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\def\rightheadline{EJDE--1998/26\hfil Existence of axisymmetric weak solutions
\hfil\folio}
\def\leftheadline{\folio\hfil Dongho Chae \&  Oleg Yu Imanuvilov
 \hfil EJDE--1998/26}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol.\ {\eightbf 1998}(1998), No.~26, pp.~1--17.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break 
ftp (login: ftp) 147.26.103.110 or 129.120.3.113\bigskip} }

\topmatter
\title
Existence of axisymmetric weak solutions of the 3-D Euler equations
for  near-vortex-sheet initial data
\endtitle

\thanks 
{\it 1991 Mathematics Subject Classifications:} 35Q35, 76C05.\hfil\break\indent
{\it Key words and phrases:} Euler equations, axisymmetry, weak solution.
\hfil\break\indent
\copyright 1998 Southwest Texas State University  and
University of North Texas.\hfil\break\indent
Submitted October 9, 1998. Published October 15, 1998.\hfil\break\indent
Partially supported by GARC-KOSEF, BSRI-MOE, KOSEF(K950701), KIAS-M97003,
\hfil\break\indent and the SNU Research Fund.
\endthanks
\author Dongho Chae \& Oleg Yu Imanuvilov\endauthor
\address Dongho Chae\hfil\break
Department of Mathematics, Seoul National University,\hfil\break
Seoul 151-742, Korea \endaddress
\email dhchae\@math.snu.ac.kr \endemail

\address Oleg Yu Imanuvilov \hfil\break
Korean Institute for Advanced Study, \hfil\break
207-43 Chungryangri-dong Dongdaemoon-ku, Seoul, Korea \endaddress
\email oleg\@kias.kaist.ac.kr\endemail

\abstract
We study the initial value problem for the 3-D Euler equation when 
the fluid is inviscid and incompressible, and flows with  
axisymmetry and without swirl.
On the initial vorticity $\omega_0$, we assumed that  
$\omega_0/r$ belongs to $L(\log L (\Bbb R^3))^{\alpha}$ with 
$\alpha >1/2$, where $r$ is the distance to an axis of symmetry. 
To prove the existence of weak global solutions, we prove first 
a new {\it a priori} estimate for the solution.  
\endabstract
\endtopmatter

\document
\head  Introduction \endhead

 We consider the Euler equations for homogeneous inviscid
 incompressible fluid flow in ${\Bbb R}^3$
$$ \gather
\frac{\partial v}{\partial t}+ (v\cdot \nabla)v=-\nabla p\, 
, \quad \,\,\text{div}\,v =0 \,\,\quad \text{in}\,\, {\Bbb R}_+ 
\times {\Bbb R}^3\,, \tag 1 \\
v(0,\cdot)=v_0\,,\tag 2 \endgather 
$$
where $v(t,x)=(v_1(t,x),v_2(t,x),v_3(t,x))$ is the velocity of the fluid flow
and $p(t,x)$ is the pressure.
The problem of finite-time breakdown of smooth solutions to (1)-(2) for 
smooth initial data is a longstanding open problem in mathematical 
fluid mechanics. (See [6,13,14] for a detailed  discussion of this 
problem.)
The situation is similar even for the case of axisymmetry (see e.g.[11],
[4]). In the case of axisymmetry without swirl velocity ($\theta$-component
of velocity), however, we have a global unique  smooth solution
for smooth initial data [14,17]. In this case a crucial role is played by
the fact that $\omega_\theta (t,x)/r$   
(where $\omega =\text{curl}\, v$, $r={\sqrt{x_1 ^2 +x_2 ^2}}$)
is preserved along the flow, and the problem looks similar
to that of  the 2-D Euler equations.

This apparent similarity between the  axisymmetric 3-D flow without swirl and
 the 2-D flow for smooth initial data breaks down for nonsmooth
initial data. In particular, Delort [8] found the very interesting
phenomenon that for a sequence of
approximate solutions to the axisymmetric 3-D Euler equations  with
nonnegative vortex-sheet initial data, either the sequence converges strongly
in $L^2 _{\text{loc}} ([0,\infty )\times {\Bbb R}^3 )$, or the weak limit of the sequence
is not a weak solution of the equations. This is in contrast with Delort's
proof of the existence of weak solutions for the 2-D Euler equations with 
the single-signed vortex-sheet initial data, where we have  weak 
convergence for the approximate solution sequence. Due to the subtle 
concentration cancellation type of phenomena in the nonlinear term, the weak
limit itself becomes a weak solution [7,10,15].
We refer to [13, Section 4.3] for an illuminating discussion on the
differences between the the quasi 2-D Euler equations and the ``pure" 2-D 
Euler equations for weak initial data.

In this paper we prove existence of weak solutions to (1)-(2)
for the axisymmetric initial data without swirl in which the vorticity
satisfies 
$$\left\vert \frac{\omega_{0}}{r}\right\vert \left[1+ \left(\log^+
\left\vert\frac{\omega_{0}}
{r}\right\vert\right)^{\alpha} \right] \in L^1({\Bbb R}^3), 
\quad \alpha > \frac{1}{2}\,,
$$
 where $\log^+ t=\max\{ 0, \log t\}$.
The idea of proof is as follows.
We divide  ${\Bbb R}^3$ into two parts: the region near the axis 
of symmetry, and the region away from the axis.  For the latter region, 
using the 2-D structure of the equations expressed in cylindrical
coordinate system, we obtain  strong compactness for the approximate solution
sequence using arguments previously used in the 2-D problem in [3]. 
For the region near axis, we could not adapt the previous 2-D arguments. 
See the next section for explicit comparison between the nonlinear terms 
in the pure 2-D Euler case and our case.
Here we use a new {\it a priori} estimate for the axisymmetric flow, 
combined with Delort's argument in [8] to overcome these difficulties. 

To the authors' knowledge this {\it a priori} estimate (See Lemma 2.1) is
completely new for the 3-D Euler equations with axisymmetry. On the other hand,
the results obtained in this paper improve substantially the results
in [5], where the authors proved existence of weak solutions for 
$$
\left\vert \frac{\omega_{0}}{r}\right\vert \in
L^1 ({\Bbb R}^3) \cap L^{p}({\Bbb R}^3), \quad p > \frac{6}{5}\,.  
$$
It would be very interesting to study (1)-(2) with initial data
in $L^1 ({\Bbb R}^3 )$.


\heading 1. Preliminaries \endheading

By a weak solution of the Euler equations with an initial data $v_0$,
we mean the vector field
$v\in L^{\infty}([0,T]; (L^2 _{\text{loc}} ({\Bbb R}^3))^3)$ with 
$\text{div}\, v=0$ such that
$$
\int_0^T\!\!\int_{{\Bbb R}^3}[ v\cdot  \varphi _t
+v\otimes v : \nabla \varphi ]\, dx\,dt+\int_{{\Bbb R}^3}v_0 \cdot
\varphi (0,x)\,dx=0\,,
$$
for all $\varphi \in C^\infty([0,T];[C^\infty_0({\Bbb R}^3)]^3)$ with 
$ \text{div}\,\varphi\equiv 0$ and $\varphi (T,x)\equiv 0$ Here we have
used the notation
$v\otimes v : \nabla \varphi =\sum _{i,j =1} ^3 v_i v_j (\varphi _i )_{x_j}$.

