\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 163, pp. 1--25.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/163\hfil Vorticity of 2D Navier-Stokes equation]
{Sharp asymptotic estimates for vorticity \\
solutions of the 2D Navier-Stokes equation}

\author[Y. You\hfil EJDE-2008/163\hfilneg]
{Yuncheng You}

\address{Yuncheng You \newline
Department of Mathematics and Statistics \\
University of South Florida \\
Tampa, FL 33620, USA}
\email{you@math.usf.edu}

\thanks{Submitted November 5, 2008. Published December 18, 2008.}
\subjclass[2000]{35B40, 35B65, 35K15, 35Q30, 76D05, 76D17}
\keywords{Navier-Stokes equation; vorticity; regularity;
asymptotic dynamics; \hfill\break\indent
Biot-Savart law; similarity solution; Oseen vortex}

\begin{abstract}
The asymptotic dynamics of high-order temporal-spatial derivatives
of the two-dimensional vorticity and velocity of an
incompressible, viscous fluid flow in $\mathbb{R}^2$ are studied,
which is equivalent to the 2D Navier-Stokes equation. It is known
that for any integrable initial vorticity, the 2D vorticity
solution converges to the Oseen vortex. In this paper, sharp
exterior decay estimates of the temporal-spatial derivatives of
the vorticity solution are established. These estimates are then
used and combined with similarity and $L^p$ compactness to show
the asymptotical attraction rates of temporal-spatial derivatives
of generic 2D vorticity and velocity solutions by the Oseen
vortices and velocity solutions respectively. The asymptotic estimates and
the asymptotic attraction rates of all the derivatives obtained in
this paper are independent of low or high Reynolds numbers.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

Vorticity is a vector field defined by $\omega = \nabla \times v$ for a
flow with velocity $v$. For any two-dimensional flow, if we orient
the $z$-axis of a Cartesian coordinate system normal to the
velocity field, then the one-dimensional vorticity is expressed by
$$
    \omega = \omega_{z} = \frac{\partial v_y}{\partial x}
- \frac{\partial v_x}{\partial y}.
$$
For an incompressible, viscous fluid motion in the domain of
$\mathbb{R}^2$, the velocity field of the fluid is governed by the
2D Navier-Stokes equation:
\begin{equation} \label{NSE}
    \begin{gathered}
    \frac{\partial u}{\partial t} - \nu \Delta u  +  (u \cdot \nabla) u 
     + \nabla p = 0, \\
     \nabla \cdot u = 0,
    \end{gathered}
\end{equation}
for $t > 0$, $x \in \mathbb{R}^2$, where $u = u(t, x)$ is a
two-dimensional vector field of the fluid velocity, $p(t, x)$ is
the pressure, and we assume that the external force on the fluid
is zero. The kinematic viscosity coefficient $\nu$ is a positive
constant. The initial value condition for $u(t, x)$ is
$$
    u(0, x) = u_0 (x), \quad \text{for }  x \in \mathbb{R}^2.
$$
Taking the curl of the equation \eqref{NSE}, we can consider the
equation for the vorticity $\omega (t, x)$ as well,
\begin{equation} \label{Veqn}
    \frac{\partial \omega}{\partial t} = \nu \Delta \omega - (u \cdot \nabla) \omega, 
    \quad t > 0, \quad     x \in \mathbb{R}^2.
\end{equation}
By the Biot-Savart law, cf.  \cite{gG94, MB02,  MP94}, the
velocity $u(t, x)$ can be reconstructed in terms of the vorticity
$\omega (t, x)$,
\begin{equation} \label{BSL}
    u(t, x) = \int_{\mathbb{R}^2}  G(x - y) \omega (t, y) \,dy,
    \quad t > 0, \; x \in \mathbb{R}^2,
\end{equation}
where
\begin{equation} \label{Ker}
    G(x) = G(x_1, x_2) = \frac{x^{\bot}}{2 \pi | x |^2}
    = \frac{(- x_2, x_1)}{2 \pi | (x_1, x_2) |^2}.
\end{equation}
 The system of equations \eqref{Veqn} and \eqref{BSL} is
equivalent to the Navier-Stokes equation \eqref{NSE}. We shall
call  \eqref{Veqn}--\eqref{BSL} the two-dimensional
\emph{vorticity equation}, which is the vorticity formulation of
the Navier-Stokes equation.

In \cite{cO11}, it was found that there is a special type of
solutions  to \eqref{Veqn}--\eqref{BSL},
\begin{equation} \label{OsV}
    \omega^{*} (t, x) = \frac{\kappa}{4\pi \nu t} \exp
    \big( - \frac{| x |^2}{4\nu t} \big),
\end{equation}
where $\kappa$ is a constant called the strength of the vortex.
These special solutions are called the \emph{Oseen vortices},
whose initial data are $\omega^{*}(0, x) = \kappa \delta (x)$ in the
distributional sense, where $\delta (x)$ is the Dirac function.

