^{2}\phi^{2}\partial_{r}+\omega_{r}^{h}\omega_{z}^{h}\phi^{2}\partial_{z})u_{r}$
$\displaystyle\quad\quad-2(\partial_{r}u_{r}+\partial_{z}u_{z})h_{\theta}\partial_{z}(\omega_{r}^{h}\phi^{2})+\partial_{z}(\omega_{r}^{h}\phi)(\partial_{z}g_{r}-\partial_{r}g_{z})\bigg{)}dy$
$\displaystyle\leq\begin{cases}C(1+\|(u_{r},u_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{h},\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+\frac{1}{8}\|(\nabla(\omega_{r}^{h}\phi),\nabla(\omega_{z}^{h}\phi))\|^{2}_{L^{2}(\mathcal{C}_{1})}\\\
\quad\quad+C(M)\|u_{r}\|^{2}_{L^{\infty}(\mathcal{C}_{1})}\int_{\mathbb{R}^{3}}\frac{h_{\theta}^{2}}{r^{2}}dy+C\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1})}^{2}\\\
C(1+\|(u_{r},u_{z})\|_{L^{\infty}(\mathcal{C}_{1})}+\|\nabla
u_{r}\|_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{h},\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+\frac{1}{8}\|\nabla(\omega_{r}^{h}\phi)\|^{2}_{L^{2}(\mathcal{C}_{1})}\\\
\quad\quad+C\|h_{\theta}\|^{2}_{L^{\infty}(\mathcal{C}_{1})}\|(\nabla
u_{r},\nabla
u_{z})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1})}^{2},\end{cases}$
$\displaystyle\int_{\mathcal{C}_{1}}|\nabla(\omega_{z}^{h}\phi)|^{2}dy=\int_{\mathcal{C}_{1}}\bigg{(}|\omega_{z}^{h}|^{2}|\nabla\phi|^{2}+\frac{1}{2}|\omega_{z}^{h}|^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})-u_{z}\partial_{r}(\omega_{r}^{h}\phi\cdot\omega_{z}^{h}\phi)$
$\displaystyle\quad\quad-\frac{u_{z}}{r}\omega_{r}^{h}\omega_{z}^{h}\phi^{2}-u_{z}\partial_{z}(\omega_{z}^{h}\phi)^{2}-2\frac{u_{r}h_{\theta}}{r}\partial_{r}(\omega_{z}^{h}\phi^{2})-\partial_{r}(\omega_{z}^{h}\phi^{2})(\partial_{z}g_{r}-\partial_{r}g_{z})\bigg{)}dy$
$\displaystyle=\int_{\mathcal{C}_{1}}\bigg{(}|\omega_{z}^{h}|^{2}|\nabla\phi|^{2}+\frac{1}{2}|\omega_{z}^{h}|^{2}(u_{r}\partial_{r}+u_{z}\partial_{z})(\phi^{2})+(\omega_{r}^{h}\omega_{z}^{h}\phi^{2}\partial_{r}+(\omega_{z}^{h}\phi)^{2}\partial_{z})u_{z}$
$\displaystyle\quad\quad+2\frac{\omega_{z}^{h}\phi^{2}}{r}\partial_{r}(u_{r}h_{\theta})-\partial_{r}(\omega_{z}^{h}\phi^{2})(\partial_{z}g_{r}-\partial_{r}g_{z})\bigg{)}dy$
$\displaystyle\leq\begin{cases}C(1+\|(u_{r},u_{z})\|_{L^{\infty}(\mathcal{C}_{1})}^{2})\|(\omega_{r}^{h},\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+\frac{1}{8}\|(\nabla(\omega_{r}^{h}\phi),\nabla(\omega_{z}^{h}\phi))\|^{2}_{L^{2}(\mathcal{C}_{1})}\\\
\quad\quad+C\|u_{r}\|^{2}_{L^{\infty}(\mathcal{C}_{1})}\int_{\mathbb{R}^{3}}\frac{h_{\theta}^{2}}{r^{2}}dy+C\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1})}^{2},\\\
C(1+\|(u_{r},u_{z})\|_{L^{\infty}(\mathcal{C}_{1})}+\|\nabla
u_{z}\|_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{h},\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}+\frac{1}{8}\|\nabla(\omega_{z}^{h}\phi)\|^{2}_{L^{2}(\mathcal{C}_{1})}\\\
\quad\quad+C\|(\nabla u_{r},\nabla
h_{\theta})\|^{2}_{L^{2}(\mathcal{C}_{1})}\|(u_{r},h_{\theta})\|^{2}_{L^{\infty}(\mathcal{C}_{1})}+C\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1})}^{2}.\end{cases}$

Thus we deduce

$\displaystyle\|\nabla\omega_{\theta}^{u}\|^{2}_{L^{2}(\mathcal{C}_{2})}\leq
C(1+\|(u_{r},u_{\theta},u_{z},h_{\theta})\|_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{\theta}^{u},\omega_{r}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}$
(4.2)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+C\|(f_{r},f_{z})\|_{L^{2}(\mathcal{C}_{1})}^{2},$
$\displaystyle\|(\nabla\omega_{r}^{u},\nabla\omega_{z}^{u})\|^{2}_{L^{2}(\mathcal{C}_{2})}\leq
C(1+\|(u_{r},u_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u})\|^{2}_{L^{2}(\mathcal{C}_{1})}+C\|f_{\theta}\|_{L^{2}(\mathcal{C}_{1})}^{2},$
(4.3)
$\displaystyle\|(\nabla\omega_{r}^{h},\nabla\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{2})}\leq
C(1+\|(u_{r},u_{z})\|^{2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{h},\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}$
(4.4)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+C\|u_{r}\|^{2}_{L^{\infty}(\mathcal{C}_{1})}+C\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1})}^{2},$

and

$\displaystyle\|\nabla(\omega_{r}^{u},\omega_{z}^{u})\|^{2}_{L^{2}(\mathcal{C}_{2})}\leq
C(1+\|(u_{r},u_{z})\|_{L^{\infty}(\mathcal{C}_{1})}+\|\nabla(u_{r},u_{z})\|_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u})\|^{2}_{L^{2}(\mathcal{C}_{1})}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad+C\|f_{\theta}\|_{L^{2}(\mathcal{C}_{1})}^{2},$
(4.5)
$\displaystyle\|\nabla(\omega_{r}^{h},\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{2})}\leq
C(1+\|(u_{r},u_{z})\|_{L^{\infty}(\mathcal{C}_{1})}+\|(\nabla u_{r},\nabla
u_{z})\|_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{h},\omega_{z}^{h})\|^{2}_{L^{2}(\mathcal{C}_{1})}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad+C\|(\nabla
u_{r},\nabla
h_{\theta})\|^{2}_{L^{2}(\mathcal{C}_{1})}\|(u_{r},h_{\theta})\|^{2}_{L^{\infty}(\mathcal{C}_{1})}+C\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1})}^{2}.$
(4.6)

As previous, utilizing the Brezis-Gallouet’s inequality and (4.2)-(4.6), we
are led to

$\displaystyle\|\omega_{\theta}^{u}\|_{L^{\infty}(\mathcal{\bar{C}}_{2})}\leq
C\bigg{(}1+(1+\|(u_{r},u_{\theta},u_{z},h_{\theta})\|^{1/2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{\theta}^{u},\omega_{r}^{h})\|_{L^{2}(\mathcal{C}_{1})}+\|(f_{r},f_{z})\|_{L^{2}(\mathcal{C}_{1})}\bigg{)}$
$\displaystyle\quad\quad\quad\quad\quad\times\ln^{1/2}\bigg{(}e+\|\Delta\omega_{\theta}^{u}\|_{L^{2}(\mathcal{\bar{C}}_{2})}\bigg{)},$
$\displaystyle\|(\omega_{r}^{u},\omega_{z}^{u})\|_{L^{\infty}(\mathcal{\bar{C}}_{2})}\leq
C\bigg{(}1+(1+\|(u_{r},u_{z})\|_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u})\|_{L^{2}(\mathcal{C}_{1})}+\|f_{\theta}\|_{L^{2}(\mathcal{C}_{1})}\bigg{)}$
$\displaystyle\quad\quad\quad\quad\quad\times\ln^{1/2}\bigg{(}e+\|(\Delta\omega_{r}^{u},\Delta\omega_{z}^{u})\|_{L^{2}(\mathcal{\bar{C}}_{2})}\bigg{)},$
$\displaystyle\|(\omega_{r}^{h},\omega_{z}^{h})\|_{L^{\infty}(\mathcal{\bar{C}}_{2})}\leq
C\bigg{(}1+(1+\|(u_{r},u_{z})\|_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{h},\omega_{z}^{h})\|_{L^{2}(\mathcal{C}_{1})}+\|u_{r}\|_{L^{\infty}(\mathcal{C}_{1})}$
$\displaystyle\quad\quad\quad\quad\quad+\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1})}\bigg{)}\ln^{1/2}\bigg{(}e+\|(\Delta\omega_{r}^{u},\Delta\omega_{z}^{u})\|_{L^{2}(\mathcal{\bar{C}}_{2})}\bigg{)}.$