We are concerned with the axisymmetric solutions to the Euler equations.
 By an axisymmetric solution of equations (1)-(2) we mean a solution of
 the form
$$
v(t,x)=v_r(r,x_3,t)e_r+v_\theta(r,x_3,t)e_\theta+v_3(r,x_3,t)e_3
$$
in the cylindrical coordinate system, using the canonical basis
$$ e_r=(\frac{x_1}{r},\frac{x_2}{r},0),\quad e_\theta=(\frac{x_2}{r},
-\frac{x_1}{r},0), \quad e_3=(0,0,1), r=\sqrt{x^2_1+x^2_2}\,.
$$
For such flows the first equation in (1) can be written as
$$ \gather
\frac{\tilde{D} v_r}{D t}
 -\frac{(v_{\theta}) ^2}{r}
 = -\frac{\partial p}{\partial r}\,, \tag 3 \\
\frac{\tilde{D}}{D t}(r v_{\theta}) =0\,,  \tag 4 \\
\frac{\tilde{D} v_3}{\partial t}
 = -\frac{\partial p}{\partial x_3}\,,  \tag 5 \endgather
$$
for each component of velocity in the cylindrical coordinate system, where
$$ \frac{\tilde{D}}{Dt} =\frac{\partial}{\partial t} + v_r
\frac{\partial}{\partial r} + v_3 \frac{\partial}{\partial x_3}\,.
$$
On the other hand, the second equation of  (1) becomes
$$
\frac{\partial}{\partial r}(r v_r ) +\frac{\partial}{\partial x_3}
(r v_3 ) =0\,. \tag 6
$$
We observe that $\theta$-component of the vorticity equation
is written as
$$
\frac{\tilde{D}}{Dt} \left( \frac{\omega _{\theta}}{r} \right)
=\frac{1}{r^4} \frac{\partial}{\partial x_3 } ( r v_{\theta} )^2 \,, \tag 7
$$
where
$$
\omega _{\theta} = \frac{\partial v_r}{\partial x_3} -
\frac{\partial v_3}{\partial r}            \tag 8
$$
is the $\theta -$component of the vorticity vector $\omega$.
If we assume that the initial velocity
$$
v_0 \in V^m =\{ v\in [\text{ H}^m ({\Bbb R}^3 )]^3 : \ \text{div} \ v =0 \}
$$
with $m\geq 4$ is axisymmetric, then due to the symmetry properties of
the Euler equations, and by  the existence of local unique 
classical solutions [12],
the solution remains axisymmetric during its existence. Here we used the standard Sobolev space
$$
\text{H}^{m}({\Bbb R}^3)=\{u \in L^2({\Bbb R}^3 )\,  : \, D^\alpha u\in L^2({\Bbb R}^3), \,\,\vert\alpha
\vert\le m\}\,.
$$
Furthermore, if $v_0$ has no \lq\lq swirl" component, i.e. $v_{0,\theta}$=0, 
then (4) and (7) imply that
$$
\frac{\tilde{D}}{Dt} \left( \frac{\omega _{\theta}}{r} \right) =0 \quad 
\forall t>0\,.  \tag 9
$$
We observe that in this case the vorticity
becomes  $\omega (t,x)=\omega_\theta  (t, r, x_3 ) e_\theta$.
Thus, we have, in particular,
$$ 
|\omega (t,x) | = |\omega_\theta  (t, r, x_3 )|\,,
$$
where $|\cdot |$ denotes the Euclidean norm in ${\Bbb R}^3$ in the left hand
side, and the absolute value in the right hand side of the equation.
In [17] Saint-Raymond proved existence of a global unique smooth solution
for smooth $v_0$ without swirl.  

Below we show explicitly the difference between 
the nonlinear terms for the 2-D Euler equations and those for  3-D Euler equations with axisymmetry and 
without swirl.
In the weak formulation of the 2-D Euler equations, if we use a test 
function of the form 
$\varphi =(-\frac{\partial \psi}{\partial x_2} ,
\frac{\partial \psi}{\partial x_1} )$ in order to satisfy 
$\text{div}\,\varphi =0$, then
$$
\int _0 ^T\!\!\int _{{{\Bbb R}^2}}[ v\otimes v : \nabla \varphi ] \,dx\,dt 
= \int _0^T\!\!\int _{{{\Bbb R}^2}} \left[(v_1 ^2 -v_2 ^2 )\frac{\partial ^2 
\psi}{\partial x_1\partial x_2} -v_1 v_2 \left(\frac{\partial ^2 \psi}{\partial x_1 ^2}-
\frac{\partial ^2 \psi}{\partial x_2 ^2} \right)\right]\,dx\,dt\,. 
$$
On the other hand, in the  axisymmetric 3-D Euler equation without swirl, if we use 
as a test function $\varphi (t, x) =\varphi _r(t, r, x_3)e_r + \varphi _3(t, r, x_3)e_3$ with 
$$
\varphi _r =\frac{1}{r}\frac{\partial \psi}{\partial x_3}, \ \ 
\varphi _3 =-\frac{1}{r}\frac{\partial \psi}{\partial r} 
$$
to satisfy $\frac{\partial(r \varphi _r )}{\partial r} + \frac{\partial(r\varphi _3 )}{
\partial x_3} =0$, then
$$ \align
\int _0 ^T\!\!\int _{{{\Bbb R}}^3} [v\otimes v : \nabla \varphi ] \,dx\,dt
= 2\pi \int _0 ^T\!\!\int _{{\Bbb R}\times {\Bbb R}_+} 
\bigg[&(v_r ^2 -v_3 ^2 )\frac{\partial ^2 \psi}{\partial r 
\partial x_3} -v_r v_3\left (\frac{\partial ^2 \psi}{\partial r ^2}-
\frac{\partial ^2 \psi}{\partial x_3 ^2}\right ) \\ 
&+ \frac{v_r v_3}{r} \frac{\partial \psi}{\partial r} -\frac{v_r ^2}{r} \frac{\partial
\psi}{\partial x_3}\bigg]\,dr\,dx_3\, dt \,. \endalign
$$
Here we have extra two nonlinear terms compared to the 2-D case, which have 
apparent singularities on the axis of symmetry.

Before closing this section, we provide a brief introduction to the Orlicz 
spaces. For more details see [1,9], and for applications to the 
2-D Euler equations, see [3,16].
By an N-function we mean a real valued function $A(t)$,  $t\geq 0$
which is continuous, increasing, convex, and satisfies 
$$
\lim_{t\rightarrow 0} \frac{A(t)}{t} =0,\quad
\lim_{t\rightarrow \infty} \frac{A(t)}{t}=+\infty \,.
$$
We say that  $A(t)$ satisfies  $\Delta_2 $-condition near infinity
if there exist  $k>0$, $t_0 \geq 0$
such that
$$
A(2t)\leq k A(t)  \ \ \ \forall t \geq t_0\,.
$$
We denote $A(t)\succ B(t)$ if for every $k>0$
$$
\lim_{t\rightarrow \infty} \frac{A(kt)}{B(t)} =\infty\,.
$$
Let $\Omega $ be a domain in ${\Bbb R}^n$.
Then  the Orlicz class $K_A (\Omega )$ is defined as the set
all functions $u$ such that $\int_{\Omega}  A(| u(x) |) \,dx <\infty$.
On the other hand, the Orlicz space $L_A (\Omega )$ is defined as
the linear hull of the Orlicz class $K_A (\Omega )$. The set $L_A (\Omega )$ 
is a Banach space equipped with the Luxembourg norm
$$
\Vert u \Vert_A
=\inf\big\{k :  \int_{\Omega}  A(\frac{u}{k} )\,dx \leq 1 \big\}\,.
$$
In general $K_A (\Omega ) \subset L_A (\Omega )$, but in case
the domain $\Omega$ is bounded in ${\Bbb R}^n$, and the N-function
$A$ satisfies the $\Delta _2 $-condition near infinity we have
$K_A (\Omega ) = L_A (\Omega )$ (see [1]).   
For example $L^p (\Omega ), \ 1<p<\infty$ is an Orlicz
space with N-functions given by $A(t)=t^p$.

Recall that for a bounded domain $\Omega $ we have
the continuous imbedding, [1],
$$
L_A (\Omega  )\hookrightarrow L_B (\Omega ) \quad \text{if}\quad A(t)\succ B(t).
$$
Also recall the following duality relations [3, Lemma 4].
(Below $X^*$ denotes the dual of $X$)

\proclaim{Lemma 1.1} Let $\Omega$ be a bounded domain in ${\Bbb R}^n$, and 
$\alpha >0$. Let $A(\cdot), B(\cdot )$ be N-functions given by 
$A(t)=t(\log^+ t)^{\alpha}$, $B(t)=\exp(t^{q/\alpha} ) -1$, 
where  $t\geq 0$. Then, we have
$$
L_B(\Omega )=L^*_A (\Omega )\,.
$$
\endproclaim

By the Orlicz-Sobolev space  $W^m L_A (\Omega )$ we mean a   subspace
of the Orlicz space $L_A (\Omega )$ consisting of functions $u$
such that the distributional derivatives
$D^{\alpha} u$ are contained in $L_A (\Omega )$ for all multi-index
$\alpha$'
with $|\alpha | \leq m$, equipped with a Banach space norm
$$
\| u\| _{m,A}
= \max_{\vert\alpha\vert\le m}  \| D^{\alpha } u \| _A \, .
$$

The following lemma corresponds to a special case of the  general result 
by  Donaldson and Trudinger [9].