It has been proved in \cite{GK88} that for any initial vorticity
$\omega_0 \in L^1(\mathbb{R}^2)$ with small Renolds numbers
$\int_{\mathbb{R}^2} |\omega_0(x)|dx/\nu$, the vorticity equation
\eqref{Veqn}--\eqref{BSL}  has a unique solution
\begin{gather*}
    \omega  \in C([0, \infty), L^1(\mathbb{R}^2)) \cap C((0, \infty),
    L^{\infty} (\mathbb{R}^2)), \\
 u \in C([0, \infty), L^1(\mathbb{R}^2) \times L^1(\mathbb{R}^2) )
 \cap C((0, \infty), L^{\infty} (\mathbb{R}^2) \times L^{\infty}
(\mathbb{R}^2)),
\end{gather*}
with the asymptotic convergence properties at specific decaying rates:
\begin{gather} \label{Vconv0}
    \lim_{t \to \infty}   t^{1 - \frac{1}{p}} \| \omega (t, \cdot )
- \omega^{*} (t, \cdot ) \|_{p} = 0, \quad \text{for }  1 \leq p \leq
\infty, \\
 \label{Uconv0}
    \lim_{t \to \infty}   t^{\frac{1}{2} - \frac{1}{q}} \| u (t, \cdot )
 - u^{*} (t, \cdot ) \|_{q} = 0, \quad \text{for }  2 < q \leq \infty,
\end{gather}
where $\omega^{*} (t, x)$ is the Oseen vortex \eqref{OsV} with the coefficient
of strength
\begin{equation} \label{kw0}
    \kappa = \kappa (\omega_0) =  \int_{\mathbb{R}^2}   \omega_0 (x) \,dx,
\quad \omega_0 \in L^1 (\mathbb{R}^2),
\end{equation}
and the corresponding Oseen velocity $u^{*} (t, x)$ is given by
\begin{equation} \label{OsU}
    u^{*}(t, x) = \int_{\mathbb{R}^2}   G(x - y) \omega^{*} (t, y) \,dy.
\end{equation}
This asymptotical convergence result \eqref{Vconv0} and \eqref{Uconv0}
was then extended in \cite{aC94, GW05}.

As indicated in \cite{BdV07, BA94, lB04, aC94, bF04, GK88, cH01,
cH06, Met91}, it is relatively easier to understand the
asymptotics of the velocity solutions of the Navier-Stokes
equation \eqref{NSE} and the associated Euler equation by studying
the long-time dynamics of the vorticity solutions of the vorticity
equation, especially for two-dimensional flows due to the scalar
vorticity. Once the vorticity field on $\mathbb{R}^2$ is known or
approximately known, one can always reconstruct the velocity field
precisely or approximately by the singular integral with the Riesz
potential specified in the Biot-Savart law. Therefore, the study
of the regularity and asymptotical dynamics of the 2D and 3D
vorticity equations is important in the mathematical theory of
fluid mechanics. Some of the established results, related topics,
and numerical simulations in recent years can be found in
\cite{ABD05, BJ05, BdV07, BA94, lB04, aC94, FS96, bF04, GW05,
GK88, cH01, cH06, tK94, KR05, aM86, RGE02, SS00, sT99}.

In this work we shall prove certain sharp estimates on the
temporal-spatial  decaying dynamics of all the derivatives of the
vorticity solution $\omega(t, x)$ and velocity solution $u(t, x)$ to
the vorticity equation \eqref{Veqn}--\eqref{BSL} and then indicate
the applications to the global attraction regularity of the Oseen
vortex and velocity solutions. We emphasize that the asymptotic
estimates and the asymptotic attraction rates of all the
temporal-spatial derivatives obtained in this paper are
independent of low or high Reynolds numbers.

This application of the estimates shows that the set of all the
Oseen  vortices,
$$
    \Psi = \{ \omega^{*} (t, x): \kappa \in \mathbb{R}\},
$$
is a global attractor for the 2D vorticity semiflow in the full
regularity sense with respect to $L^p$ topology. Moreover, it also
shows that the attraction rates of $\Psi$ for the temporal-spatial
derivatives are getting faster and faster in the specific and
uniform order of $O(t ^{- (\alpha + |\beta|/2)})$, which depends on the
time derivative order $\alpha$ and the spatial derivative order
$\beta$. Therefore, it reveals that regularity asymptotics of the 2D
vorticity solutions on $\mathbb{R}^2$ is determined by the unique
Oseen vortex pattern completely.

Below we shall use the notation $\partial_{t}^{\alpha} = (\partial / \partial
t)^{\alpha}$  for $\alpha \in Z^{+}$, $\partial_{x}^{\beta} = \partial^{|\beta|}
/ (\partial x_1^{\beta_1} \partial x_2^{\beta_2})$ for $\beta = (\beta_1, \beta_2)
\in Z^{+} \times Z^{+}$, where $Z^{+}$ stands for the set of all
nonnegative integers. As notational remarks, we have
$$
| \beta | = \beta_1 + \beta_2, \quad \beta! = \beta_1 ! \beta_2 !, \quad
\binom{\beta}{\gamma} = \frac{\beta!}{\gamma! (\beta - \gamma)!}
 \leq 2^{| \beta |}\quad \text{for }  \gamma \leq \beta.
$$
By convention, if $p = \infty$, then $1/p = 0$. We use $| \cdot |$
to denote either an absolute value or a vector norm. The norm of
$L^p (\mathbb{R}^2)$ will be denoted by $\| \cdot \|_p, 1 \leq p
\leq \infty $. For simplicity, we shall always use $\| \cdot \|$
for the norm of $L^1 (\mathbb{R}^2)$ instead of using $\| \cdot \|_1$.