and

$\displaystyle\|(\omega_{r}^{u},\omega_{z}^{u})\|_{L^{\infty}(\mathcal{\bar{C}}_{2})}\leq
C\bigg{(}1+(1+\|(u_{r},u_{z})\|^{1/2}_{L^{\infty}(\mathcal{C}_{1})}+\|(\nabla
u_{r},\nabla
u_{z})\|^{1/2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{u},\omega_{z}^{u})\|_{L^{2}(\mathcal{C}_{1})}$
$\displaystyle\quad\quad+\|f_{\theta}\|_{L^{2}(\mathcal{C}_{1})}\bigg{)}\ln^{1/2}\bigg{(}e+\|(\Delta\omega_{r}^{u},\Delta\omega_{z}^{u})\|_{L^{2}(\mathcal{\bar{C}}_{2})}\bigg{)},$
$\displaystyle\|(\omega_{r}^{h},\omega_{z}^{h})\|_{L^{\infty}(\mathcal{\bar{C}}_{2})}\leq
C\bigg{(}1+(1+\|(u_{r},u_{z})\|^{1/2}_{L^{\infty}(\mathcal{C}_{1})}+\|(\nabla
u_{r},\nabla
u_{z})\|^{1/2}_{L^{\infty}(\mathcal{C}_{1})})\|(\omega_{r}^{h},\omega_{z}^{h})\|_{L^{2}(\mathcal{C}_{1})}$
$\displaystyle\quad\quad+\|(\nabla u_{r},\nabla
h_{\theta})\|_{L^{2}(\mathcal{C}_{1})}\|(u_{r},h_{\theta})\|_{L^{\infty}(\mathcal{C}_{1})}+\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1})}\bigg{)}\ln^{1/2}\bigg{(}e+\|(\Delta\omega_{r}^{u},\Delta\omega_{z}^{u})\|_{L^{2}(\mathcal{\bar{C}}_{2})}\bigg{)}.$

Then scaling back to find

$\displaystyle\lambda^{2}\|\omega_{\theta}^{u}\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}\leq
C\bigg{(}1+(1+\lambda^{1/2}\|(u_{r},u_{\theta},u_{z},h_{\theta})\|^{1/2}_{L^{\infty}(\mathcal{C}_{1,\lambda})})\lambda^{1/2}\|(\omega_{r}^{u},\omega_{\theta}^{u},\omega_{r}^{h})\|_{L^{2}(\mathcal{C}_{1,\lambda})}$
$\displaystyle\quad\quad+\lambda^{3/2}\|(f_{r},f_{z})\|_{L^{2}(\mathcal{C}_{1,\lambda})}\bigg{)}\ln^{1/2}\bigg{(}e+\lambda^{5/2}\|\Delta\omega_{\theta}^{u}\|_{L^{2}(\mathcal{C}_{2,\lambda})}\bigg{)},$
$\displaystyle\lambda^{2}\|(\omega_{r}^{u},\omega_{z}^{u})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}\leq
C\lambda^{1/2}\bigg{(}1+(1+\lambda\|(u_{r},u_{z})\|_{L^{\infty}(\mathcal{C}_{1,\lambda})})\|(\omega_{r}^{u},\omega_{z}^{u})\|_{L^{2}(\mathcal{C}_{1,\lambda})}$
$\displaystyle\quad\quad+\lambda^{3/2}\|f_{\theta}\|_{L^{2}(\mathcal{C}_{1,\lambda})}\bigg{)}\ln^{1/2}\bigg{(}e+\lambda^{5/2}\|(\Delta\omega_{r}^{u},\Delta\omega_{z}^{u})\|_{L^{2}(\mathcal{C}_{2,\lambda})}\bigg{)},$
$\displaystyle\lambda^{2}\|(\omega_{r}^{h},\omega_{z}^{h})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}\leq
C\bigg{(}1+(1+\lambda\|(u_{r},u_{z})\|_{L^{\infty}(\mathcal{C}_{1,\lambda})})\lambda^{1/2}\|(\omega_{r}^{h},\omega_{z}^{h})\|_{L^{2}(\mathcal{C}_{1,\lambda})}$
$\displaystyle\quad\quad+\lambda\|u_{r}\|_{L^{\infty}(\mathcal{C}_{1,\lambda})}+\lambda^{3/2}\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1,\lambda})}\bigg{)}\ln^{1/2}\bigg{(}e+\lambda^{5/2}\|(\Delta\omega_{r}^{h},\Delta\omega_{z}^{h})\|_{L^{2}(\mathcal{C}_{2,\lambda})}\bigg{)},$

and

$\displaystyle\lambda^{2}\|(\omega_{r}^{u},\omega_{z}^{u})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}\leq
C\lambda^{1/2}\bigg{(}1+(1+\lambda^{1/2}\|(u_{r},u_{z})\|^{1/2}_{L^{\infty}(\mathcal{C}_{1,\lambda})}+\lambda^{3/2}\|f_{\theta}\|_{L^{2}(\mathcal{C}_{1,\lambda})}$
(4.7) $\displaystyle\quad+\lambda\|(\nabla u_{r},\nabla
u_{z})\|^{1/2}_{L^{\infty}(\mathcal{C}_{1,\lambda})})\|(\omega_{r}^{u},\omega_{z}^{u})\|_{L^{2}(\mathcal{C}_{1,\lambda})}\bigg{)}\ln^{1/2}\bigg{(}e+\lambda^{5/2}\|(\Delta\omega_{r}^{u},\Delta\omega_{z}^{u})\|_{L^{2}(\mathcal{C}_{2,\lambda})}\bigg{)},$
$\displaystyle\lambda^{2}\|(\omega_{r}^{h},\omega_{z}^{h})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}\leq
C\bigg{(}1+(1+\lambda^{1/2}\|(u_{r},u_{z})\|^{1/2}_{L^{\infty}(\mathcal{C}_{1,\lambda})}+\lambda\|(\nabla
u_{r},\nabla u_{z})\|^{1/2}_{L^{\infty}(\mathcal{C}_{1,\lambda})})$
$\displaystyle\quad\times\lambda^{1/2}\|(\omega_{r}^{h},\omega_{z}^{h})\|_{L^{2}(\mathcal{C}_{1,\lambda})}+\lambda^{\frac{3}{2}}\|(\nabla
u_{r},\nabla
h_{\theta})\|_{L^{2}(\mathcal{C}_{1,\lambda})}\|(u_{r},h_{\theta})\|_{L^{\infty}(\mathcal{C}_{1,\lambda})}+\lambda^{3/2}\|\bar{\textbf{g}}\|_{L^{2}(\mathcal{C}_{1,\lambda})}\bigg{)}$
$\displaystyle\quad\times\ln^{1/2}\bigg{(}e+\lambda^{5/2}\|(\Delta\omega_{r}^{h},\Delta\omega_{z}^{h})\|_{L^{2}(\mathcal{\bar{C}}_{2})}\bigg{)}.$
(4.8)

By the a priori bound in (1.20), we have

$\displaystyle\|\omega_{\theta}^{u}\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}$
$\displaystyle\leq$ $\displaystyle
C(M)\frac{(\ln\lambda)^{3/4}}{\lambda^{5/4}},$
$\displaystyle\|(\omega_{r}^{u},\omega_{z}^{u})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}$
$\displaystyle\leq$ $\displaystyle C(M)\frac{\ln\lambda}{\lambda},$ (4.9)
$\displaystyle\|(\omega_{r}^{h},\omega_{z}^{h})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}$
$\displaystyle\leq$ $\displaystyle C(M)\frac{\ln\lambda}{\lambda}.$

This verifies (1.25).

We further derive the decay of $\nabla\omega_{\theta}^{u}$ under the
additional assumption that $\textbf{f}\in H^{2}(\mathbb{R}^{3})$ and
$\textbf{g}\in H^{3}(\mathbb{R}^{3})$. Recall that

$(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{\theta}^{u}-\frac{u_{r}\omega_{\theta}^{u}}{r}+\frac{1}{r}\partial_{z}(h_{\theta}^{2}-u_{\theta}^{2})=(\partial^{2}_{r}+\frac{1}{r}\partial_{r}+\partial^{2}_{z}-\frac{1}{r^{2}})\omega_{\theta}^{u}+(\partial_{z}f_{r}-\partial_{r}f_{z}).$
(4.10)

Choose the domain

$\displaystyle\mathcal{C}_{3}=\\{(r,\theta,z):\frac{7}{8}<r<\frac{9}{8},0\leq\theta\leq
2\pi,|z|\leq\frac{1}{4}\\}.$

Take a cut-off function $\phi(y)\in C_{0}^{\infty}(\mathcal{C}_{2})$,
satisfying $0\leq\phi\leq 1$, $\phi(y)=1$ for $y\in\mathcal{C}_{3}$ and
$|\nabla\phi|\leq C$, for some positive constant $C$. Taking $\partial_{r}$ to
the equation in (4.10) and multiplying the resulting equation by
$\partial_{r}\omega_{\theta}^{u}\phi^{2}$, and integrating over
$\mathcal{C}_{2}$, we deduce that