\proclaim{Lemma 1.2}
Let $\Omega \subset {\Bbb R}^2$ be a bounded domain, 
and $B(t)=\exp(t^2)-1$, then we have a continuous imbedding
$$ 
H^1_0(\Omega)\hookrightarrow  L_B(\Omega ).
$$
Moreover, for any N-function $A(t)$ with $A(t)\prec B(t)$ we have a compact imbedding
$$ 
H^1_0(\Omega)\hookrightarrow  \hookrightarrow  L_A(\Omega ).
$$
\endproclaim

Combining dual of the compact imbedding in Lemma 1.2, and Lemma 1.1 we have

\proclaim{Corollary 1.1} Let $\Omega\subset {\Bbb R}^2$  be a bounded domain 
and  $A(t)=t(\log^+ t)^{\alpha}$ with $\alpha > \frac12$. Then we have the 
compact imbedding
$$
L_A(\Omega)\hookrightarrow\hookrightarrow \text{H}^{-1}(\Omega).
$$
\endproclaim

\heading  2. Main Results \endheading


Our main result is as follows:
\proclaim {Theorem 2.1} Suppose $\alpha > \frac{1}{2}$ is given. 
Let $v_0\in V^0 $ be an axisymmetric initial data
with $v_{0,\theta}\equiv 0,$  and 
$\vert \frac{\omega_{0}}{r}\vert \left[1+ (\log^+
\vert\frac{\omega_{0}}{r}\vert)^{\alpha} \right] \in L^1({\Bbb R}^3)$.  
Then there exists a weak solution of problem
(1)-(2). Moreover, the solution satisfies 
$$
\| v(t,\cdot)\|_{V^0} \leq \|v_0 \|_{V^0},
$$ 
and
$$
\int_{{\Bbb R}^3} \left\vert \frac{\omega (t,\cdot)}{r}\right\vert \left[1+ \left( \log^+
\left\vert\frac{\omega (t,\cdot)}{r}\right\vert \right)^{\alpha} \right]\, dx \leq
\int_{{\Bbb R}^3} \left\vert \frac{\omega_{0}}{r}\right\vert \left[1+ \left(\log^+
\left\vert\frac{\omega_{0}}{r}\right\vert \right)^{\alpha} \right]\,  dx 
$$
for almost every $t\in [0, \infty)$.
\endproclaim

In this section our aim is to prove the above theorem.
Below we denote 
$$
Q=[0,T]\times  {\Bbb R}^3 , \quad G=\{ (r, x_3 )\in {\Bbb R}^2 \, |\, r>0, 
x_3 \in {\Bbb R} \}.
$$
We start from establishment of the following a priori estimate.
\proclaim{Lemma 2.1} Let $v(t,x)\in C([0,T];[C^1(\overline{{\Bbb R}^3})\bigcap 
H^1({\Bbb R}^3)]^3)\bigcap C([0,T]; V^0) $
be the classical  solution of the Euler equations for the axisymmetric initial
 data $v_0$ without  the swirl
component, and  with the vorticity satisfying 
$\frac{\omega_{0}}{r}\in L^1({\Bbb R}^3)$.
Then the following estimate holds:
$$
\int^T_0\!\!\int_{{\Bbb R}^3}\frac{1}{1+x^2_3}\left
(\frac{v_r}{r}\right)^2\,dx\,dt\le C\left(\Vert v_0\Vert_{
V^0}^2+\left\Vert \frac{\omega_{0}}{r}\right\Vert_{L^1({\Bbb R}^3)}\right)\,.
\tag 10
$$\endproclaim

\noindent{\bf Proof}. The velocity conservation law for the Euler equations
implies the estimate
$$
\Vert \sqrt{r} v_r\Vert_{L^\infty(0,T;L^2(G))}+\Vert \sqrt{r} v_3\Vert_{
L^\infty(0,T;L^2(G))}\le C\Vert v_0\Vert_{V^0}.\tag 11
$$
Moreover, (9) immediately yields the estimate for $L^1$-norm of vorticity
$$
\Vert \omega(t,\cdot)\Vert_{L^1(G)}
\le\Vert \omega_0\Vert_{L^1(G)}.
$$
We set $\rho(x_3)=\int^{x_3}_{-\infty} 1/(1+\tau^2)\, d\tau$.
Multiplying (9) by $2\pi r \rho (x_3)$ scalarly in \hfill\break
$L^2(0,T;L^2(G))$ and integrating by parts, we obtain
$$\aligned
0=&\int_{{\Bbb R}^3}\frac{\rho\omega_\theta}{r}\,dx\bigg\vert^T_0
-\int_0^T\!\!\int_{G}2\pi \rho' v_3 \omega_\theta \,dr\,dx_3\,dt \\
=&\int_{{\Bbb R}^3}\frac{\rho\omega_\theta}{r}dx\bigg\vert^T_0
+\int_0^T\!\!\int_G
2\pi \rho' v_3\left(\frac{\partial v_3}{\partial r}-\frac{\partial v_r}{
\partial x_3}\right)\,dr\,dx_3\,dt \\
=&\int_{{\Bbb R}^3}\frac{\rho\omega_\theta}{r}dx\bigg\vert^T_0
-\int_0^T\!\!\int^{+\infty}_{-\infty}\pi \rho' v^2_3(t,0,x_3)\,dx_3\,dt \\
&+\int_0^T\!\!\int_G 2\pi\left(\rho''v_3v_r+\rho' v_r\frac{\partial v_3}{
\partial x_3}\right)\,dr\,dx_3\,dt\,,\endaligned \tag 12
$$
where we used the regularity assumption of solution $v$, and the 
integration by parts used above can be
justified easily. Indeed,
$$\align
\int_0^T\!\!\int_G &2\pi \rho' v_3\left(\frac{\partial v_3}{\partial r}-
\frac{\partial v_r}{\partial x_3}\right) \,dr\,dx_3\,dt\\
=&\lim_{r_k\rightarrow +\infty}2\pi\int_0^T\!\!\int^{+\infty}_{-\infty}\!\!
\int_0^{r_k} \rho'v_3\frac{\partial v_3}{\partial r}\,dr\,dx_3\,dt\\
&-\lim_{b_k\rightarrow +\infty}2 \pi \int_0^T \int_{-b_k}^{b_k}\int_0^{\infty} 
 \rho' v_3\frac{\partial v_r}{\partial x_3}\,dr\,dx_3\,dt\\
=&-\int_0^T\!\!\int^{+\infty}_{-\infty}\pi\rho'v^2_3(t,0,x_3)\,dx\,dt
 +\lim_{r_k\rightarrow +\infty}\int_0^T\!\!\int^{+\infty}_{-\infty} \pi
 \rho' v^2_3(t,r_k,x_3)\,dx_3\,dt\\
&-\lim_{b_k\rightarrow +\infty}
 \int_0^T\!\!\int^\infty_0 2\pi \rho' v_3v_rdrdt\bigg\vert^{b_k}_{-b_k}\\
&+\lim_{b_k \rightarrow +\infty}\int_0^T\!\!\int^{b_k}_{-b_k}\int_0^\infty 2
\pi\left(\rho''v_3v_r+\rho' v_r\frac{\partial v_3}{
\partial x_3}\right)\,dr\,dx_3\,dt\,.
\endalign
$$
for all sequence $r_k\rightarrow +\infty$. Since 
$v\in C([0,T];(C^1(\overline{R^3}))^3)$,
$$\align
\int_{(0,T)\times{\Bbb R}^3} \vert v\vert^2\,dx\,dt
=&2\pi \int_0^{+\infty}\left (\int_0^T\!\!\int_{-\infty}^{+\infty} 
\vert v\vert^2\,dx_3\,dt\right)r\,dr \\
=&2\pi\int_{-\infty}^{\infty}\left(\int_0^T\!\!\int_0^{+\infty}
\vert v\vert^2r\,dr\,dt\right)\,dx_3< \infty \,,\endalign
$$
and $\lim_{x_3\rightarrow \infty} \rho'(x_3)=0$
one can find a sequence $r_k \rightarrow +\infty$ and $b_k\rightarrow +\infty$
such that 
$$
\int_0^T\!\!\int_{-\infty}^{\infty} \rho' v^2_3(t,r_k,x_3) \,dx_3\,dt 
\rightarrow 0,\quad \lim_{b_k\rightarrow +\infty}\int_0^T\!\!\int_0^\infty 
2\pi\rho v_3v_r\,dt\,dr\bigg\vert^{b_k}_{-b_k}\rightarrow 0\,.
$$
From (6) we have
$$
\frac{\partial v_3}{\partial x_3}=-\frac{v_r}{r} -\frac{\partial v_r}{
\partial r}\,.\tag 13
$$
Therefore (12) and  (13) imply 
$$\multline
0=\int_{{\Bbb R}^3}\rho\frac{\omega_\theta}{r}dx\bigg\vert^T_0-\int_0^T\!\!\int^{+\infty}_{
-\infty} \pi\rho' v^2_3(t,0,x_3)\,dx_3\,dt\\
+\int_0^T\!\!\int_G 2\pi\left(\rho'' v_3v_r-\rho'\frac{(v_r)^2}{r}-\rho'v_r
\frac{\partial v_r}{\partial r}\right)\,dr\,dx_3\,dt\,. \endmultline\tag 14
$$
Since, by assumption,  $v(t,x)$ is a smooth and axisymmetric vector field
$$
v_r(t,0,x_3)=0\quad \forall \,t\in {\Bbb R}^1_{+},\,\, x_3\in
 {\Bbb R}^1.
$$
Thus integration by parts in (14), which can be justified similarly to the 
above,  implies
$$\multline
\int_0^T\!\!\int^{+\infty}_{-\infty}\pi\rho' v^2_3(t,0,x_3)\,dx_3\,dt+
\int_0^T\!\!\int_G 2 \pi\rho'(x_3) \frac{(v_r)^2}{r}\,dr\,dx_3\,dt\\
=\int_0^T\!\!\int_G2\pi \rho'' v_3 v_r \,dr\,dx_3\,dt+\int_{{\Bbb R}^3}
\rho\frac{\omega_\theta}{r}dx\bigg\vert^T_0\,.
\endmultline
$$
Since $\rho'(x_3)>0,\vert \rho(x_3)\vert<C$ for all $x_3\in {\Bbb R}^1$ we obtain
the inequality
$$\multline
2\pi \int_0^T\!\!\int_G\left\vert\rho' \frac{(v_r)^2}{r}\right\vert dr dx_3 dt\le 2\pi
\int^T_0 \int_G \vert\rho'' v_3v_r\vert \,dr\,dx_3\,dt
+C\left\Vert \frac{\omega_{0}}{r}\right\Vert_{L^1({\Bbb R}^3)}\\
\le 2\pi
\left(\int_0^T\!\!\int_G\rho'\frac{( v_r)^2}{r}\,dr\,dx_3\,dt\right)^\frac 12
\left(\int_0^T\!\!\int_Grv_3^2\frac{(\rho'')^2}{\rho'}\,dr\,dx_3\,dt\right)^\frac 12
+C\left\Vert \frac{\omega_{0}}{r}
\right\Vert_{L^1({\Bbb R}^3)}\,.\endmultline 
$$
Hence by the Cauchy-Bunyakovskii inequality we have
$$
\int_0^T\!\!\int_G \vert\rho'\vert \frac{(v_r)^2}{r} \,dr\,dx_3\,dt\le C\left(
\int_0^T\!\!\int_{{\Bbb R}^3} v^2_3 \frac{\vert\rho''\vert^2}{\rho'}
\,dx\,dt +\left\Vert\frac{\omega_{0,\theta}}{r}\right\Vert_{L^1({\Bbb R}^3)}\right). \tag 15
$$
Since $\sup_{x_3 \in {\Bbb R}}\vert \rho''(x_3)\vert^2/\vert \rho'(x_3)
\vert\le C$, inequalities
(11) and (15) imply the estimate (10). \hfil $\blacksquare$ \medskip