For any function $g$ defined on $\mathbb{R}^2$, we shall use $g^r$
and $g_r$ to denote the functions
$$
    g^r (x) = \vartheta_r (x) g(x) \quad \text{and} \quad  g_r (x)
  = (1 - \vartheta_r (x)) g(x),
$$
respectively, where $\vartheta_r$ is the characteristic function
of the set $\mathbb{R}^2 \cap \{| x | > r \}$.

The following convolution property
\cite[Th\'{e}or\`{e}me IV.15]{hB87} will be used:
for any $f \in L^1 (\mathbb{R}^n)$ and
$g \in L^p (\mathbb{R}^n), 1 \leq p \leq \infty$, one has $f \ast
g \in L^p (\mathbb{R}^n)$ and
\begin{equation} \label{convln}
    \| f \ast g \|_{L^p (\mathbb{R}^n)} \leq \| f \|_{L^1 (\mathbb{R}^n)}
  \| g \|_{L^p (\mathbb{R}^n)}.
\end{equation}
One of the basic $L^p$ estimates for convolutions is the
Hardy-Littlewood-Sobolev inequality \cite[Theorem 4.5.3]{lH90}:
 Let integer $n \geq 1$ and
$$
    \Phi (x) = | x |^{- n/a}, \quad x \in \mathbb{R}^n.
$$
If $1 < a < \infty, 1 < p < q < \infty$, and $1/p + 1/a = 1 + 1/q$,
then there exists a uniform constant $S_{n, p, q} > 0$, such that
\begin{equation} \label{HLS}
    \| \Phi \ast g \|_q \leq S_{n, p, q}   \| g \|_p, \quad
 \text{for }  g \in L^p (\mathbb{R}^n).
\end{equation}

In Section 2, we prove the exterior decay estimates of the $L^p$ norms
for vorticity and velocity. In Section 3, based on the mild solution
formula and semigroup expressions of the derivatives of vorticity solutions,
the exterior decay rates are established for the linear terms and
the nonlinear integrands by decomposition methods. Then in Section 4,
these nested estimates are utilized to prove the global decay rates of
temporal-spatial derivatives of vorticity solutions in 
Theorem \ref{Thm: wspdc} and
Theorem \ref{Thm: tsdrdc} via the bootstrap approach. Finally in Section 5, an
application result on the regularity of asymptotical attraction of
the Oseen vortex and velocity solutions is shown.

\section{Exterior Decay Estimates of Vorticity and Velocity}

By the optimal smoothing inequalities \cite[Theorems 1 and
2]{CL96} and the ultracontractivity (or called $L^1$ to $L^\infty$
regularity) of the parabolic semigroups shown in \cite{lG76, DS84,
SY02} for unbounded  domains, a solution $(\omega(t, x), u(t, x))$ of
the 2D vorticity equations \eqref{Veqn}--\eqref{BSL} with
integrable initial data $\omega_0(x)$ satisfies the following
estimates:
\begin{equation}  \label{estinfty}
    \| \omega(t, \cdot) \|_{\infty} \leq \frac{1}{4\pi \nu t}   \| \omega_0 \|,
\quad \| u(t, \cdot) \|_{\infty} \leq \frac{1}{\sqrt{2\pi^2 \nu t}}
\| \omega_0 \|, \quad t > 0.
\end{equation}
Based on these estimates, it has been shown in \cite[Theorem
3]{CL96}  that for the 2D Navier-Stokes vorticity, the following
pointwise decay holds,
\begin{equation} \label{ptdec}
    | \omega(t, x) |\leq \int_{\mathbb{R}^2} H(t, x, y) | \omega_0 (y) | \,dy,
\quad t> 0, \; x \in \mathbb{R}^2,
\end{equation}
where, for any given $0 < b < 1$,
\begin{equation} \label{kerineq}
    H(t, x, y) \leq \frac{C_{b}}{t} \exp
\big(-   \frac{b | x - y |^2}{4\nu t} \big),
\end{equation}
with the constant $C_{b}$ given by
$$
    C_{b} = \frac{1}{4\pi \nu} \exp
\Big(\frac{b \| \omega_0 \|^2}{2 \pi^2 \nu (1 - b)} \Big).
$$

First we prove an estimate of the $L^p$ norm of vorticity in the exterior
of a bounded ball.

\begin{lemma} \label{Lm: wtheta}
Let $\omega(t, x)$ be a vorticity solution of
\eqref{Veqn}--\eqref{BSL} with initial data $\omega_0 \in
L^1(\mathbb{R}^2)$. Then for any $p \in [1, \infty]$ and any given
$R > 1$, the following estimate is satisfied by $\omega$,
\begin{equation} \label{wgR}
    \| \omega^{R} (t, \cdot) \|_p \leq L_0 (p, \|\omega_0\|)
\Big( \frac{1}{R}   t^{-( \frac{1}{2} - \frac{1}{p} )} \| \omega_0 \|
 +  t^{-( 1 - \frac{1}{p} )} \| \omega_0^{R/2} \| \Big),
\end{equation}
where $L_0 (p, \|\omega_0\|)$ is a constant depending only on $p$ and
$\|\omega_0\|$.
\end{lemma}