$\displaystyle\int_{\mathcal{C}_{2}}|\nabla(\partial_{r}\omega_{\theta}^{u}\phi)|^{2}dy+\int_{\mathcal{C}_{2}}2\frac{|\partial_{r}\omega_{\theta}^{u}|^{2}\phi^{2}}{r^{2}}dy$
$\displaystyle=\int_{\mathcal{C}_{2}}|\partial_{r}\omega_{\theta}^{u}|^{2}|\nabla\phi|^{2}dy+\int_{\mathcal{C}_{2}}\frac{2}{r^{3}}\omega_{\theta}^{u}\partial_{r}\omega_{\theta}^{u}\phi^{2}dy+\int_{\mathcal{C}_{2}}u_{z}\frac{1}{r}\partial_{z}\omega_{\theta}^{u}\partial_{r}\omega_{\theta}^{u}\phi^{2}dy$
$\displaystyle+\int_{\mathcal{C}_{2}}2u_{r}\partial_{r}(\partial_{r}\omega_{\theta}^{u}\phi)\partial_{r}\omega_{\theta}^{u}\phi
dy-\int_{\mathcal{C}_{2}}\frac{1}{r^{2}}u_{r}\omega_{\theta}^{u}\partial_{r}\omega_{\theta}^{u}\phi^{2}dy+\int_{\mathcal{C}_{2}}\frac{1}{r}u_{r}|\partial_{r}\omega_{\theta}^{u}|^{2}\phi^{2}dy$
$\displaystyle+\int_{\mathcal{C}_{2}}u_{z}[\partial_{z}(\partial_{r}\omega_{\theta}^{u}\phi)\partial_{r}\omega_{\theta}^{u}\phi-|\partial_{r}\omega_{\theta}^{u}|^{2}\phi\partial_{z}\phi+\partial_{z}\omega_{\theta}^{u}\partial_{r}(\partial_{r}\omega_{\theta}^{u}\phi)\phi+\partial_{z}\omega_{\theta}^{u}\partial_{r}\omega_{\theta}^{u}\phi\partial_{r}\phi]dy$
$\displaystyle-\int_{\mathcal{C}_{2}}(u_{r}\partial_{r}+u_{z}\partial_{z})(\partial_{r}\omega_{\theta}^{u}\phi)\partial_{r}\omega_{\theta}^{u}\phi-(u_{r}\partial_{r}+u_{z}\partial_{z})\phi|\partial_{r}\omega_{\theta}^{u}|^{2}\phi
dy$
$\displaystyle-\int_{\mathcal{C}_{2}}u_{r}\bigg{[}\frac{1}{r}\omega_{\theta}^{u}\partial_{r}(\partial_{r}\omega_{\theta}^{u}\phi)\phi+\frac{1}{r}\omega_{\theta}^{u}\partial_{r}\omega_{\theta}^{u}\phi\partial_{r}\phi\bigg{]}dy+\int_{\mathcal{C}_{2}}\frac{2}{r^{2}}(u_{\theta}\omega_{r}^{u}-h_{\theta}\omega_{r}^{h})\partial_{r}\omega_{\theta}^{u}\phi^{2}dy$
$\displaystyle+2\int_{\mathcal{C}_{2}}h_{\theta}[\partial_{r}(\partial_{r}\omega_{\theta}^{u}\phi)\frac{\omega_{r}^{h}\phi}{r}+\frac{1}{r}\partial_{r}\omega_{\theta}^{u}\phi\omega_{r}^{h}\partial_{r}\phi]-u_{\theta}[\partial_{r}(\partial_{r}\omega_{\theta}^{u}\phi)\frac{\omega_{r}^{u}\phi}{r}+\frac{1}{r}\partial_{r}\omega_{\theta}^{u}\phi\omega_{r}^{u}\partial_{r}\phi]dy$
$\displaystyle-\int_{\mathcal{C}_{2}}(\partial_{z}f_{r}-\partial_{r}f_{z})(\partial_{r}(\partial_{r}\omega_{\theta}^{u}\phi)\phi+\partial_{r}\omega_{\theta}^{u}\phi\partial_{r}\phi+\frac{1}{r}\partial_{r}\omega_{\theta}^{u}\phi^{2})dy$
$\displaystyle\leq$ $\displaystyle
C(1+\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2})}+\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2})}^{2})\|\nabla\omega_{\theta}^{u}\|_{L^{2}(\mathcal{C}_{2})}^{2}+C\|\bar{\textbf{f}}\|_{L^{2}(\mathcal{C}_{2})}^{2}$
$\displaystyle+C(1+\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2})}^{2})\|(\omega_{r}^{u},\omega_{\theta}^{u},\omega_{r}^{h})\|^{2}_{L^{2}(\mathcal{C}_{2})}+\frac{1}{8}\|\nabla(\partial_{r}\omega_{\theta}^{u}\phi)\|_{L^{2}(\mathcal{C}_{2})}^{2}$
$\displaystyle\leq$ $\displaystyle
C(1+\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2})}+\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2})}^{2}+\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2})}^{3})\|(\omega_{r}^{u},\omega_{\theta}^{u},\omega_{r}^{h})\|^{2}_{L^{2}(\mathcal{C}_{2})}$
$\displaystyle+(1+\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2})}+\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2})}^{2})\|(f_{r},f_{z})\|_{L^{2}(\mathcal{C}_{2})}^{2}+C\|\bar{\textbf{f}}\|_{L^{2}(\mathcal{C}_{2})}^{2}$
$\displaystyle+\frac{1}{8}\|\nabla(\partial_{r}\omega_{\theta}^{u}\phi)\|_{L^{2}(\mathcal{C}_{2})}^{2},$

where the second inequality follows from (4.2).

Set

$\displaystyle\bar{\mathcal{C}}_{3}\coloneqq\\{(r,z):\frac{7}{8}<r<\frac{9}{8},|z|\leq\frac{1}{4}\\}.$

Utilizing the Brezis-Gallouet inequality in Lemma 2.1 and the above localized
energy estimates, we conclude

$\displaystyle\|\partial_{r}\omega_{\theta}^{u}\|_{L^{\infty}(\bar{\mathcal{C}}_{3})}$
$\displaystyle\leq$ $\displaystyle
C(1+\|\nabla(\partial_{r}\omega_{\theta}^{u})\|_{L^{2}(\mathcal{C}_{3})})\ln^{1/2}(e+\|\Delta(\partial_{r}\omega_{\theta}^{u})\|_{L^{2}(\mathcal{C}_{3})})$
$\displaystyle\leq$ $\displaystyle
C\bigg{(}1+(1+\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2})}^{\frac{1}{2}}+\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2})}+\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2})}^{\frac{3}{2}})\|(\omega_{r}^{u},\omega_{\theta}^{u},\omega_{r}^{h})\|_{L^{2}(\mathcal{C}_{2})}$
$\displaystyle\quad+(1+\|({\textbf{u}},\textbf{h})\|^{\frac{1}{2}}_{L^{\infty}(\mathcal{C}_{2})}+\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2})})\|(f_{r},f_{z})\|_{L^{2}(\mathcal{C}_{2})}+\|\bar{\textbf{f}}\|_{L^{2}(\mathcal{C}_{2})}\bigg{)}$
$\displaystyle\quad\times\ln^{1/2}(e+\|\Delta(\partial_{r}\omega_{\theta}^{u})\|_{L^{2}(\mathcal{C}_{3})}).$

Then scaling back to the domains $\mathcal{C}_{2,\lambda}$ and

$\displaystyle\mathcal{C}_{3,\lambda}=\\{(r,\theta,z):\frac{7\lambda}{8}<r<\frac{9\lambda}{8},0\leq\theta\leq
2\pi,|z|\leq\frac{\lambda}{4}\\},$

we derive that

$\displaystyle\lambda^{3}\|\partial_{r}\omega_{\theta}^{u}\|_{L^{\infty}(\mathcal{C}_{3,\lambda})}$
$\displaystyle\leq$ $\displaystyle
C\bigg{(}1+(1+\lambda^{\frac{1}{2}}\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}^{\frac{1}{2}}+\lambda\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}+\lambda^{\frac{3}{2}}\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}^{\frac{3}{2}})$
$\displaystyle\quad\times\lambda^{\frac{1}{2}}\|(\omega_{r}^{u},\omega_{\theta}^{u},\omega_{r}^{h})\|_{L^{2}(\mathcal{C}_{2,\lambda})}+(1+\lambda^{\frac{1}{2}}\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})}^{\frac{1}{2}}+\lambda\|({\textbf{u}},\textbf{h})\|_{L^{\infty}(\mathcal{C}_{2,\lambda})})$
$\displaystyle\quad\times\lambda^{\frac{3}{2}}\|(f_{r},f_{z})\|_{L^{2}(\mathcal{C}_{2,\lambda})}+\lambda^{\frac{5}{2}}\|\bar{\textbf{f}}\|_{L^{2}(\mathcal{C}_{2,\lambda})}\bigg{)}\ln^{\frac{1}{2}}(e+\lambda^{\frac{7}{2}}\|\Delta(\partial_{r}\omega_{\theta}^{u})\|_{L^{2}(\mathcal{C}_{2,\lambda})})$
$\displaystyle\leq$ $\displaystyle
C(M)\lambda^{\frac{3}{2}}(\frac{\ln\lambda}{\lambda})^{\frac{3}{4}}\lambda^{\frac{1}{2}}(\ln\lambda)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C(M)\lambda^{-\frac{5}{4}}(\ln\lambda)^{\frac{5}{4}},$

where we use the boundedness of $\|\nabla^{4}{\bf
u}\|_{L^{2}(\mathbb{R}^{3})}$ to control
$\|\Delta(\partial_{r}\omega_{\theta}^{u})\|_{L^{2}(\mathcal{C}_{2,\lambda})}$,
which follows from the $L^{p}$ theory to steady Stokes system and a bootstrap
argument under the assumption that ${\bf f}\in H^{2}(\mathbb{R}^{3})$ and
${\bf g}\in H^{3}(\mathbb{R}^{3})$.