  Now, let  $v_0^\varepsilon$ be an axisymmetric
initial datum without the swirl component such that
$$
v_0^\varepsilon \rightarrow v_0\,\,\text{in}\,\, V^0,\quad
v_0^\varepsilon\in (C^\infty({\Bbb R}^3))^3\,,
\quad
\frac{\omega_{0,\theta}^\varepsilon}{r}\rightarrow
\frac{\omega_{0,\theta}}{r}\,\,\text{in}\,\, L^1({\Bbb R}^3)\,.\tag 16
$$
Such an approximation $v_0^\varepsilon$  for any axisymmetric function, 
$v_0 \in V^0$ without swirl was constructed in [17] for example. 
In [17] also, it was  proved that in this case there exists a unique solution 
of the problem (1)-(2), 
$v_\varepsilon (t,\cdot)\in C([0,T];[C^2({\Bbb R}^3)]^3)\cap L^2(0,T;
H^1({\Bbb R}^3))$.
Without  loss of generality, passing to a subsequence if it is necessary,
we may assume that
$$ v^\varepsilon\rightarrow v\,\, \text{weakly in}\,\, 
[L^2((0,T)\times {\Bbb R}^3)]^3\,.
%\quad v^\varepsilon \rightarrow v\,\, \text{a.e. in }\,Q=(0,T)\times 
%{\Bbb R}^3.
\tag 17
$$

We have
\proclaim{Lemma 2.2} Let $\{  v^{\varepsilon} (x,t)\}_{\varepsilon\in (0,1)}$ be a sequence of
smooth solutions of (1)-(2) associated with the initial datum
$\{  v^{\varepsilon}_0 \}$ with axisymmetry and without swirl, and satisfying (16) 
and (17). Then, for each
$\varphi\in C([0,T]; C_0({\Bbb R}^3)),$ we have
$$\int_Q[(v_r^\varepsilon)^2-(v_3^\varepsilon)^2]\varphi \,dx\,dt\rightarrow
\int_Q[(v_r)^2-(v_{3})^2]\varphi \,dx\,dt                \,\,
\text{as}\,\, \varepsilon \rightarrow
+0\,.\tag 18$$ \endproclaim

\noindent{\bf Remark.}
The above lemma is very similar to Delort's in [8],
where he proved it in particular
under the assumptions on the sequence $\{ v^{\varepsilon} (x,t)\}$ that
$$ \gather
\omega ^{\varepsilon} _{\theta} (x,t) \geq 0 \ \ \text{almost everywhere
in $(0,T)\times {\Bbb R}^3$, and} \\
\{ \omega ^{\varepsilon} _{\theta}\} \ \text{is uniformly bounded in
$L^{\infty} (0, \infty; L^1 (\overline{{\Bbb R}}_+\times {\Bbb R},
(1+r^2 )\,dr\,dx_3 ))$}\,. \endgather
$$ 
In our case, however, we only need to assume 
$\frac{\omega _0}{r} \in L^1 ({\Bbb R}^3)$, and $\{v^{\varepsilon}\} $ 
is the associated sequence of approximated solutions. \medskip


\noindent{\bf Proof of Lemma 2.2}. We follow Delort's arguments. Denote
$$
(\Delta^{-1} f)(x)=-\frac{1}{4\pi}\int_{{\Bbb R}^3}
\frac{f(y)}{\vert x- y\vert}\, dy\,.
$$
Relation $v^\varepsilon=- \Delta^{-1} \text{\bf curl}\, \omega^\varepsilon$ 
implies
$$ \gather
v_1^\varepsilon=\Delta^{-1} \partial_3 \omega_2^\varepsilon\,, \\
v_2^\varepsilon=-\Delta^{-1}\partial_3\omega_1^\varepsilon\,, \\
v_3^\varepsilon=\Delta^{-1}(\partial_2\omega_1^\varepsilon-\partial_1
\omega_2^\varepsilon)\,. \endgather
$$
Let $\Phi\in  C^\infty_0({\Bbb R}^3),$ in $ \Phi\ge 0, \Phi\equiv 1\,\,
\text{for all}\,\, (t,x)\in \text{supp}\, \varphi$. Set
 $$ \gather 
\varphi v^\varepsilon_1=\varphi\Delta^{-1}\partial_3(\Phi \omega_2^\varepsilon)+
w^\varepsilon_1\,, \\
\varphi v^\varepsilon_2=-\varphi\Delta^{-1}\partial_3(\Phi \omega_2^\varepsilon)+
w^\varepsilon_2\,, \\
\varphi v^\varepsilon_3=\varphi\Delta^{-1}(\partial_2(\Phi \omega_1^\varepsilon)
-\partial_1(\Phi \omega_2^\varepsilon))+
w^\varepsilon_3\,, \endgather
$$
 where
$$ \gather
 w^\varepsilon_1=\varphi [\Phi,\Delta^{-1}\frac{\partial}{\partial x_3}]
\omega^\varepsilon_2\,, \\
 w^\varepsilon_2=-\varphi [\Phi,\Delta^{-1}\frac{\partial}{\partial x_3}]
\omega^\varepsilon_1\,, \\
w^\varepsilon_3=\varphi([\Phi,\Delta^{-1}\frac{\partial}{\partial x_2}]
\omega^\varepsilon_1-[\Phi,\Delta^{-1}\frac{\partial}{\partial x_1}]
\omega^\varepsilon_2)\,, \endgather
$$  
where $[A,B]=AB-BA$ is the commutator of operators $A,B$, and $\partial _i
=\partial /\partial_{x_i}.$
 Note that $w^\varepsilon_i$ are uniformly bounded in
$L^\infty(0,T; \text{H}^1_{\text{loc}}({\Bbb R}^3))\cap
H^1(0,T; H^{-4}_{\text{loc}}({\Bbb R}^3))$ for each $i=1,2,3.$
Really let us prove this claim for example for $w^\varepsilon_1.$
Denote $z_\varepsilon=\Delta^{-1} \frac{\partial}{\partial x_3} 
(\Phi \omega^\varepsilon_2)$. 
Then $\Delta z_\varepsilon=\frac{\partial}{\partial x_3}
(\Phi \omega_2^\varepsilon)$, and
$$
\Delta (\Phi v^\varepsilon_1)=\partial_3(\Phi \omega^\varepsilon_2)
-\partial_3\Phi \omega^\varepsilon_2+ 
2\sum_{i=1}^3\frac{\partial}{\partial x_i}
(v^\varepsilon_1\frac{\partial \Phi}{\partial x_i})-v^\varepsilon_1\Delta \Phi\,.
$$
Denote $u_\varepsilon=\Phi v_1^\varepsilon-z_\varepsilon$. Then
$$
\Delta u_\varepsilon=
-\frac{\partial(\partial_3\Phi v_1^\varepsilon)}{\partial x_3}
+\frac{\partial^2\Phi}{\partial x_3^2} v^\varepsilon_1+
\frac{\partial(\partial_3\Phi v^\varepsilon_3)}{\partial x_1}
-\frac{\partial^2 \Phi}{\partial x_1\partial x_3}v^\varepsilon_3
+2\sum_{i=1}^3\frac{\partial}{\partial x_i}
(v^\varepsilon_1\frac{\partial \Phi}{\partial x_i})
-v^\varepsilon_1\Delta \Phi
$$
and 
$$
\multline
u_\varepsilon= \Delta^{-1}\left(-\frac{\partial(\partial_3 \Phi v_1^\varepsilon)}
{\partial x_3}+\frac{\partial(\partial_3 \Phi v^\varepsilon_3)}{\partial x_1} 
+2\sum_{i=1}^3\frac{\partial}{\partial x_i}
(v^\varepsilon_1\frac{\partial \Phi}{\partial x_i})
\right) \\
\qquad  +\Delta^{-1}\left(\frac{\partial^2\Phi}{\partial x_3^2} 
v^\varepsilon_1-\frac{\partial^2 \Phi}{\partial x_1\partial x_3}
v^\varepsilon_3-v_1 ^{\varepsilon} \Delta \Phi \right),
\endmultline
$$
where the first component of $u_\varepsilon$ is bounded in $L^\infty
(0,T;H^1(
{\Bbb R}^3))$, and the second one is bounded in  $ L^\infty(0,T; H^1_{\text{loc}}({\Bbb R}^3))$ due to compactness
of $\text{ supp}\, \Phi$ in $[0,T]\times{\Bbb R}^3$. Since the function $\varphi$ also has a compact support in $[0,T]\times {\Bbb R}^3,
$ the first part of our statement is proved. 
To prove the
uniform boundness of $\frac{\partial w^\varepsilon_1}{\partial t}$ in $L^2(0,T; H^{-4}_{\text{loc}}({\Bbb R}^3))$ we first recall that 
for any smooth solution $v$ of the 3-D Euler equations with initial data $v_0$, we have
in general
$$
\| v(t_1 )- v(t_2 ) \|_{H^{-3} (B_r )} \leq C (r) \|v_0 \|_{V^0} ^2 \ |t_1 -t_2 |
$$
for all  $t_1 , t_2 $ with $ 0<  t_1 \leq t_2 < T$, where $B_r$ is a ball with the center $0$ and radius $r$
(see e.g. [5]). This estimate implies immediately that
$$
\left\Vert \frac{\partial v^\varepsilon}{\partial t}\right\Vert _{L^\infty(0,T;H^{-3}(B_r ))} \leq C(r),
\tag 19
$$
where $C$ is independent of $\varepsilon$.
Taking the time derivative of $u_\varepsilon$ we have 
$$
\left\Vert \frac{\partial w^\varepsilon}{\partial t}\right\Vert_{
L^2(0,T;H^{-4}(B_r))}\le C(r)\left(\left\Vert \frac{\partial v^\varepsilon}{
\partial t}\right\Vert_{L^2(0,T;H^{-3}(B_r ))}+\Vert v^\varepsilon\Vert_{L^2(0,T;V^0)}
\right) \leq C(r). 
$$