\begin{proof}
By \eqref{ptdec}--\eqref{kerineq} where $b \in (0, 1)$ is arbitrarily fixed, 
for any given $R > 1$ and for $| x | > R$, we have
\begin{equation} \label{xgR}
    \begin{split}
    | \omega (t, x) |
&\leq \int_{\mathbb{R}^2 \cap \{|y| \leq R/2\}} \frac{C_{b}}{t} \exp
 \Big( -   \frac{b |x - y|^2}{4\nu t} \Big) | \omega_0(y) |\,dy \\
&\quad + \int_{\mathbb{R}^2 \cap \{|y| > R/2\}} \frac{C_{b}}{t}
\exp \Big( -   \frac{b |x - y|^2}{4\nu t} \Big) | \omega_0(y) |\,dy
   \end{split}
\end{equation}
and we shall denote the two integrals in \eqref{xgR} by $J_1 (t, x)$
and $J_2 (t, x), |x| > R$, respectively. Note that for the function
$$
    \varphi (t) = \frac{1}{t^{1/2}}
\exp \big( -   \frac{\zeta^2}{t} \big), $$
where $\zeta$ is a constant,
one can check that
\begin{equation} \label{phimax}
    \max_{t > 0}  \varphi (t) = \varphi (2\zeta^2) = \frac{1}{\sqrt{2}   \zeta}   e^{- 1/2}.
\end{equation}
By \eqref{phimax}, the first integral in \eqref{xgR} satisfies the estimate
\begin{align*}
    J_1 (t, x) &\leq C_{b} \int_{\{|y| \leq R/2\}} \frac{1}{t}
\exp \Big( -   \frac{b (|x| - |y|)^2}{4\nu t} \Big) |\omega_0 (y) | \,dy \\
    & \leq C_{b} \int_{\{|y| \leq R/2\}} \frac{1}{t^{1/2}}
\Big[ \frac{1}{t^{1/2}} \exp \Big( -   \frac{b (|x|/2)^2}{4\nu t} \Big) \Big] 
|\omega_0 (y) | \,dy \\
    & = C_{b} \int_{\{|y| \leq R/2\}} \frac{1}{t^{1/2}}\exp
\Big( -   \frac{b |x|^2}{32\nu t} \Big)
 \Big[ \frac{1}{t^{1/2}} \exp \Big( -   \frac{b |x|^2}{32\nu t} \Big) \Big]
|\omega_0 (y) | \,dy \\
    & \leq C_1 (b) \frac{1}{t^{1/2} | x |} \exp
\Big( -   \frac{b | x |^2}{32\nu t} \Big) \|\omega_0 \|,
\end{align*}
due to that $| x | > R > R/2 \geq | y |$, where
$$
    C_1(b) = \frac{4\sqrt{\nu}   C_{b}   e^{- 1/2}}{\sqrt{b}}.
$$
Hence it follows that, for $p \geq 1$,
\begin{equation} \label{J1p}
    \begin{split}
    \| J_1 (t, \cdot) \|_p
&\leq \frac{C_1 (b)}{t^{1/2}} \Big( \int_{\{| x | > R\}}  \frac{1}{| x |^p}
 \exp \Big( -   \frac{p  b   | x |^2}{32\nu t} \Big) \,dx \Big)^{1/p} \| \omega_0 \| \\
& \leq \frac{C_1 (b)}{t^{1/2}R} \Big( \int_{\{| x | > R\}}  \exp
 \Big( -   \frac{p  b   | x |^2}{32\nu t} \Big) \,dx \Big)^{1/p} \|\omega_0 \| \\
    & \leq \frac{C_2 (b, p)}{R} \; t^{- (\frac{1}{2} - \frac{1}{p})} \|\omega_0 \|,
    \end{split}
\end{equation}
where
$$
    C_2(b, p) = C_1(b) \Big( \int_{\mathbb{R}^2} \exp
\big(-   \frac{p  b   | \xi |^2}{32\nu} \big) \,d\xi \Big)^{1/p}.
$$
According to the convolution property \eqref{convln}, the second integral
$J_2(t, x), | x | > R$, in \eqref{xgR} is estimated as follows,
\begin{equation} \label{J2p}
    \begin{split}
    \| J_2(t, \cdot) \|_p
& \leq \frac{C_{b}}{t} \Big( \int_{\{| y | > R/2\}}
 \exp \big( -   \frac{p  b   | x - y |^2}{4\nu t} \big) \,dy \Big)^{1/p}
   \|\omega_0^{R/2} \| \\
& \leq \frac{C_{b}}{t} \Big( \int_{\mathbb{R}^2}  \exp
 \big( -   \frac{p  b  | y |^2}{4\nu t} \big) \,dy \Big)^{1/p} \|\omega_0^{R/2} \| \\
& \leq C_3(b, p)  t^{- (1 - \frac{1}{p})} \|\omega_0^{R/2} \|,
    \end{split}
\end{equation}
where
$$
    C_3(b, p) = C_{b} \Big( \int_{\mathbb{R}^2}  \exp
\big( -   \frac{p  b  | \xi |^2}{4\nu} \big) \,d\xi \Big)^{1/p}.
$$
Finally, let $L_0(p, \|\omega_0\|) = \max \{C_2(b, p), C_3(b, p) \}$.
Then the estimate \eqref{wgR} follows from \eqref{J1p} and \eqref{J2p}.
The lemma is proved.
\end{proof}