Thus

$\displaystyle\|\partial_{r}\omega_{\theta}^{u}\|_{L^{\infty}(\mathcal{C}_{3,\lambda})}\leq
C(M)\frac{(\ln\lambda)^{5/4}}{\lambda^{7/4}}.$

Similarly, there holds

$\displaystyle\|\partial_{z}\omega_{\theta}^{u}\|_{L^{\infty}(\mathcal{C}_{3,\lambda})}\leq
C(M)\frac{(\ln\lambda)^{5/4}}{\lambda^{7/4}}.$

Then

$\displaystyle\|\nabla\omega_{\theta}^{u}\|_{L^{\infty}(\mathcal{C}_{3,\lambda})}\leq
C(M)\frac{(\ln\lambda)^{5/4}}{\lambda^{7/4}}.$

Step 2: we employ the Biot-Savart law to get better decay estimates of
$|\omega_{r}^{u}|+|\omega_{z}^{u}|$ and $|\omega_{r}^{h}|+|\omega_{z}^{h}|$.

Since $-\Delta(u_{r}{\textbf{e}}_{r}+u_{z}{\textbf{e}}_{z})={\rm
curl\,}(\omega_{\theta}^{u}{\textbf{e}}_{\theta})$, then according to the
formula (2.105) in [28, Page 77],

$\displaystyle\nabla(u_{r}{\textbf{e}}_{r}+u_{z}{\textbf{e}}_{z})=\mathbb{K}\ast(\omega_{\theta}^{u}{\textbf{e}}_{\theta})+\mathbb{C}(\omega_{\theta}^{u}{\textbf{e}}_{\theta}),$
(4.11)

where $\mathbb{C}(\omega_{\theta}^{u}{\textbf{e}}_{\theta})$ is a matrix whose
entries are components of $\omega_{\theta}^{u}{\textbf{e}}_{\theta}$ and
$\mathbb{K}$ is a Calderon-Zymund kernel.

Applying Lemma 2.2 to (4.11) and note the decay of $\omega_{\theta}^{u}$, we
can get

$|\nabla u_{r}|+|\nabla u_{z}|\leq C\frac{(\ln\lambda)^{7/4}}{\lambda^{5/4}}.$
(4.12)

Now go back to (4.7) and (4.8), we get better decay on
$|\omega_{r}^{u}|+|\omega_{z}^{u}|$ and $|\omega_{r}^{h}|+|\omega_{z}^{h}|$,

$\displaystyle|\omega_{r}^{u}|+|\omega_{z}^{u}|$ $\displaystyle\leq$
$\displaystyle
C\lambda^{-3/2}\bigg{(}\lambda^{1/2}(\lambda^{-1/2}(\ln\lambda)^{1/2})^{1/2}+\lambda(\lambda^{-5/4}(\ln\lambda)^{7/4})^{1/2}\bigg{)}(\ln\lambda)^{1/2}$
(4.13) $\displaystyle\leq$ $\displaystyle
C\lambda^{-9/8}(\ln\lambda)^{11/8},\quad\text{ for large $\lambda$},$
$\displaystyle|\omega_{r}^{h}|+|\omega_{z}^{h}|$ $\displaystyle\leq$
$\displaystyle C\lambda^{-2}\lambda(\ln\lambda)^{7/8}(\ln\lambda)^{1/2}$
(4.14) $\displaystyle\leq$ $\displaystyle
C\lambda^{-1}(\ln\lambda)^{11/8},\quad\text{ for large $\lambda$}.$

Step 3: Improved decay rate of $\nabla h_{\theta}$ by weighted energy
estimates. By slightly modifying the proof in Lemma 3.2 in [35] (taking
$\delta=(\frac{1}{2})^{-}$ in that lemma), we obtain the following estimates,
which will be used in the proof of Theorem 1.3.

######  Lemma 4.1.

Let $({\bf u},{\bf h},p)$ be an axially symmetric smooth D-solutions to
inhomogeneous stationary MHD equations with _f_ and _g_ satisfying (1.19) and
(1.23), where ${\bf h}(r,z)=h_{\theta}(r,z){\bf e}_{\theta}$. Then the
following estimates hold

$\displaystyle\int_{\mathbb{R}^{3}}|\omega^{u}_{\theta}|^{2}dx+\int_{\mathbb{R}^{3}}r^{(\frac{3}{2})^{-}}|\nabla\omega^{u}_{\theta}|^{2}dx+\int_{\mathbb{R}^{3}}r^{(\frac{5}{2})^{-}}|\partial_{z}\nabla\omega^{u}_{\theta}|^{2}dx\leq
C(M).$

######  Lemma 4.2.

Let $({\bf u},{\bf h},p)$ be an axially symmetric smooth D-solutions to
inhomogeneous stationary MHD equations with _f_ satisfying (1.19), (1.23) and
_g_ satisfying (1.24), where ${\bf h}(r,z)=h_{\theta}(r,z){\bf e}_{\theta}$,
suppose that

$\displaystyle|u_{r}(r,z)|+|u_{\theta}(r,z)|+|u_{z}(r,z)|+|h_{\theta}(r,z)|$
$\displaystyle\leq$ $\displaystyle C(1+r)^{-\delta},$ (4.15)
$\displaystyle|\nabla u_{r}(r,z)|+|\nabla u_{z}(r,z)|$ $\displaystyle\leq$
$\displaystyle C(1+r)^{-1-\gamma},$ (4.16)

holds for some $\delta,\gamma\in[0,1]$. Then the following estimates holds

$\displaystyle\int_{\mathbb{R}^{3}}r^{1+\delta\land\gamma}(|\nabla\omega_{r}^{u}|^{2}+|\nabla\omega_{z}^{u}|^{2})dx\leq
C(M),$ (4.17)
$\displaystyle\int_{\mathbb{R}^{3}}r^{1+\delta\wedge\gamma}(|\nabla\omega_{r}^{h}|^{2}+|\nabla\omega_{z}^{h}|^{2})dx\leq
C(M),$ (4.18)
$\displaystyle\int_{\mathbb{R}^{3}}r^{1+\delta\wedge\gamma+2\delta}(|\partial_{z}\nabla\omega_{r}^{u}|^{2}+|\partial_{z}\nabla\omega_{z}^{u}|^{2})dx\leq
C(M),$ (4.19)
$\displaystyle\int_{\mathbb{R}^{3}}r^{1+\delta\wedge\gamma+2\delta}(|\partial_{z}\nabla\omega_{r}^{h}|^{2}+|\partial_{z}\nabla\omega_{z}^{h}|^{2})dx\leq
C(M),$ (4.20)

where $\delta\wedge\gamma=\mathrm{min}\\{\delta,\gamma\\}$.

In particular, by Lemma 2.3, we obtain the following decay rate:

$\displaystyle|\omega_{r}^{u}(r,z)|+|\omega^{u}_{z}(r,z)|$ $\displaystyle\leq$
$\displaystyle
C(M)r^{-\frac{7}{8}-\frac{1}{8}(3(\delta\wedge\gamma)+2\delta)},$
$\displaystyle|\omega_{r}^{h}(r,z)|+|\omega^{h}_{z}(r,z)|$ $\displaystyle\leq$
$\displaystyle
C(M)r^{-\frac{7}{8}-\frac{1}{8}(3(\delta\wedge\gamma)+2\delta)}.$

###### Proof.

According to Lemma 3.8 in [35], we have (4.17) and (4.19). Similarly, we can
prove (4.18). It remains to prove (4.20).

We start to derive the weighted estimates of
$\nabla\partial_{rz}^{2}h_{\theta}$.

$\displaystyle\partial_{rz}^{2}\bigg{[}(u_{r}\partial_{r}+u_{z}\partial_{z})h_{\theta}-\frac{u_{r}}{r}h_{\theta}\bigg{]}=\bigg{(}\partial_{r}^{2}+\frac{1}{r}\partial_{r}+\partial_{z}^{2}-\frac{1}{r^{2}}\bigg{)}\partial_{rz}^{2}h_{\theta}$
(4.21)
$\displaystyle\quad\quad-\frac{1}{r^{2}}\partial_{rz}^{2}h_{\theta}+\frac{2}{r^{3}}\partial_{z}h_{\theta}+\partial_{rz}^{2}(\partial_{z}g_{r}-\partial_{r}g_{z})$

take $\eta^{2}r^{a_{2}}\partial_{rz}^{2}h_{\theta}$ as a test function to
(4.21) and integrating over $\mathbb{R}^{3}$, we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\nabla\partial_{rz}^{2}h_{\theta}|^{2}dx+\frac{1}{2}\int_{\mathbb{R}^{3}}\bigg{[}\partial_{r}(\eta^{2}r^{a_{2}-1})-\partial^{2}_{r}(\eta^{2}r^{a_{2}})\bigg{]}|\partial_{rz}^{2}h_{\theta}|^{2}dx$
$\displaystyle+2\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}-2}|\partial_{rz}^{2}h_{\theta}|^{2}dx-\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}\partial_{rzz}^{3}h_{\theta}(u_{r}\partial_{r}+u_{z}\partial_{z})\partial_{r}h_{\theta}dx$
$\displaystyle-\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}\partial_{rzz}^{3}h_{\theta}(\partial_{r}u_{r}\partial_{r}+\partial_{r}u_{z}\partial_{z})h_{\theta}dx-\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}\partial^{3}_{rzz}h_{\theta}\bigg{(}\frac{u_{r}}{r}-\frac{\partial_{r}u_{r}}{r}\bigg{)}\frac{h_{\theta}}{r}dx$
$\displaystyle+\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}\partial^{3}_{rzz}h_{\theta}\frac{u_{r}}{r}\partial_{r}h_{\theta}dx+2\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}-2}\partial^{3}_{rzz}h_{\theta}\frac{h_{\theta}}{r}dx$
$\displaystyle-\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}\partial_{rzz}^{3}h_{\theta}\partial_{r}(\partial_{z}g_{r}-\partial_{r}g_{z})dx$
$\displaystyle\eqqcolon$
$\displaystyle-\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\nabla\partial_{rz}^{2}h_{\theta}|^{2}dx+\sum_{i=1}^{2}D_{2i}+\sum_{j=1}^{6}E_{2j}.$

Using previous estimates, we can bound these terms as follows.