Hence to prove (18) we need only to pass to the limit
in the following equation.
$$\multline
A^\varepsilon=\int_Q[(\Delta^{-1}\partial_3(\Phi \omega^\varepsilon_2))^2+
(\Delta^{-1}\partial_3(\Phi \omega^\varepsilon_1))^2
-(\Delta^{-1}\partial_2(\Phi \omega^\varepsilon_1))^2-
(\Delta^{-1}\partial_1(\Phi \omega^\varepsilon_2))^2\\
+2
(\Delta^{-1}\partial_2(\Phi \omega^\varepsilon_1))
(\Delta^{-1}\partial_1(\Phi \omega^\varepsilon_2))]\varphi \,dx\,dt.\endmultline
$$
After simplifications we have:
$$\multline
A^\varepsilon=(\Phi\omega_2^\varepsilon,\varphi\Delta^{-2}(\partial^2_1-\partial^2_3)
(\Phi \omega_2^\varepsilon))_{L^2(Q)}+
(\Phi\omega_1^\varepsilon,\varphi\Delta^{-2}(\partial^2_2-\partial^2_3)
(\Phi \omega_1^\varepsilon))_{L^2(Q)}\\
-
2(\Phi \omega_1^\varepsilon, \varphi \Delta^{-2} \partial_1\partial_2
(\Phi \omega^\varepsilon_2))_{L^2(Q)}+A_0^\varepsilon=A_1^\varepsilon+A^\varepsilon_0,
\endmultline
$$
where
$$\multline
A_0^\varepsilon=
(\Phi \omega_2^\varepsilon, [\Delta^{-1}\partial_3,\varphi]
(\Delta^{-1}\partial_3(\Phi\omega_2^\varepsilon)))_{L^2(Q)}
+(\Phi \omega_1^\varepsilon, [\Delta^{-1}\partial_3,\varphi]
(\Delta^{-1}\partial_3(\Phi\omega_1^\varepsilon)))_{L^2(Q)}\\
-(\Phi \omega_1^\varepsilon, [\Delta^{-1}\partial_2,\varphi]
(\Delta^{-1}\partial_2(\Phi\omega_1^\varepsilon)))_{L^2(Q)}
-(\Phi \omega_2^\varepsilon, [\Delta^{-1}\partial_1,\varphi]
(\Delta^{-1}\partial_1(\Phi\omega_2^\varepsilon)))_{L^2(Q)}  \\
+2(\Phi \omega_1^\varepsilon, [\Delta^{-1}\partial_2,\varphi]
(\Delta^{-1}\partial_1(\Phi\omega_2^\varepsilon)))_{L^2(Q)}.\endmultline
$$
Since each sequence $[\Delta^{-1}\partial_j,\varphi](\Delta^{-1}\partial_k(
\Phi \omega_\ell^\varepsilon))$ belongs to  a compact set in 
$\text{H}^1_{\text{loc}}({\Bbb R}^3)$, we
obtain
$$
A_0^\varepsilon \rightarrow A_0 \,\, \text{as} \,\, \varepsilon
\rightarrow 0
$$
for a subsequence.
In [8] Delort proved that the term $A_1^\varepsilon$ can be rewritten
as follows
$$
A_1^\varepsilon=\int_0^T\!\!\int_G\int _G K(t,r,x_3,r',x'_3)(
\Phi \omega^\varepsilon_\theta
)(t,r,x_3) (\Phi \omega^\varepsilon_\theta)(t,r',x'_3)
\,dr\,dx_3\,dr'\,dx_3'\,dt\,,
$$
where the function $K(t,r,x_3,r',x'_3)$ satisfies 
$$
K\in C^\infty \,\,\text{on}\,\, \{(r,x_3,r',x'_3)\in
\overline{{\Bbb R}}_+\times {\Bbb R}\times
\overline{{\Bbb R}}_+\times {\Bbb R}
; (r,x_3)\neq (r',x'_3)\}
$$
and $K$ is locally bounded on $\overline{{\Bbb R}}_+\times {\Bbb R}\times \overline{
{\Bbb R}}_+\times {\Bbb R}$.

Let $\eta(\tau)\in C_0^\infty({\Bbb R}^1)$, $\eta (\tau)\ge 0$ for any $\tau \in 
{\Bbb R}^1$
and $\eta\equiv 1$ in some neighborhood of $0$. Set
$$\multline
A^\varepsilon_1= I^{\varepsilon,\delta}_1+I^{\varepsilon,\delta}_2=
\int_0^T\!\!\int_G\int_G K(t,r,x_3,r',x'_3)\left(1-\eta\left(\frac{r}{\delta}
\right)\right)
\left(1-\eta\left(\frac{r'}{\delta}\right)\right)\times
\\
\left(1-\eta\left(\frac{\vert r'-r\vert+
\vert x_3 -x'_3\vert}{\delta}\right)\right) (\Phi\omega^\varepsilon_\theta)
(t,r,x_3) (\Phi \omega^\varepsilon_\theta)(t,r',x'_3)\,dr\,dx_3dr'dx_3'dt\\
+\int_0^T\!\!\int_G\int_G K(t,r,x_3,r',x_3')\left[\eta\left(\frac{r}{\delta}\right)
\left(1-\eta\left(\frac{r'}{\delta}\right)\right)
\left(1-\eta\left(\frac{\vert r'-r\vert+
\vert x_3 -x'_3\vert}{\delta}\right)\right)\right. \\
+
\eta\left(\frac{r'}{\delta}\right)
\left (1-\eta\left(\frac{\vert r-r'
\vert+\vert x_3-x_3'\vert}{\delta}\right)\right)\\+\left.
\eta\left(\frac{\vert r-r'\vert+\vert x_3-x_3'\vert}{\delta}\right)\right]
(\Phi \omega^\varepsilon_\theta)(t,r,x_3)(\Phi \omega^\varepsilon_\theta)(t,r',x_3')
\,dr\,dx_3\,dr'\,dx'_3 dt.\endmultline\tag 20
$$
Our aim is to prove that for any $\kappa >0$ there exist $\varepsilon_0>0$ and
$\delta _0 >0$ such that
$$
\vert I^{\varepsilon,\delta}_2\vert\le \kappa \quad \forall \varepsilon \in
(0, \varepsilon_0),\,\, \delta\in (0,\delta_0).
\tag 21 $$
We start from the following estimate
$$\multline
\vert I^{\varepsilon,\delta}_2\vert\le \hat C\big[\int^T_0
\int_{\vert r\vert\le c\delta}
\int^{+\infty}_{-\infty} \vert \Phi \omega^\varepsilon_\theta\vert dx_3drdt
\left(\left\Vert \frac{\omega_0}{r}\right\Vert_{L^1({\Bbb R}^3)}+1\right)
 \\
+\int_0^T\!\!\int_{(G\times G)
\cap \{\vert r-r'\vert+\vert x_3-x_3'\vert<c\delta\}}\vert
(\Phi\omega^\varepsilon_\theta)
(t,r,x_3)(\Phi\omega^\varepsilon_\theta)(t,r',x'_3)\vert
\,dr\,dx_3\,dr'\,dx'_3dt\big].\endmultline \tag 22
$$