Next we shall estimate the exterior decay of the velocity solution
$u(t, x)$ in \eqref{BSL}. The key to this estimation is the Riesz
potential corresponding to the kernel $G(x-y)$ in \eqref{Ker}. For
any given $\theta \in (0, n)$, define a Riesz potential
$\Gamma_{\theta}$ to be
\begin{equation} \label{Rp}
    \Gamma_{\theta} (f) (x) = \int_{\mathbb{R}^n}
\frac{1}{| x - y |^{n - \theta}}   f(y) \,dy, \quad f \in L^1(\mathbb{R}^n)
 \cap L^{\infty} (\mathbb{R}^n).
\end{equation}

\begin{lemma} \label{Lm: Gpest}
Let $1 < p < q < \infty$ and $\theta = n(\frac{1}{p} -
\frac{1}{q})$. Then there exist positive constants $M_1 (n,
\theta, q)$ and  $M_2(n, \theta, q)$ such that for any given
$R > 1$, it holds that
\begin{equation} \label{gest}
    \| \Gamma_\theta (f)^R (\cdot) \|_q \leq
\frac{M_1}{R^{n - \theta - \frac{n}{q}}}  \| f \|
+ M_2 \| f^{R/2} \|_p,  \quad f \in L^1(\mathbb{R}^n) \cap L^{\infty}
(\mathbb{R}^n).
\end{equation}
\end{lemma}

\begin{proof}
    By the similar decomposition as in the proof of Lemma \ref{Lm: wtheta},
we have
$$
    | \Gamma_{\theta} (f) (x) | \leq \int_{\{|y| \leq R/2\}}
  \frac{1}{| x - y |^{n - \theta}}   |f(y)|\,dy + \int_{\{|y|
 > R/2\}} \frac{1}{| x - y |^{n - \theta}}   |f(y)|\,dy,
$$
where the two integrals are denoted by $I_1(x)$ and $I_2(x)$ respectively.
First we can get
\begin{equation} \label{I1pt}
    I_1(x) \leq \frac{2^{n - \theta}}{| x |^{n - \theta}}   \| f \|, \quad
\text{for }  | x | > R.
\end{equation}
Since $\theta = n(\frac{1}{p} - \frac{1}{q})$ implies 
$n - \theta - \frac{n}{q} = n(1 - \frac{1}{p}) > 0$, it follows that
\begin{equation} \label{Ism}
    \| I_1^R (\cdot) \|_q \leq 2^{n - \theta} \| f \|
\Big( \int_{\{|x| > R\}} \frac{1}{| x |^{q(n - \theta)}} \,dx \Big)^{1/q}
 = \frac{M_1 (n, \theta, q)}{R^{n - \theta - \frac{n}{q}}}  \| f \|,
\end{equation}
where
$$
M_1 (n, \theta, q) = 2^{n - \theta}
\Big( \frac{\Omega_n}{q(n - \theta) - n} \Big)^{1/q},
$$
in which $\Omega_n$ is the surface area of the unit sphere in
$\mathbb{R}^n$.

By the Hardy-Littlewood-Sobolev inequality \eqref{HLS}, we readily have
\begin{equation} \label{Ilg}
    \| I_2^R (\cdot) \|_q \leq M_2(n, \theta, q) \| f^{R/2} \|_p,
\end{equation}
with $M_2 (n, \theta, q) = S_{n, p, q}$ in \eqref{HLS}.
Combining \eqref{Ism} and \eqref{Ilg}, we obtain \eqref{gest}.
\end{proof}

\begin{lemma} \label{Lm: Ginfest}
For any given $\theta \in (0, n)$, there exist positive constants
$M_{1,\infty} (n, \theta)$ and $M_{2,\infty} (n, \theta)$ such that
for any given $R > 1$, it holds that
\begin{equation} \label{gestinf}
    \| \Gamma_\theta (f)^R (\cdot) \|_{\infty}
\leq \frac{M_{1,\infty}}{R^{n - \theta}}   \| f \| + M_{2,\infty}
 \| f^{R/2} \|^{\frac{\theta}{n}}
\| f^{R/2} \|_{\infty}^{1 - \frac{\theta}{n}}, \quad
 f \in L^1(\mathbb{R}^n) \cap L^{\infty} (\mathbb{R}^n).
\end{equation}
\end{lemma}