$\displaystyle|D_{21}|$ $\displaystyle\leq$ $\displaystyle C\int_{r\geq
r_{0}}r^{a_{2}-2}|\partial_{rz}^{2}h_{\theta}|^{2}dx\leq
C\int_{\mathbb{R}^{3}}r^{1+\gamma}|\partial_{rz}^{2}h_{\theta}|^{2}dx,\quad\text{if
}a_{2}\leq 3+\gamma,$ $\displaystyle|D_{22}|$ $\displaystyle\leq$
$\displaystyle
C\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}-2}|\partial^{2}_{rz}h_{\theta}|^{2}dx\leq
C\int_{\mathbb{R}^{3}}r^{1+\gamma}|\partial_{rz}^{2}h_{\theta}|^{2}dx,\quad\text{if
}a_{2}\leq 3+\gamma,$ $\displaystyle|E_{21}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{16}\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\nabla\partial_{rz}^{2}h_{\theta}|^{2}dx+\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}(|u_{r}|+|u_{z}|)^{2}|\partial_{rz}^{2}h_{\theta}|^{2}dx$
$\displaystyle\leq$
$\displaystyle\frac{1}{16}\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\nabla\partial^{2}_{rz}h_{\theta}|^{2}dx+C\int_{\mathbb{R}^{3}}r^{1+\gamma}|\partial^{2}_{rz}h_{\theta}|^{2}dx,\quad\text{if
}a_{2}\leq 1+\gamma+2\delta,$ $\displaystyle|E_{22}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{16}\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\nabla\partial_{rz}^{2}h_{\theta}|^{2}dx+\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}(|\nabla
u_{r}|+|\nabla u_{z}|)^{2}|\nabla h_{\theta}|^{2}dx$ $\displaystyle\leq$
$\displaystyle\frac{1}{16}\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\nabla\partial^{2}_{rz}h_{\theta}|^{2}dx+C\|\nabla
h_{\theta}\|_{L^{2}}^{2},\quad\text{if }a_{2}\leq 2(1+\gamma),$
$\displaystyle|E_{23}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{16}\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\nabla\partial_{rz}^{2}h_{\theta}|^{2}dx+\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}\bigg{(}\frac{|u_{r}|^{2}}{r^{2}}+|\nabla
u_{r}|^{2}\bigg{)}|\frac{h_{\theta}}{r}|^{2}dx$ $\displaystyle\leq$
$\displaystyle\frac{1}{16}\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\nabla\partial_{rz}^{2}h_{\theta}|^{2}dx+C\int_{\mathbb{R}^{3}}\frac{h_{\theta}^{2}}{r^{2}},\quad\text{if
}a_{2}\leq 2\min\\{1+\gamma,1+\delta\\},$ $\displaystyle|E_{24}|$
$\displaystyle\leq$
$\displaystyle\frac{1}{16}\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\nabla\partial_{rz}^{2}h_{\theta}|^{2}dx+C\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}\frac{u_{r}^{2}}{r^{2}}|\nabla
h_{\theta}|^{2}dx,\quad\text{if }a_{2}\leq 2(1+\delta),$
$\displaystyle|E_{25}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{16}\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\nabla\partial_{rz}^{2}h_{\theta}|^{2}dx+C\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}-4}\frac{h_{\theta}^{2}}{r^{2}}dx$
$\displaystyle\leq$
$\displaystyle\frac{1}{16}\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\nabla\partial_{rz}^{2}h_{\theta}|^{2}dx+C\int_{\mathbb{R}^{3}}\frac{h_{\theta}^{2}}{r^{2}}dx,\quad\text{if
}a_{2}\leq 4,$ $\displaystyle|E_{26}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{16}\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\nabla\partial_{rz}^{2}h_{\theta}|^{2}dx+C\int_{\mathbb{R}^{3}}\eta^{2}r^{a_{2}}|\partial_{r}(\partial_{z}g_{r}-\partial_{r}g_{z})|^{2}dx.$

We infer that

$\displaystyle\int_{\mathbb{R}^{3}}r^{1+2\delta+\gamma}|\nabla\partial_{rz}^{2}h_{\theta}|^{2}dx<\infty.$
(4.22)

Similarly, there also holds

$\displaystyle\int_{\mathbb{R}^{3}}r^{1+2\delta+\gamma}|\nabla\partial_{zz}^{2}h_{\theta}|^{2}dx<\infty.$
(4.23)

Therefore

$\displaystyle\int_{\mathbb{R}^{3}}r^{1+2\delta+\gamma}(|\nabla\partial_{z}\omega^{h}_{r}|^{2}+|\nabla\partial_{z}\omega^{h}_{z}|^{2})dx<\infty.$

∎

By (1.20) and (4.12), choose $\delta=(\frac{1}{2})^{-}$ and
$\gamma=(\frac{1}{4})^{-}$ in Lemma 4.2, and we are led to

$\displaystyle\int_{\mathbb{R}^{3}}r^{(\frac{5}{4})^{-}}(|\nabla\omega^{h}_{r}|^{2}+|\nabla\omega^{h}_{z}|^{2})dx<\infty,$
(4.24)
$\displaystyle\int_{\mathbb{R}^{3}}r^{(\frac{9}{4})^{-}}(|\nabla\partial_{z}\omega^{h}_{r}|^{2}+|\nabla\partial_{z}\omega^{h}_{z}|^{2})dx<\infty,$
(4.25)

and

$\displaystyle|\omega^{h}_{r}(r,z)|+|\omega_{z}^{h}(r,z)|\leq
C(M)r^{-(\frac{35}{32})^{-}}.$ (4.26)

Note that $\omega_{r}^{h}=-\partial_{z}h_{\theta}$ and
$\partial_{r}h_{\theta}=\omega_{z}^{h}-\frac{h_{\theta}}{r}$, we also have
(1.28). The proof of Theorem 1.2 is finished.

##  5 Proof of Theorem 1.3

Note that the scaling technique by employing the Brezis-Gallouet inequality
can not be applied to get any decay along the axis, the only way we know is to
do the weighted energy estimates with the weight $\rho=\sqrt{r^{2}+z^{2}}$.

The proof is organized as follows. In Step 1, some weighted energy estimates
for $\Pi\coloneqq\frac{h_{\theta}}{r}$ were derived by using the weight
$\rho=\sqrt{r^{2}+z^{2}}$ as in [35]. In Step 2, we further infer that
$|\Pi(r,z)|^{2}\leq\frac{C}{r^{2}|z|^{(\frac{31}{84}+\frac{41}{84}\tau)^{-}}}$
and

$\displaystyle|h_{\theta}(r,z)|$ $\displaystyle\leq$ $\displaystyle
C(M)(\rho+1)^{-(\frac{31}{168}+\frac{41}{168}\tau)^{-}},\quad\forall(r,z)\in\mathbb{R}_{+}\times\mathbb{R},\rho=\sqrt{r^{2}+z^{2}}.$

In Step 3, where the special case $u_{\theta}\equiv 0$ is considered, it will
be shown that

$\displaystyle|\omega_{\theta}^{u}(r,z)|$ $\displaystyle\leq$ $\displaystyle
C(M)(1+\rho)^{-(\frac{13}{64}(1+\tau))^{-}},\quad\forall(r,z)\in\mathbb{R}_{+}\times\mathbb{R},\rho=\sqrt{r^{2}+z^{2}},$

using similar arguments in the previous two steps on the equation of
$\Omega\coloneqq\frac{\omega_{\theta}^{u}}{r}$. In Step 4, the decay of
$(u_{r},u_{z})$ will be obtained from $\omega_{\theta}^{u}$ by the Biot-Savart
law. Finally, we complete the proof by an iteration of $\tau$.

###### Proof of Theorem 1.3.

Step 1. We first notice that the quantity $\Pi\coloneqq\frac{h_{\theta}}{r}$
satisfies the following elliptic equation

$\displaystyle(u_{r}\partial_{r}+u_{z}\partial_{z})\Pi=\bigg{(}\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\partial_{z}^{2}\bigg{)}\Pi+\frac{1}{r}(\partial_{z}g_{r}-\partial_{r}g_{z}).$
(5.1)

Following the argument developed in [35], we have the following weighted
estimates for $\Pi$.