Let us consider the system of ordinary differential equations
$$
\frac{d X_{ \varepsilon } (t,\alpha)}{dt}=v^\varepsilon (t,X_{ \varepsilon } 
(t,\alpha)),\quad X_{ \varepsilon } (t,\alpha)\vert_{t=0}=\alpha\,.
\tag 23
$$
Using (23), one can write out the solution of (9) as 
$$
\left(\frac{\omega^\varepsilon_\theta}{r}\right)(t,X_{ \varepsilon } 
(t,\alpha))=
\left(\frac{\omega^\varepsilon_{0,\theta}}{r}\right)(\alpha),\quad
\alpha\in{{\Bbb R}}^3\,.
$$
Or, equivalently 
$$
\frac{\omega^\varepsilon_\theta}{r}(t,\alpha)=
\left(\frac{\omega^\varepsilon_{0,\theta}}{r}\right)(X_{ \varepsilon } ^{-1
}(t,\alpha))\,.\tag 24
$$
Let us denote 
$$
\frak O^{(t)}_{\delta,\varepsilon}=X_\varepsilon \left(t,\{ (r, x_3)\in \overline{G}\, |
\, r\le \delta,\,\,(r,x_3 )\in \text{supp}\, \Phi (t,\cdot)\}\right )\,.
$$
Since, by assumption, $\text{supp}\,\Phi$  is compact in $[0,T]\times {\Bbb R}^3$
and the mapping
$X^{-1}_\varepsilon(t,\cdot)$ conserves a volumes, 
we have
$$
\sup_{t\in [0,T]} \mu (\frak{ O}^{(t)}_{\delta,\varepsilon}) 
 \rightarrow  0\quad\text{as}\,\,
\delta\rightarrow +0\,,
\tag 25
$$
uniformly in $\varepsilon$.

Taking into account that $\text{det}(\nabla X_{ \varepsilon } (t,\alpha))\equiv 1,$ one 
can estimate the first term of the right hand side of (22) as follows:
$$
\int_0^T\!\!\int_{\vert r\vert\le c\delta}
\int^{+\infty}_{-\infty
}
2\pi\vert \Phi\omega^\varepsilon_\theta\vert \,dr\,dx_3\,dt\le C \int_0^T\!\!\int_{
\frak{O}_{\delta,\varepsilon}^{(t)}
}
\left\vert \left(\frac{\omega^\varepsilon_{0,\theta}}{r}\right)(t,x)
\right\vert \,dx\,dt=B_{\varepsilon,\delta}\,.
$$
Note that
$$
B_{\varepsilon,\delta}\le C \sup_{
t\in(0,T)}
\int_{
\frak{O}_{\delta,\varepsilon}^{(t)}}\left(
\left\vert\frac{\omega_{0,\theta}}{r}
\right\vert 
+\left\vert\frac{\omega_{0,\theta}}{r}
-\frac{\omega^\varepsilon_{0,\theta}}{r}\right\vert\right)\,dx\,.
$$
Hence, by (16),(25) for any $\kappa >0$ there exists $\varepsilon_0>0,\delta_0>0$ such that
$$
\vert B_{\varepsilon, \delta}\vert\le \frac{\kappa}{4a}\quad
\forall\varepsilon\in(0,\varepsilon_0),\,\, \delta\in(0,\delta_0)\,,\tag 26
$$
where $a=\hat C(\left\Vert\frac{\omega_0}{r}\right\Vert_{L^1({\Bbb R}^3)}+1).$
On other hand, from
$$
\int_{(G\times G)\cap \{\vert r-r'\vert+\vert x_3-x_3'\vert\le c\delta\}}
\vert (\Phi \omega^\varepsilon_\theta)(t,r,x_3)
(\Phi\omega^\varepsilon_\theta)(t,r',x_3')\vert \,dr\,dx_3\,dr'\,dx_3'\,dt
$$
after the change of variables we obtain
$$\multline
\int_0^T\!\!\int_{G}
 \vert (\Phi\omega^\varepsilon_\theta)
(t,r',x'_3)\vert (\int_{\{\vert\tilde r\vert+\vert \tilde x_3\vert\le c\delta\}}
\vert (\Phi\omega^\varepsilon_\theta)(t,\tilde r+r',\tilde x_3+x'_3)\vert d\tilde rd\tilde x_3)
\,dr'\,dx'_3 dt\\
\le C
\left\Vert \frac{\omega^\varepsilon_{0}}{r}\right\Vert_{L^1({\Bbb R}^3)}
\int^T_0(\sup_{(r',x'_3)\in G}
\int_{\{\vert\tilde r\vert+
\vert \tilde x_3\vert\le c\delta\}}
\vert (\Phi\omega^\varepsilon_\theta)(t,\tilde r+r',\tilde x_3+
x_3')\vert \,d\tilde r\,d\tilde x_3)\,dt\\
\le C
\left\Vert \frac{\omega^\varepsilon_{0}}{r}\right\Vert_{L^1({\Bbb R}^3)}
\int^T_0(\sup_{(r',x'_3)\in G}
\int_{\{\vert\tilde r\vert+
\vert \tilde x_3\vert\le c\delta\}}\vert \omega^\varepsilon_\theta
(t,\tilde r+r',\tilde x_3+
x_3')\vert \,d\tilde r\, d\tilde x_3 )\,dt\\
\le C
\left\Vert \frac{\omega^\varepsilon_{0}}{r}\right\Vert_{L^1({\Bbb R}^3)}
%\times \\ \quad \times
\int^T_0\sup_{(r',x'_3)\in G}\int_{\{(r,x_3)\in G \vert
\vert r-r'\vert+
\vert x_3-x'_3\vert\le c\delta\}}\left\vert
\frac{\omega^\varepsilon_{0,\theta}}{r}(X_{ \varepsilon } ^{-1}(t,x))
\right\vert \,dx \,dt\,.
\endmultline\tag 27
$$

Set $\mu (\{x\in {\Bbb R}^3 \vert
\vert r-r'\vert+\vert x_3-x_3'\vert\le c\delta\})=\gamma(\delta).$
Then
$$\multline
\int_{(G\times G)\cap \{\vert r-r'\vert+\vert x_3-x_3'\vert\le c\delta\}}
\vert (\Phi\omega^\varepsilon_\theta)(t,r,x_3)\vert
\vert (\Phi\omega^\varepsilon_\theta)(t,r',x'_3)\vert \,dr\,dx_3\,dr'\,dx_3'\,dt\\
\le \hat C T\left\Vert\frac{\omega^\varepsilon_{0}}{r}\right\Vert_{L^1({\Bbb R}^3)}
\sup \Sb \frak B\\
\mu(\frak B)\le \gamma(\delta)\endSb\int_{\frak B}
\left\vert\frac{\omega^\varepsilon_{0}}{r}\right\vert \,dx\,.
\endmultline\tag 28
$$
Since $\gamma(\delta)\rightarrow 0$ as $\delta\rightarrow 0$ for any $\kappa >0$, 
one can find $\delta_0>0$ and $\varepsilon_0>0$ such that right hand side of (28) is less than or equal to
$\frac{\kappa}{4}$ for all $\delta\in (0,\delta_0)$ and $\varepsilon\in(0,\varepsilon_0)$.
Then, taking into account (28) , we obtain (21).