\begin{proof}
Using the same notation as in the proof of Lemma \ref{Lm: Gpest},
by \eqref{I1pt} we can get
$$
    \| I_1^R(\cdot) \|_{\infty} \leq \frac{M_{1,\infty}}{R^{n - \theta}}
  \| f \|,
$$
with $M_{1,\infty} = 2^{n - \theta}$. For $I_2(x), | x | > R$, we get
\begin{align*}
    I_2 (x)
& = \int_{\{|y| > R/2\}} \frac{1}{| x - y |^{n - \theta}}   |f(y)|\,dy \\
 & = \int_{\{|y| > R/2\} \cap \{|x-y| > W\}} \frac{1}{| x - y |^{n - \theta}}
  |f(y)|\,dy \\
&\quad + \int_{\{|y| > R/2\} \cap \{|x-y| \leq W\}}
  \frac{1}{| x - y |^{n - \theta}} |f(y)|\,dy \\
& \leq \frac{1}{W^{n - \theta}}   \| f^{R/2} \|
  + \int_{\{|z| \leq W\}} \frac{1}{|z|^{n - \theta}}\,dz   \| f^{R/2}
 \|_{\infty}\\
& = \frac{1}{W^{n - \theta}}   \| f^{R/2} \|
 + \frac{\Omega_n}{\theta}   W^{\theta}   \| f^{R/2} \|_{\infty}.
\end{align*}
In the above inequality we choose constant
$$
    W = \Big( \frac{\| f^{R/2} \|}{ \| f^{R/2} \|_{\infty}} \Big)^{1/n}
$$
to obtain
$$
    \| I_2^R(\cdot) \|_{\infty} \leq M_{2,\infty}
\| f^{R/2} \|^{\frac{\theta}{n}}
\| f^{R/2} \|_{\infty}^{1 - \frac{\theta}{n}},
$$
where $M_{2, \infty} = 1 + \Omega_n / \theta$.
Thus the estimate \eqref{gestinf} is proved.
\end{proof}

By the Biot-Savart law \eqref{BSL}, Lemma \ref{Lm: Gpest} and
Lemma \ref{Lm: Ginfest} directly lead to the exterior decay estimates
of the velocity field $u(t, x)$, as stated in the following lemma.

\begin{lemma} \label{Lm: uspdec}
Let $1 < p < q < \infty$ satisfy $\frac{1}{2} = \frac{1}{p} -
\frac{1}{q}$. Let $\omega(t, x)$ and $u(t, x)$ be the vorticity and
velocity solution of the vorticity equation
\eqref{Veqn}--\eqref{BSL} with $\omega_0 \in L^1 (\mathbb{R}^2)$.
Then there exist positive constants $L_1 (p), L_2(p),
L_{1,\infty}$, and $L_{2, \infty}$ such that for any given $R >
1$, the following estimates hold,
\begin{equation} \label{upest}
    \| u^R (t, \cdot) \|_q \leq \frac{L_1 (p)}{R^{2(\frac{1}{2}
 - \frac{1}{q})}} \| \omega (t, \cdot) \| + L_2 (p) \|
\omega^{R/2} (t, \cdot) \|_p,
\end{equation}
and
\begin{equation} \label{uinfest}
    \| u^R (t, \cdot) \|_{\infty} \leq \frac{L_{1, \infty}}{R}
\| \omega (t, \cdot) \| + L_{2, \infty} \| \omega^{R/2} (t, \cdot) \|^{1/2} \|
 \omega^{R/2} (t, \cdot) \|_{\infty}^{1/2}.
\end{equation}
\end{lemma}

\begin{proof}
Simply let $n = 2, \theta = 1$ in Lemma \ref{Lm: Gpest} and
Lemma \ref{Lm: Ginfest} and add the coefficient $1/(2\pi)$ to the
Riesz potential \eqref{Rp}. Then the conclusion follows.
\end{proof}

\section{Exterior Decay Estimates of Spatial Derivatives}

The vorticity equation \eqref{Veqn} is a semilinear evolutionary equation.
By the semigroup theory, the vorticity solution $\omega(t, x)$ satisfies the
 variation-of-constant formula for all mild solutions \cite[Chapter 3]{SY02},
 which is the following nonlinear integral equation
\begin{equation} \label{voc}
    \omega(t, \cdot) = e^{At} \omega_0 - \int_0^t  e^{A(t - s)} 
    (u(s,\cdot) \cdot \nabla) \omega(s, \cdot) ds, \quad t \geq 0,
\end{equation}
where $e^{At}, t \geq 0$, is the $C_0$-semigroup on the Banach
space $Y = C_B(\mathbb{R}^2)$, generated by the closed linear
operator $A = \nu \Delta : \mathcal{D} (A) = C_B^2(\mathbb{R}^2) \to
C_B (\mathbb{R}^2)$ and given by
\begin{equation} \label{sg}
    (e^{At}g) (x) = \frac{1}{4\pi \nu t} \int_{\mathbb{R}^2}
\exp \Big( -   \frac{| x - y |^2}{4\nu t} \Big) g(y) dy, \quad
t \geq 0, \; g \in Y.
\end{equation}
Here $C_B^k (\mathbb{R}^2)$ stands for the Banach space of
functions whose derivatives up to the $k$-th order are all
continuous and bounded, with $C_B^0(\mathbb{R}^2)$ denoted by
$C_B(\mathbb{R}^2)$.