Suppose that

$\displaystyle|u_{r}(r,z)|+|u_{z}(r,z)|\leq
C(1+\rho)^{-\tau},\quad\rho=\sqrt{r^{2}+z^{2}}$ (5.2)

for some $\tau\in[0,1]$, and _g_ satisfies (1.31) and (1.31), then we have

$\displaystyle\int_{\mathbb{R}^{3}}|\Pi(r,z)|^{2}rdrdz<\infty,$ (5.3)
$\displaystyle\int_{\mathbb{R}^{3}}\rho^{1+\tau}|\nabla\Pi(r,z)|^{2}rdrdz<\infty,$
(5.4)
$\displaystyle\int_{\mathbb{R}^{3}}\rho^{1+3\tau}|\nabla\partial_{z}\Pi(r,z)|^{2}rdrdz<\infty.$
(5.5)

Step 2. Derive the decay rate for $h_{\theta}$.

By a priori estimates, we have

$\displaystyle\int_{\mathbb{R}^{3}}\frac{|h_{\theta}(r,z)|^{2}}{r^{2}}+|\nabla
h_{\theta}(r,z)|^{2}dx<\infty.$ (5.6)

Combining the results in (4.18) and (5.3)-(5.6), then

$\displaystyle\int_{\mathbb{R}^{3}}|\Pi(r,z)|^{2}dx<\infty,$ (5.7)
$\displaystyle\int_{\mathbb{R}^{3}}(r^{2}+|z|^{1+\tau})|\nabla\Pi(r,z)|^{2}dx<\infty,$
(5.9)
$\displaystyle\int_{\mathbb{R}^{3}}(r^{3+\gamma}+|z|^{1+3\tau})|\nabla\partial_{z}\Pi(r,z)|^{2}dx<\infty,$

where $\gamma$ can be any constant less than $\frac{1}{4}$. Fix $d>1$, then
for each $n\in\mathbb{N}$,

$\displaystyle\int_{2^{n}}^{2^{n+1}}\int_{d}^{\infty}|\Pi(r,z)|^{2}rdrdz<\infty.$

By mean value theorem, there exists $z_{n}\in[2^{n},2^{n+1}]$ such that

$\displaystyle\int_{d}^{\infty}|\Pi(r,z_{n})|^{2}rdr\leq\frac{C}{z_{n}}.$

Then for any $z$, choose $z_{n}>z$ and

$\displaystyle\int_{d}^{\infty}|\Pi(r,z)|^{2}rdr$ $\displaystyle=$
$\displaystyle\int_{d}^{\infty}|\Pi(r,z_{n})|^{2}rdr-2\int_{d}^{\infty}\int_{z}^{z_{n}}\Pi(r,t)\partial_{t}\Pi(r,t)rdrdt=:I_{1}+I_{2},$
$\displaystyle|I_{2}|$ $\displaystyle\leq$
$\displaystyle\bigg{(}\int_{d}^{\infty}\int_{z}^{z_{n}}|\Pi(r,t)|^{2}rdrdt\bigg{)}^{1/2}\bigg{(}\int_{d}^{\infty}\int_{z}^{z_{n}}|\partial_{t}\Pi(r,t)|^{2}rdrdt\bigg{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle\frac{C}{|z|^{\frac{1}{2}(1+\tau)}}.$

Letting $z_{n}\to\infty$, then $I_{1}\to 0$ and

$\displaystyle\int_{d}^{\infty}|\Pi(r,z)|^{2}rdr\leq\frac{C}{|z|^{\frac{1}{2}(1+\tau)}}.$
(5.10)

Similarly, one can find $z_{n}\in[2^{n},2^{n+1}]$ such that

$\displaystyle\int_{d}^{\infty}|\nabla\Pi(r,z_{n})|^{2}rdr$
$\displaystyle\leq$ $\displaystyle\frac{C}{z_{n}^{2}},$
$\displaystyle\int_{d}^{\infty}|\nabla\Pi(r,z)|^{2}rdr$ $\displaystyle=$
$\displaystyle\int_{d}^{\infty}|\nabla\Pi(r,z_{n})|^{2}rdr-2\int_{d}^{\infty}\int_{z}^{z_{n}}\nabla\Pi(r,t)\cdot\partial_{t}\nabla\Pi(r,t)rdrdt$
$\displaystyle\eqqcolon$ $\displaystyle J_{1}+J_{2},$ $\displaystyle|J_{2}|$
$\displaystyle\leq$
$\displaystyle\bigg{(}\int_{d}^{\infty}\int_{z}^{z_{n}}|\nabla\Pi(r,t)|^{2}rdrdt\bigg{)}^{1/2}\bigg{(}\int_{d}^{\infty}\int_{z}^{z_{n}}|\partial_{t}\nabla\Pi(r,t)|^{2}rdrdt\bigg{)}^{1/2}$
$\displaystyle\leq$
$\displaystyle\begin{cases}\frac{C}{d^{\frac{5+\gamma}{2}}},\\\
\frac{C}{|z|^{1+2\tau}}.\end{cases}$

Letting $n\to\infty$, $J_{1}\to 0$. Recall that $\gamma=(\frac{1}{4})^{-}$,
then
$J_{2}\leq\min\bigg{\\{}\frac{C}{d^{\frac{5+\gamma}{2}}},\frac{C}{|z|^{1+2\tau}}\bigg{\\}}$
and by interpolation

$\displaystyle\int_{d}^{\infty}|\nabla\Pi(r,z)|^{2}rdr\leq\bigg{(}\frac{C}{d^{(\frac{21}{8})^{-}}}\bigg{)}^{\frac{16}{21}}\bigg{(}\frac{C}{|z|^{1+2\tau}}\bigg{)}^{\frac{5}{21}}\leq\frac{C}{d^{2}|z|^{(\frac{5}{21}(1+2\tau))^{-}}}.$

Finally,

$\displaystyle|\Pi(d,z)|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{r_{1}-d}\int_{d}^{r_{1}}|\Pi(r,z)|^{2}dr+(|\Pi(r,z)|^{2}-\frac{1}{r_{1}-d}\int_{d}^{r_{1}}|\Pi(r,z)|^{2}dr)$
$\displaystyle\eqqcolon$ $\displaystyle H_{1}+H_{2},$ $\displaystyle|H_{2}|$
$\displaystyle=$
$\displaystyle\bigg{|}|\Pi(d,z)|^{2}-|\Pi(d_{*},z)|^{2}\bigg{|}\leq
2\int_{d}^{r_{1}}|\Pi(r,z)\partial_{r}\Pi(r,z)|dr$ $\displaystyle\leq$
$\displaystyle\frac{C}{d}\bigg{(}\int_{d}^{\infty}|\Pi(r,z)|^{2}rdr\bigg{)}^{1/2}\bigg{(}\int_{d}^{\infty}|\nabla\Pi(r,z)|^{2}rdr\bigg{)}^{1/2}$
$\displaystyle\leq$
$\displaystyle\frac{C}{d}\bigg{(}\frac{C}{|z|^{\frac{1}{2}(1+\tau)}}\bigg{)}^{1/2}\bigg{(}\frac{C}{d^{2}|z|^{(\frac{5}{21}(1+2\tau))^{-}}}\bigg{)}^{\frac{1}{2}}\leq\frac{C}{d^{2}|z|^{(\frac{31}{84}+\frac{41}{84}\tau)^{-}}},$

It follows from (5.11) that $\displaystyle\lim_{r_{1}\to\infty}H_{1}=0$ and

$\displaystyle|h_{\theta}(d,z)|\leq
C(M)|z|^{-(\frac{31}{168}+\frac{41}{168}\tau)^{-}}.$ (5.11)

Together with previous decay results on $r$, we have

$\displaystyle|h_{\theta}(r,z)|$ $\displaystyle\leq$ $\displaystyle
C(M)(\rho+1)^{-(\frac{31}{168}+\frac{41}{168}\tau)^{-}},\quad\forall(r,z)\in\mathbb{R}_{+}\times\mathbb{R},\rho=\sqrt{r^{2}+z^{2}}.$
(5.12)

In general case where $u_{\theta}\neq 0$, we only have $\tau=0$, which implies
that $|h_{\theta}|\leq C(M)(\rho+1)^{-\frac{31}{168}}$. We remark here that we
do not obtain any decay rate for $\Pi$, since the weighted estimate for
$\nabla\partial_{r}\Pi$ is not available.

Step 3. For the special case where the flow has zero swirl $u_{\theta}\equiv
0$, we can further derive the decay rates of ${\bf u}$.