On the other hand, we have 
$$\multline
K(t,r,x_3,r'x_3')\left(1-\eta\left(\frac{r}{\delta}\right)
\right)\left(1-\eta\left(\frac{r'}{\delta}\right)\right)
\left(1-\eta\left(\frac{\vert r-r'\vert+\vert
x_3-x_3'\vert}{\delta}\right) \right)\\
\in C^\infty([0,T]\times \overline G\times \overline  G).\endmultline$$ Hence
$$
I^{\varepsilon,\delta}_1\rightarrow I_1^\delta\quad \text{as}\,\,\,
 \varepsilon
\rightarrow 0.\tag 29
$$
Thus by (21) and (29), 
$$
A_1^\varepsilon\rightarrow A_1.
$$
The proof of the lemma is complete. \hfil $\blacksquare$ \medskip

Let us introduce a class of axisymmetric vector fields without a swirl 
component, $L^2 _a({\Bbb R}^3  )=\{ v\in (L^2 ({\Bbb R}^3 ) )^3 \, | \,  v=v(r, x_3),  v_{\theta}=0 \}.$
For a given N-function $A(t)$, following [3], we introduce 
$$
\Cal{Q}_A ({\Bbb R}^3 )=\{ v\in L^2 _a ({\Bbb R}^3 ) 
\cap W^1 L_A ({\Bbb R}^3 )\, | \, \text{\bf  div}\, v =0 , 
\text{\bf curl}\, v \in L_A ({\Bbb R}^3 )   \}
$$
equipped with the Banach space norm
$\Vert v\Vert _{{\Cal Q}_A ({\Bbb R}^3 ) }= 
( \Vert v\Vert _{L^2({\Bbb R}^3)}^2+ 
\Vert \text{\bf  curl}\, v \Vert^2_{L_A ^2({\Bbb R}^3)} )^{1/2}$. 
Here the derivatives are in the distribution
sense. We can extend our definition to $\Cal{Q}_A (\Omega )$ for any 
axisymmetric domain $\Omega$ in ${\Bbb R}^3$. 

Now, we establish the following compactness lemma, which is an axisymmetric 
analogue of Lemma 6. of [3].

\proclaim{Lemma 2.3} 
Let $A(t)$ be an N-function satisfying the $\Delta _2-$condition, and satisfies
$ A(t)\succ t(\log ^+ t )^{\frac{1}{2}} $.  Then  for any bounded sequence 
$\{ v^{\varepsilon} \}$ in 
$\Cal{Q}_A ( {\Bbb R}^3)$ there exists a subsequence, denoted by the same 
notation, $\{ v^{\varepsilon} \}$ and $v\in\Cal{Q}_A ( {\Bbb R}^3)$ such that 
$$ 
\lim_{\varepsilon\rightarrow 0} \int_{{\Bbb R}^3} \rho |v^{\varepsilon}|^2 \, dx
 = \int_{{\Bbb R}^3}  \rho |v|^2 \,dx
$$
 for any given axisymmetric  test function $\rho \in C_0 ^{\infty} 
 ({\Bbb R}^3)$ with $\text{supp}\, \rho 
\subset\{ (r, x_3 )\in  {\Bbb R}^2 \, | \, r> 0 \} $.
\endproclaim

\noindent{\bf Proof.} Let $\{ v^{\varepsilon}\}$ be a uniformly bounded 
sequence in $\Cal{Q}_A ( {\Bbb R}^3)$.  Then, there
exists a subsequence, denoted by $\{ v^{\varepsilon}\}$, and $v$ in  
$\Cal{Q}_A ( {\Bbb R}^3)$ such that
$$
v^{\varepsilon} \rightarrow v \quad \text{weakly in $L^2  ( {\Bbb R}^3 )$}\,.
\tag 30
$$
For such $v^{(\varepsilon)} $ we introduce stream functions 
$\psi^{(\varepsilon)}=\psi^{(\varepsilon)} (r,x_3 )$ such that
$$
v_r^{(\varepsilon)}=-\frac{1}{r}\frac{\partial\psi^{(\varepsilon)}}{\partial x_{3}},\quad v_3^{(\varepsilon)}=
\frac{1}{r}\frac{\partial \psi^{(\varepsilon)}}{\partial r}.
$$
 Let a function $\rho \in C_0 ^{\infty} ({\Bbb R}^3) $ and  a bounded domain 
 $W$  with  $\text{supp}\,\rho  \subset \overline W \subset G $ be given. 
Then, by integration by part we obtain
$$
\align
\int_ {{\Bbb R}^3}\rho |v^{(\varepsilon)}|^2 \,dx
&= \int _ {{\Bbb R}^3}\rho ((v_r^{(\varepsilon)} ) ^2 
+(v_3^{(\varepsilon)}) ^2)dx \\
&=2\pi \int_ {{\Bbb R}^3}\left( -\rho v_r^{(\varepsilon)} 
\frac{1}{r} \frac{\partial \psi^{(\varepsilon)}}{\partial x_3} 
+\rho v_3^{(\varepsilon)} \frac{1}{r} \frac{\partial\psi^{(\varepsilon)}}{
\partial r} \right)r\,dr\,dx_3 \\
&=2\pi\int _ {{\Bbb R}^3}\left( \frac{\partial \rho}{\partial x_3} v_r^{(\varepsilon)} \psi^{(\varepsilon)} -
\frac{\partial \rho}{\partial r} v_3^{(\varepsilon)} \psi^{(\varepsilon)} \right. \\
&\quad \quad \left. +\rho \frac{\partial v_r^{(\varepsilon)}}{\partial x_3} \psi^{(\varepsilon)}
-\rho \frac{\partial v_3^{(\varepsilon)}}{\partial r} \psi^{(\varepsilon)}
\right)\,dr\,dx_3\\
&= 2\pi \int _G \left( \frac{\partial \rho}{\partial x_3} v_r^{(\varepsilon)} \psi ^{(\varepsilon)}-\frac{\partial \rho}{\partial r} v_3^{(\varepsilon)} \psi ^{(\varepsilon)}
  \right)\,dr\,dx_3 \\
&\quad \quad + 2\pi \int_G \omega_{\theta}^{(\varepsilon)} \psi ^{(\varepsilon)}\rho \,dr\,dx_3 
=\{ 1\}^{(\varepsilon)} +\{2\}^{(\varepsilon)}\,.
\tag 31
\endalign
$$
Since 
$$\| \nabla \psi^\varepsilon \|_{L^2 (W )} \leq C(W) \left\Vert \frac{\nabla \psi^\varepsilon}{r} \right\Vert_{L^2 (W )} =
C(W)  \| v^\varepsilon\|_{L^2 (W )} \leq C,
$$
 we obtain by Rellich's compact imbedding lemma that 
$$
\rho_1 \psi ^{\varepsilon} \rightarrow \rho_1 
\psi \quad \text{strongly in} \quad L^2(W) \quad \forall \rho_1 \in C_0 
^{\infty} (W )
$$
after choosing a subsequence.
This, combined with (30), provides easily  that $\{ 1\} ^\varepsilon \rightarrow \{ 1\}$  in (31) as $\varepsilon \rightarrow 0$.

To prove $\{ 2\} ^\varepsilon \rightarrow \{ 2\}$ we observe that
$$
\rho \psi ^{\varepsilon} \rightarrow \rho \psi \,, \tag 32
$$
and
$$
\|\omega _{\theta} ^\varepsilon\|_{L_{t(\log^+ t)^{\frac12}}(W)} \leq C
\|\omega _{\theta} ^\varepsilon\|_{L_A(W)} \leq C_2\,, \tag 33
$$
where $B(t)= \exp (t^2 )-1$. Since $A(t)=t(\log ^+ t )^{\alpha }\succ t(\log ^+ t )^{\frac12}$  by hypothesis, 
applying Corollary 1.1, we find that there exists a subsequence $\{ \omega ^{\varepsilon}
_{\theta} \}$  and $\omega_{\theta}$  in $\text{H}^{-1} (W)\hookleftarrow\hookleftarrow L_{A}(W)$ such that
$$
\omega ^{\varepsilon}_{\theta}  \rightarrow \omega _{\theta}   
\quad \text{in}\quad \text{H}^{-1} (W)\,.
\tag 34
$$
We decompose our estimate
$$
\align
\left\vert \int_G ( \omega_{\theta}^{\varepsilon} \psi ^{\varepsilon} -
 \omega_{\theta} \psi  )\rho \,dr\,dx_3 \right\vert 
&\leq 
\left\vert \int_W ( \omega_{\theta}^{\varepsilon} -\omega_{\theta} )\psi ^{\varepsilon} \rho \,dr\,dx_3\right\vert +\left\vert  \int_W (\psi ^\varepsilon -\psi)\omega_{\theta} \rho \,dr\,dx_3 \right\vert \\
&=J_1^{\varepsilon} + J_2 ^{\varepsilon}.
\endalign
$$
From (32) and (34) we obtain 
$$
J_1^{\varepsilon}\leq C\|\psi ^{\varepsilon} \|_{H^1 (W)} \| \omega_{\theta}^{\varepsilon} - \omega_{\theta}\|_{\text{H}^{-1} (W)}\rightarrow 0
$$ 
after choosing a subsequence, if necessary.
 On the other hand, the convergence 
$J_2 ^{\varepsilon}\rightarrow 0$ for another subsequence, 
if necessary, follows from (32). 
This completes the proof of the lemma. \hfil $\blacksquare$