In this section we shall study the spatial derivatives of the linear
 term and the nonlinear integrand in \eqref{voc}. We have
\begin{equation} \label{dsg}
    \partial_x^{\beta} (e^{At}g)(x) = \frac{1}{4 \pi \nu t}
\int_{\mathbb{R}^2} \partial_x^{\beta} \exp
\Big( -   \frac{| x - y |^2}{4\nu t} \Big) g(y) dy.
\end{equation}
The following two regularity estimates for vorticity and velocity solutions
$\omega(t, x)$ and $u(t, x)$ of \eqref{Veqn}--\eqref{BSL} have been
shown in \cite{GS03, tK94},
\begin{gather} \label{Nw}
    \| \partial_t^{\alpha} \partial_x^{\beta} \omega (t, \cdot) \|_p
\leq N_{\omega}(\alpha, |\beta|, p)   t^{- (\alpha + \frac{|\beta|}{2}
 + 1 - \frac{1}{p})} \| \omega_0 \|, \quad \text{for }  p \in [1, \infty],
\\
 \label{Nu}
    \| \partial_t^{\alpha} \partial_x^{\beta} u(t, \cdot) \|_q \leq N_{u}(\alpha.
     |\beta|, q)
 t^{- (\alpha + \frac{|\beta|}{2} + \frac{1}{2} - \frac{1}{q})} \| \omega_0 \|,
\quad \text{for } q \in (2, \infty],
\end{gather}
where $N_{\omega}(\alpha, |\beta|, p)$ and $N_u (\alpha, |\beta|, p)$ are positive
constants depending on $\alpha, |\beta|, p$, and $\| \omega_0 \|$.
Now we prove a key lemma, which shows the exterior decay estimates of
these spatial derivatives in \eqref{dsg}.

\begin{lemma} \label{Lm: sgspd}
For $p \in [1, \infty]$ and any index $\beta = (\beta_1, \beta_2)$ with
$| \beta | = k \geq 0$, there exist positive constants $L_3(k, p)$
and $L_4(k)$ such that for any given $R > 1$,
\begin{equation} \label{spddec}
    \| \partial_x^{\beta}(e^{At}g)^R \|_p \leq L_3(k, p) R^{- (3 - \frac{2}{p} )}
 t^{- \frac{1}{2} (k - 1)} \| g \| + L_4(k)   t^{- \frac{k}{2} }
 \| g^{R/2} \|_p,
\end{equation}
for any $g \in L^1(\mathbb{R}^2) \cap L^{\infty} (\mathbb{R}^2)$.
\end{lemma}