In this case, one has

$\displaystyle(u_{r}\partial_{r}+u_{z}\partial_{z})\omega_{\theta}^{u}-\frac{u_{r}}{r}\omega^{u}_{\theta}+\frac{1}{r}\partial_{z}(h_{\theta}^{2})=\bigg{(}\partial_{r}^{2}+\frac{1}{r}\partial_{r}+\partial_{z}^{2}-\frac{1}{r^{2}}\bigg{)}\omega^{u}_{\theta}-(\partial_{z}f_{r}-\partial_{r}f_{z}),$

and

$\displaystyle(u_{r}\partial_{r}+u_{z}\partial_{z})\Omega+\partial_{z}\Pi^{2}=\bigg{(}\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\partial_{z}^{2}\bigg{)}\Omega+\frac{1}{r}(\partial_{z}f_{r}-\partial_{r}f_{z}),$

for $\Omega\coloneqq\frac{\omega^{u}_{\theta}(r,z)}{r}$. We run the same
argument as in [35] for $\Omega$ to derive the estimate
$\int_{\mathbb{R}^{3}}\rho^{d_{1}}|\nabla\Omega|^{2}dx$, there is an
additional term

$\displaystyle\bigg{|}\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{1}}\Omega\partial_{z}\Pi^{2}dx\bigg{|}$
$\displaystyle\leq$ $\displaystyle
2\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{1}}|\Pi||\Omega||\partial_{z}\Pi|dx$
$\displaystyle\leq$ $\displaystyle
C\|\Pi\|_{L^{\infty}}\|\Omega\|_{L^{2}}(\int_{\mathbb{R}^{3}}\rho^{2d_{1}}|\partial_{z}\Pi|^{2}dx)^{1/2}$
$\displaystyle<$ $\displaystyle\infty,\quad\text{if
}d_{1}\leq\frac{1+\tau}{2},$

where $\phi$ is a cut-off function satisfying $\phi\in
C^{\infty}_{0}(\mathbb{R}^{3})$, $\phi=\phi(\rho)$, $0\leq\phi\leq 1$,
$\phi(\rho)=1$ on $2\rho_{0}\leq\rho\leq\rho_{1}$, $\phi(\rho)=0$ on
$\rho\leq\rho_{0}$ or $\rho\geq 2\rho_{1}$, such that
$|\nabla\phi|\leq\frac{C}{\rho_{0}}$ on $\rho_{0}<\rho<2\rho_{0}$ and
$|\nabla\phi|\leq\frac{C}{\rho_{1}}$ on $\rho_{1}<\rho<2\rho_{1}$, and
$\nabla\phi=0$ elsewhere. Thus

$\displaystyle\int_{\mathbb{R}^{3}}\rho^{\frac{1+\tau}{2}}|\nabla\Omega(r,z)|^{2}dx<\infty.$
(5.13)

To derive the estimate of $\nabla\partial_{z}\Omega$, we see that
$\partial_{z}\Omega$ satisfies

$\displaystyle\partial_{z}[(u_{r}\partial_{r}+u_{z}\partial_{z})\Omega+\partial_{z}\Pi^{2}]=\bigg{(}\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\partial_{z}^{2}\bigg{)}\partial_{z}\Omega+\partial_{z}(\frac{1}{r}(\partial_{z}f_{r}-\partial_{r}f_{z})).$
(5.14)

Multiplying (5.14) by $\phi^{2}\rho^{d_{2}}\partial_{z}\Omega$ and integrating
over $\mathbb{R}^{3}$, then we get

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{2}}|\nabla\partial_{z}\Omega|^{2}dx-\frac{1}{2}\int_{\mathbb{R}^{3}}[\partial_{r}^{2}(\phi^{2}\rho^{d_{2}})+\partial_{z}^{2}(\phi^{2}\rho^{d_{2}})]|\partial_{z}\Omega|^{2}dx$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{3}}\partial_{r}(\phi^{2}\rho^{d_{2}})\frac{|\partial_{z}\Omega|^{2}}{r}dx-\int_{\mathbb{R}^{3}}\partial_{z}(\phi^{2}\rho^{d_{2}})\partial_{z}\Omega(u_{r}\partial_{r}+u_{z}\partial_{z})\Omega
dx$
$\displaystyle-\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{2}}\partial_{z}^{2}\Omega(u_{r}\partial_{r}+u_{z}\partial_{z})\Omega
dx-\int_{\mathbb{R}^{3}}\partial_{z}(\phi^{2}\rho^{d_{2}})\partial_{z}\Omega\cdot\partial_{z}\Pi^{2}dx$
$\displaystyle-\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{2}}\partial_{z}^{2}\Omega\cdot\partial_{z}\Pi^{2}dx+\int_{\mathbb{R}^{3}}\partial_{z}(\phi^{2}\rho^{d_{2}})\partial_{z}\Omega(\frac{(\partial_{z}f_{r}-\partial_{r}f_{z})}{r})dx$
$\displaystyle+\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{2}}\partial_{z}^{2}\Omega\frac{(\partial_{z}f_{r}-\partial_{r}f_{z})}{r}dx$
$\displaystyle\eqqcolon$
$\displaystyle\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{2}}|\nabla\partial_{z}\Omega|^{2}dx+\sum_{i=1}^{6}K_{i}+\sum_{j=1}^{2}Q_{j}.$

These terms can be bounded as follows.

$\displaystyle|K_{1}|+|K_{2}|$ $\displaystyle\leq$ $\displaystyle
C\int_{\rho\geq\rho_{0}}\rho^{d_{2}-2}|\nabla\Omega|^{2}dx\leq\infty,\quad\text{if
}d_{2}\leq\frac{5+\tau}{2},$ $\displaystyle|K_{3}|$ $\displaystyle\leq$
$\displaystyle
C\int_{\rho\geq\rho_{0}}\rho^{d_{2}-1}|(u_{r},u_{z})||\nabla\Omega|^{2}dx\leq\infty,\quad\text{if
}d_{2}\leq\frac{3}{2}(1+\tau),\ $ $\displaystyle|K_{4}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{8}\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{2}}|\nabla\partial_{z}\Omega|^{2}dx+C\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{2}}|(u_{r},u_{z})|^{2}|\nabla\Omega|^{2}dx,\quad\text{if
}d_{2}\leq\frac{1}{2}+\frac{5\tau}{2},$ $\displaystyle|K_{5}|$
$\displaystyle\leq$ $\displaystyle
C\int_{\mathbb{R}^{3}}\phi\rho^{d_{2}-1}|\nabla\Omega||\Pi||\partial_{z}\Pi|dx$
$\displaystyle\leq$ $\displaystyle
C\int_{\mathbb{R}^{3}}\rho^{\frac{1+\tau}{2}}|\nabla\Omega|^{2}dx+C\int_{\mathbb{R}^{3}}\rho^{2(d_{2}-1)-\frac{1+\tau}{2}}|\Pi|^{2}|\partial_{z}\Pi|^{2}dx$
$\displaystyle\leq$ $\displaystyle
C\int_{\mathbb{R}^{3}}\rho^{1+\tau}|\nabla\Omega|^{2}dx+C\int_{\mathbb{R}^{3}}\rho^{1+\tau}|\partial_{z}\Pi|^{2}dx,\quad\text{if
}d_{2}\leq\frac{7}{4}+\frac{3}{4}\tau,$ $\displaystyle|K_{6}|$
$\displaystyle\leq$
$\displaystyle\frac{1}{8}\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{2}}|\partial_{z}^{2}\Omega|^{2}dx+C\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{2}}\Pi^{2}|\partial_{z}\Pi|^{2}dx$
$\displaystyle\leq$
$\displaystyle\frac{1}{8}\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{2}}|\partial_{z}^{2}\Omega|^{2}dx+C\int_{\mathbb{R}^{3}}\rho^{1+\tau}|\partial_{z}\Pi|^{2}dx,\quad\text{if
}d_{2}\leq 1+\tau,$

and

$\displaystyle|Q_{1}|$ $\displaystyle\leq$ $\displaystyle
C\int_{\rho\geq\rho_{0}}\rho^{2(d_{2}-1)-(1+\tau)}|\frac{\partial_{z}f_{r}-\partial_{r}f_{z}}{r}|^{2}dx+C\int_{\rho\geq\rho_{0}}\rho^{\frac{1+\tau}{2}}|\nabla\Omega|^{2}dx,\quad\text{if
}d_{2}\leq\frac{4+\tau}{2},$ $\displaystyle|Q_{2}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{8}\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{2}}|\nabla\partial_{z}\Omega|^{2}dx+C\int_{\mathbb{R}^{3}}\phi^{2}\rho^{d_{2}}|\frac{\partial_{z}f_{r}-\partial_{r}f_{z}}{r}|^{2}dx.$

Letting $\rho_{1}\to\infty$, we obtain

$\displaystyle\int_{\mathbb{R}^{3}}\rho^{1+\tau}|\nabla\partial_{z}\Omega|^{2}dx<\infty.$
(5.15)

Combining the weighted energy estimate in Lemma 4.1 and (5.13)-(5.15), we find

$\displaystyle\int_{\mathbb{R}^{3}}r^{2}|\Omega|^{2}dx<\infty,$
$\displaystyle\int_{\mathbb{R}^{3}}(r^{3+\delta}+|z|^{\frac{1+\tau}{2}})|\nabla\Omega|^{2}dx<\infty,$
$\displaystyle\int_{\mathbb{R}^{3}}(r^{3+3\delta}+|z|^{1+\tau})|\nabla\partial_{z}\Omega|^{2}dx<\infty.$

Same as Step 2, fix any $d>1$, we have

$\displaystyle\int_{d}^{\infty}|\Omega(r,z)|^{2}rdr\leq\frac{C}{d|z|^{\frac{1+\tau}{4}}}.$

For each $n\in\mathbb{N}$,

$\displaystyle\int_{2^{n}}^{2^{n+1}}\int_{d}^{\infty}|z|^{\frac{1+\tau}{2}}|\nabla\Omega(r,z)|^{2}rdrdz<\infty.$