Using Lemma 2.3 we establish  the following

\proclaim{Lemma 2.4} Suppose a sequence $\{ v^{\varepsilon} \}$ and $v$ be 
given as in (1.7), Lemma 2.2.
Let $\eta(r)\in C^\infty ({\Bbb R}_+),$ $\eta(r) \ge 0$,
$\eta (r)=1, \,\, r \in [1,\infty]\,\, \text{and}\,\, \eta(r)=0 $ for $r
<\frac 12$. Then for any $\delta >0$ and 
$\varphi \in C^{\infty} ([0, T];C_0 ^{\infty} ({\Bbb R}^3))$ we have
$$
\int_Q\eta\left(\frac{r}{\delta}\right)\vert v^\varepsilon -
v\vert^2 \varphi \,dx\,dt\rightarrow 0\,\,\, \text{as} \,\,{\varepsilon}\rightarrow
+0\,,\tag 35
$$
after choosing a subsequence.
\endproclaim

\noindent{\bf Proof.} Let $W$ be any given bounded domain in $G$ whose 
closure does not intersect with the axis of symmetry.  
By conservation of $L ^2 ({\Bbb R}^3)$ norm of velocity
we have 
$$
\| v^{\varepsilon}(t,\cdot) \|^2 _{L^2(W)}  \le  C(W) \| v^{\varepsilon}(t,\cdot) \|^2 _{L^2 ({\Bbb R}^3) } 
= C(W) \| v^{\varepsilon}_0 \|^2 _{L^2 ({\Bbb R}^3) } 
\leq C(W, v_0 )\,.
\tag 36
$$
 On the other hand, the conservation of $\frac{\omega ^{\varepsilon}_\theta (t, x)}{r}$ along the flow, (9),
implies 
$$
 \left \Vert
{\omega^\varepsilon}(t,\cdot)\right\Vert_{L_A(  W)}
\le C (W)\left\Vert\frac{\omega^\varepsilon}{r}(t,\cdot)\right
\Vert_{L_A({ {\Bbb R}^3)}}=C(W)
\left\Vert\frac{\omega^\varepsilon_{0}}{r}\right
\Vert_{L_A( {{\Bbb R}^3)} }
\le C(W, v_0 ),
\tag 37
$$
where $A=A(t)=t (\log^+t)^{\alpha}$.
Combining (36) and (37), we find that 
$$
\sup_{t\in[0,T]} \|v^{\varepsilon} (t, \cdot ) \|_{\Cal{Q}_A (W)}  \leq C.
\tag 38
$$
From the  estimate  (38), combined with  (19), together with Lemma 2.3,  
 we deduce by  using   the standard compactness lemma that there is a 
 subsequence $\{ v^{\varepsilon} (t,r,x_3)\}$
 such that
$$
v^{\varepsilon}\rightarrow v\,\,\text{strongly in}\,\, L^2 ([0,T]\times W)\,.
$$
Now (35) follows from this immediately.
The lemma is proved. \hfil$\blacksquare$ \smallskip

\noindent{\bf Proof of Theorem 1.1} To prove the theorem we have only to 
show that
$$
I^\varepsilon=\int_{Q} v_i^\varepsilon v_j^\varepsilon \varphi \,dx\,dt
\rightarrow I=\int_Q v_i v_j \varphi \,dx\,dt\,,\tag 39
$$
for all $i,j\in \{1,2,3\},$ and $\varphi\in C^\infty([0,T]; C^\infty_0
({\Bbb R}^3)).$
Let $\eta(\tau)\in C^\infty({\Bbb R}^1_+), 0\le\eta\le 1, \eta(\tau)=1
$ for all $\tau\in [1,+\infty)$ and $\eta(\tau)=0\,\,\,
\text{for}\,\, \tau\in[0,\frac 12]$.
For any $\delta >0$ we set
$$
I^\varepsilon=I^{\varepsilon,\delta}_1+I^{\varepsilon,\delta}_2=
\int_Q\eta\left(\frac{r}{\delta}\right)v_i^\varepsilon v_j^\varepsilon \varphi dx+
\int_Q\left(
1-\eta\left(\frac{r}{\delta}\right)
\right)v_i^\varepsilon v_j^\varepsilon \varphi\, dx\,.
$$
By Lemma 2.4
$$
I^{\varepsilon,\delta}_1\rightarrow
\int_Q\eta\left(\frac{r}{\delta}\right)v_i v_j \varphi dx\,\,\text{as}\,\,
\varepsilon \rightarrow 0\,.   \tag 40
$$
Hence the statement of theorem will be proved, if we show that for any $
\kappa >0$ there exists $\delta_0>0$ such that for all $\delta\in(0,\delta_0)$ one can find 
$\varepsilon_0(\delta)>0$ that 
$$
\vert I^{\varepsilon,\delta}_2\vert\le \kappa\quad \forall\, 
\varepsilon\in(0,\varepsilon_0)\,.\tag 41
$$
Indeed,  in case either $i$ or $j$ equals $1$ or $2$, by Lemma 2.1 we have
$$\align 
\left\vert\int_Q\left(1-\eta\left(\frac{r}{\delta}\right)\right)
v_i^\varepsilon v_j^\varepsilon \varphi \,dx\right\vert
\le& 2\pi\int_0^T\!\!\int^{+\infty}_{-
\infty}\int_0 ^{\delta} r \vert v_r^\varepsilon v^\varepsilon_j\varphi\vert \,dr 
\,dx_3\,dt\\
\le& C\delta\int_0^T\!\!\int^{+\infty}_{-\infty}\int_0 ^{\delta}\vert
\varphi v^\varepsilon_r v^\varepsilon_j\vert \,dr\,dx_3\,dt  \tag 42 \\
\le& C\delta\left(\int_0^T\!\!\int _{-\infty} ^{+\infty}
\int^{+\infty}_0 \frac{1}{1+x_3 ^2}
\frac{\vert v_r^\varepsilon\vert^2}{r} \,dr\,dx_3\,dt\right)^\frac 12 \Vert
v^\varepsilon_0\Vert_{V^0}\\
\le& C\delta \left (\Vert v_0\Vert^2_{V^0}+\left\Vert
 \frac{\omega_{0}}{r}
\right\Vert_{L^1({\Bbb R}^3)}+1\right)^\frac 12(\Vert v_0\Vert_{V^0}+1)\,.
\endalign
$$
Hence taking parameter  $\varepsilon_0=1$ and parameter $\delta_0$ sufficiently small,  we obtain (41).

Let us consider the case $i=j=3$. 
Then
$$
\vert I^{\varepsilon,\delta}_2\vert \le \hat C
\int_Q\left\vert 1-\eta\left(
\frac{r}{\delta}\right)\right\vert
(v^\varepsilon_3)^2 \,dx\,dt.\tag 43
$$
Set $\rho(r)=1-\eta(r).$
Let $\delta_1 >0$ be such that
$$\left
\vert \int_Q \rho\left(\frac{r}{\delta}\right)((v_r)^2-
(v_3)^2) \,dx\,dt\right\vert \le \frac{\kappa}{4\hat C}\quad\forall \,
\delta\in(0,\delta_1).\tag 44
$$
In the above we also proved that for each $\kappa>0$ there exists $\delta_2 >0$
such that 
$$
\int_Q (v_r^\varepsilon)^2\rho\left(\frac{r}{\delta}\right)\,dx\,dt\le \frac{\kappa}{4\hat C}\,\,
\forall\, \delta\in(0,\delta_2), \,\, \varepsilon\in (0,1).\tag 45
$$
Let $\delta_0=\min\{ \delta_1,\delta_2\}$.

Note that by Lemma 2.2 for $\delta\in(0,\delta_0)$
$$
\int_Q[(v^\varepsilon_r)^2-(v_3^\varepsilon)^2]\rho\left(\frac{r}{\delta}
\right)
 \,dx\,dt\rightarrow
\int_Q[(v_r)^2-(v_3)^2]\rho
\left(\frac{r}{\delta}\right)\,dx\,dt\,,
$$
as $\varepsilon \rightarrow + 0$.

Thus for every $\kappa>0$ and $\delta\in(0,\delta_0)$ one can find $\varepsilon_0(\delta)$ that
$$\left\vert
\int_Q[(v^\varepsilon_r)^2-(v_3^\varepsilon)^2-
 (v_r)^2+(v_3)^2]\rho
\left(\frac{r}{\delta}\right)\,dx\,dt\right\vert \le \frac{\kappa}{4\hat C}\quad 
\forall \, \varepsilon\in(0,\varepsilon_0(\delta)).\tag 46
$$

Inequalities (43)-(46) imply (41).
Since now (36) is proved for an arbitrary $i,j\in\{ 1,2,3\}$,  the proof of 
the existence part of Theorem 2.1
is complete. The inequalities for the energy and the vorticity follow immediately
by the energy conservation for velocity and the conservation of 
$\frac{\omega_{\theta}}{r}$ for the smooth approximate solutions, and taking
 limit for suitable subsequence. This completes the proof of the theorem.
\hfil$\blacksquare$ \medskip

\noindent{\bf Acknowledgment.} The authors would like to express gratitude 
to the anonymous referee for a very constructive criticism with helpful 
suggestions.

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