By mean value theorem, one can find $z_{n}\in[2^{n},2^{n+1}]$ such that

$\displaystyle\int_{d}^{\infty}|\nabla\Omega(r,z_{n})|^{2}rdr$
$\displaystyle\leq$ $\displaystyle\frac{C}{|z_{n}|^{3/2}},$
$\displaystyle\int_{d}^{\infty}|\nabla\Omega(r,z)|^{2}rdr$ $\displaystyle=$
$\displaystyle\int_{d}^{\infty}|\nabla\Omega(r,z_{n})|^{2}rdr-2\int_{d}^{\infty}\int_{z}^{z_{n}}\nabla\Omega(r,t)\cdot\partial_{t}\nabla\Omega(r,t)rdrdt$
$\displaystyle\coloneqq$ $\displaystyle L_{1}+L_{2},$ $\displaystyle|L_{2}|$
$\displaystyle\leq$
$\displaystyle\bigg{(}\int_{d}^{\infty}\int_{z}^{z_{n}}|\nabla\Pi(r,t)|^{2}rdrdt\bigg{)}^{1/2}\bigg{(}\int_{d}^{\infty}\int_{z}^{z_{n}}|\partial_{t}\nabla\Pi(r,t)|^{2}rdrdt\bigg{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle\begin{cases}\frac{C}{d^{3+2\delta}},\\\
\frac{C}{|z|^{\frac{3}{4}(1+\tau)}}.\end{cases}$

Letting $n\to\infty$, $L_{1}\to 0$. Take $\delta=(\frac{1}{2})^{-}$ and
$L_{2}\leq\textrm{min}\\{\frac{C}{d^{3+2\delta}},\frac{C}{|z|^{\frac{3}{4}(1+\tau)}}\\}$,

$\displaystyle\int_{d}^{\infty}|\nabla\Omega(r,z)|^{2}rdr\leq\bigg{(}\frac{C}{d^{4-}}\bigg{)}^{\frac{1}{4}}\bigg{(}\frac{C}{|z|^{\frac{3}{4}(1+\tau)}}\bigg{)}^{\frac{3}{4}}\leq\frac{C}{d|z|^{\frac{9}{16}(1+\tau)^{-}}}.$

Hence,

$\displaystyle|\Omega(d,z)|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{r_{1}-d}\int_{d}^{r_{1}}|\Omega(r,z)|^{2}dr+\bigg{(}|\Omega(d,z)|^{2}-\frac{1}{r_{1}-d}\int_{d}^{r_{1}}|\Omega(r,z)|^{2}dr\bigg{)}$
$\displaystyle\eqqcolon$ $\displaystyle G_{1}+G_{2}.$ $\displaystyle|G_{2}|$
$\displaystyle=$
$\displaystyle\bigg{|}|\Omega(d,z)|^{2}-|\Omega(d_{*},z)|^{2}\bigg{|}$
$\displaystyle\leq$ $\displaystyle
2\int_{d}^{r_{1}}|\Omega(r,z)\cdot\partial_{r}\Omega(r,z)|dr$
$\displaystyle\leq$
$\displaystyle\frac{C}{d}\bigg{(}\int_{d}^{\infty}|\Omega(r,z)|^{2}rdr\bigg{)}^{1/2}\bigg{(}\int_{d}^{\infty}|\nabla\Omega(r,z)|^{2}rdr\bigg{)}^{1/2}$
$\displaystyle\leq$
$\displaystyle\frac{C}{d}\bigg{(}\frac{C}{d|z|^{\frac{1}{4}(1+\tau)^{-}}}\bigg{)}^{1/2}\bigg{(}\frac{C}{d|z|^{\frac{9}{16}(1+\tau)^{-}}}\bigg{)}^{1/2}\leq\frac{C}{d^{2}|z|^{\frac{13}{32}(1+\tau)^{-}}}.$

Therefore we have

$\displaystyle|\omega_{\theta}^{u}(r,z)|$ $\displaystyle\leq$ $\displaystyle
C(M)(1+\rho)^{-(\frac{13}{64}(1+\tau))^{-}},\quad\forall(r,z)\in\mathbb{R}_{+}\times\mathbb{R},\rho=\sqrt{r^{2}+z^{2}}.$

Step 4. Decay rate of ${\bf u}$.

Following the argument developed in [35], fix any
$\textbf{x}\in\mathbb{R}^{3}\setminus\\{0\\}$, define a cut-off function
$\varphi\in C_{0}^{\infty}(\mathbb{R}^{3})$ satisfying $\varphi({\bf y})\equiv
1$ for $\forall\,{\bf y}\in B_{\rho/4}({\bf x})$ and $\varphi({\bf y})\equiv
0$ for $\forall\,{\bf y}\notin B_{\rho/2}({\bf x})$, where $\rho=|{\bf x}|$
and $|\nabla\varphi({\bf y})|\leq\frac{C}{|y|}$, $|\nabla^{2}\varphi({\bf
y})|\leq\frac{C}{|y|^{2}}$ for ${\bf y}\in D:=B_{\rho/2}({\bf x})\setminus
B_{\rho/4}({\bf x})$. For ${\bf u}({\bf x})=u_{r}{\bf e}_{r}+u_{z}{\bf
e}_{z}$, since $\textrm{curl }{\bf u}=\omega_{\theta}^{u}{\bf e}_{\theta}$,
then

$\displaystyle{\bf u}({\bf x})$ $\displaystyle=$
$\displaystyle-\int_{\mathbb{R}^{3}}\nabla_{\bf y}\Gamma({\bf x},{\bf
y})\times(\omega_{\phi}^{u}({\bf y})\varphi({\bf y}){\bf e}_{\phi})d{\bf
y}-\int_{\mathbb{R}^{3}}\Gamma({\bf x},{\bf y})(\nabla_{\bf y}\varphi({\bf
y})\times{\bf e}_{\phi})\omega_{\phi}^{u}({\bf y})d{\bf y}$
$\displaystyle\int_{\mathbb{R}^{3}}\Gamma({\bf x},{\bf y})(\Delta_{\bf
y}\varphi)({\bf y}){\bf u}({\bf y})d{\bf y}+2\int_{\mathbb{R}^{3}}(\nabla_{\bf
y}\Gamma)({\bf x},{\bf y})\cdot(\nabla_{\bf y}\varphi)({\bf y}){\bf u}({\bf
y})d{\bf y}$ $\displaystyle\coloneqq$ $\displaystyle\sum_{i=1}^{4}R_{i}$

According to [35], we have $|R_{i}|\leq\frac{C}{\rho^{1/2}},\quad i=2,3,4$.
For the estimate of $R_{1}$, fix a $d\in(0,\frac{\rho}{2})$, which will be
determined later, then

$\displaystyle|R_{1}|$ $\displaystyle\leq$ $\displaystyle\sup\limits_{{\bf
y}\in B_{d}({\bf x})}|\omega_{\phi}^{u}({\bf y})|\int_{B_{d}({\bf
x})}|\nabla\Gamma({\bf x}-{\bf y})|d{\bf y}$
$\displaystyle+\bigg{(}\int_{B_{\rho/2}({\bf x})\setminus B_{d}({\bf
x})}|\nabla\Gamma({\bf x}-{\bf y})|^{2}d{\bf
y}\bigg{)}^{\frac{1}{2}}\bigg{(}\int_{B_{\rho/2}({\bf x})\setminus B_{d}({\bf
x})}|\omega_{\phi}^{u}({\bf y})|^{2}d{\bf y}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C\rho^{-(\frac{13}{64}(1+\tau))^{-}}d+Cd^{-\frac{1}{2}}.$

By choosing $d=\rho^{\frac{13}{96}(1+\tau)^{-}}$, we obtain the bound for
$|R_{1}|\leq\frac{C}{\rho^{\frac{13}{192}(1+\tau)^{-}}}$. Hence we have

$\displaystyle|(u_{r},u_{z})(r,z)|\leq
C(M)(1+\rho)^{-\frac{13}{192}(1+\tau)^{-}}.$ (5.16)

Step 5. Iteration.

At the beginning, we have $\tau=0$ in (5.2), then by using the arguments
developed in Step 1 to Step 4, we have a new $\tau$ in (5.2), which will be
denoted by $\tau_{1}=(\frac{13}{192})^{-}$. Running a second iteration of
these three steps, we get a new $\tau_{2}=\tau_{1}+\frac{13}{192}\tau_{1}$,
and after $n$ iteration, we get

$\displaystyle\tau_{n}=\tau_{1}+\frac{13}{192}\tau_{n-1}=\tau_{1}\sum_{i=0}^{n-1}(\frac{13}{192})^{i}.$

Let $n\to\infty$, $\tau_{n}\to(\frac{13}{179})^{-}$ as $n\to\infty$. In a
word, we infer the following decay rates

$\displaystyle|u_{r}(r,z)|+|u_{z}(r,z)|$ $\displaystyle\leq$ $\displaystyle
C(M)(\rho+1)^{-(\frac{13}{179})^{-}},$ (5.17)
$\displaystyle|\omega_{\theta}(r,z)|$ $\displaystyle\leq$ $\displaystyle
C(M)(\rho+1)^{-(\frac{39}{179})^{-}},$ (5.18) $\displaystyle|h_{\theta}(r,z)|$
$\displaystyle\leq$ $\displaystyle C(M)(\rho+1)^{-(\frac{3041}{15036})^{-}}.$
(5.19)

∎

Acknowledgement. Weng is partially supported by National Natural Science
Foundation of China 11701431, 11971307, 12071359. We sincerely thank the
anonymous reviewers for their careful reading and thoughtful suggestions that
help to improve this paper.

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