\begin{proof}
Similar to \eqref{phimax}, for any given $\ell > 0$ and a constant
$\zeta > 0$, the function
$$
    \psi (t) = \frac{1}{t^{\ell}} \exp \big(-   \frac{\zeta^2}{t} \big)
$$
has the maximum
\begin{equation} \label{psimax}
     \max_{t > 0} \psi (t) = \psi (\zeta^2) = \frac{e^{-1}}{\zeta^{2\ell}}.
\end{equation}
By \eqref{dsg} and direct differentiation, we have
\begin{equation} \label{sdsg}
    \begin{split}
    | \partial_x^{\beta} ( e^{At}g)^R (x) |
 & \leq \frac{1}{4 \pi \nu t} \sum_{\beta_1 = 0}^{k}
 \int_{\mathbb{R}^2} \Big| \partial_{x_1}^{\beta_1} \partial_{x_2}^{k - \beta_1}
\exp \Big( -   \frac{|x - y |^2}{4\nu t} \Big) \Big| |g(y)|\,dy \\
    & = \frac{1}{4\pi \nu t} \sum_{j = [\frac{k + 1}{2}]}^{k} C_j (k)
 \int_{\mathbb{R}^2} \frac{|x - y|^i}{t ^j} \exp
 \Big( -  \frac{|x - y|^2}{4\nu t} \Big) |g(y)| \,dy \\
    & = \frac{1}{4\pi \nu t} \sum_{j
 = [\frac{k + 1}{2}]}^{k} C_j (k) \int_{\mathbb{R}^2}
 \frac{|x - y|^{2j - k}}{t ^j} \exp \Big( -  \frac{|x - y|^2}{4\nu t} \Big)
  |g(y)|\,dy,
    \end{split}
\end{equation}
in which we have $0 \leq i \leq j$ and $j + (j - i) = k$ so that
$0 \leq i = 2j - k$, and $C_j (k)$ is a positive constant depending on $j$
and $k = | \beta |$. For each term
$I_j (t, x), j = [\frac{k + 1}{2}], \cdots, k$, in \eqref{sdsg},
a decomposition yields
\begin{equation} \label{decomp}
    I_j (t, x) = \frac{C_j (k)}{4\pi \nu t} \int_{\mathbb{R}^2}
 \frac{|x - y|^{2j - k}}{t ^j} \exp
\big( -  \frac{|x - y|^2}{4\nu t} \big) |g(y)|  dy = I_{j,1}(t, x)
+ I_{j,2} (t, x).
\end{equation}
The first integral in \eqref{decomp} satisfies the estimate
\begin{equation*}
    \begin{split}
    I_{j,1} (t, x) & = \frac{C_j (k)}{4\pi \nu t} \int_{\{| y | \leq R/2\}}
\frac{|x - y|^{2j - k}}{t ^{j}} \exp \big( -  \frac{|x - y|^2}{4\nu t} \big)
|g(y)| dy \\
    & = \frac{C_j (k)}{4\pi \nu t^{(k - 1)/2}} \int_{\{| y | \leq R/2\}}
\frac{|x - y|^{2j - k}}{t^{\frac{3}{2} + j - \frac{k}{2}}}
\exp \big( -  \frac{|x - y|^2}{4\nu t} \big) |g(y)| dy \\
    &\leq \frac{C_j (k) e^{-1}}{4\pi \nu t^{(k - 1)/2}}
\int_{\{| y | \leq R/2\}} \frac{|x - y|^{2j - k}(4\nu)^{\frac{3}{2} + j 
- \frac{k}{2}}}{|x - y|^{2(\frac{3}{2} + j - \frac{k}{2})}}  |g(y)| dy \\
    & \leq \frac{\widetilde{C}(j, k)}{| x |^3   t^{(k - 1)/2}}
\int_{\{| y | \leq R/2\}} |g(y)| dy \\
&\leq \frac{\widetilde{C}(j, k)}{| x |^3   t^{(k - 1)/2}} \| g \|, \quad
\text{for}  \;  | x | > R,
    \end{split}
\end{equation*}
where we used \eqref{psimax} and the fact that $| x | > R > R/2 \geq | y |$,
and the constant $\widetilde{C}(j, k)$ is given by
$$
    \widetilde{C}(j, k) = 8C_j (k) (\pi e)^{-1} (4\nu)^{\frac{1}{2}
(1 - k) + j}.
$$
Then it follows that
\begin{equation} \label{term1}
    \begin{split}
    \| & I_{j, 1}^R (t, \cdot) \|_p  \leq
 \frac{\widetilde{C}(j, k)}{t^{(k - 1)/2}}
\Big( \int_{\{| x | > R\}} \frac{1}{| x |^{3p}} \,dx \Big)^{1/p} \| g \| \\
    & = \widetilde{C}(j, k) \big(\frac{2\pi}{3p -2} \big)^{1/p}
R^{- (3 - \frac{2}{p})} t^{- \frac{1}{2} (k - 1)} \| g \| \\
    &= M_3 (j, k, p) R^{- (3 - \frac{2}{p})}  t^{- \frac{1}{2} (k - 1)}
 \| g \| ,
    \end{split}
\end{equation}
where $M_3 (j, k, p) = \widetilde{C}(j, k) (2\pi / (3p -2) )^{1/p}$.
The second integral in \eqref{decomp} is given by
$$
     I_{j, 2}^R (t, x) =  \frac{C_j (k)}{4\pi \nu t} \int_{\{| y | > R/2\}}
\frac{|x - y|^{2j - k}}{t ^{j}} \exp \Big(-  \frac{|x - y|^2}{4\nu t} \Big)
|g(y)|\,dy,  \; | x | > R.
$$
By the convolution property \eqref{convln} we can assert that
\begin{equation} \label{term2}
      \| I_{j, 2}^R (t, \cdot) \|_p \leq \frac{C_j (k) (4\nu)^{\frac{1}{2}
(2j - k)}}{\pi t ^{j - \frac{1}{2} (2j - k)}}
\Big( \int_{\mathbb{R}^2} | z |^{2j - k} e^{- | z |^2} dz \Big)
\| g^{R/2} \|_p = \frac{M_4 (j, k)}{t^{k/2}}   \| g^{R/2} \|_p,
\end{equation}
where, as we remarked after \eqref{sdsg}, the exponents $2j - k \geq 0$ for
$j = [\frac{k + 1}{2}], \cdots, k$, and the constant $M_4 (j, k)$ is given
 by
$$
    M_4(j, k) = C_j (k) \pi^{-1} (4\nu)^{j - \frac{k}{2}}
\Big( \int_{\mathbb{R}^2} | z |^{2j - k} e^{- | z |^2} dz \Big).
$$
Substituting \eqref{term1} and \eqref{term2} into \eqref{decomp} and
\eqref{sdsg}, we obtain
\begin{align*}
 \| \partial_x^{\beta} (e^{At}g)^R \|_p
& \leq \sum_{j = [\frac{k + 1}{2}]}^{k} \left( \| I_{j, 1}^R (t, \cdot) \|_p
+ \| I_{j, 2}^R (t, \cdot) \|_p \right) \\
    & \leq \sum_{j =[\frac{k + 1}{2}]}^{k}
\left( M_3 (j, k, p)  R^{- (3 - \frac{2}{p})}   t^{- \frac{1}{2} (k - 1)}
 \| g \| + M_4 (j, k)  t^{- \frac{k}{2}}   \| g^{R/2} \|_p \right) \\
    & =  L_3(k, p) R^{- (3 - \frac{2}{p} )}   t^{- \frac{1}{2} (k - 1)}
 \| g \| + L_4(k)   t^{- \frac{k}{2} } \| g^{R/2} \|_p,
\end{align*}
with $L_3 (k, p) = \sum_{j=[\frac{k + 1}{2}]}^{k} M_3 (j, k, p)$ and
$L_4 (k) = \sum_{j=[\frac{k + 1}{2}]}^{k} M_4 (j, k)$.
Thus \eqref{spddec} is proved for $p \in [1, \infty)$.
The proof of \eqref{spddec} for $p = \infty$ is similar and even easier.
\end{proof}

Next is another key lemma showing the regularity estimates of the spatial
derivatives of the nonlinear integrand function in \eqref{voc}, which
involves the nonlinear term $v(t, x)$ in the vorticity equation \eqref{Veqn}.
Here
\begin{equation} \label{vtxdef}
    v(t, x) = (u(t, x) \cdot \nabla) \omega(t, x).
\end{equation}

\begin{lemma} \label{Lm: nldec}
Let $\omega(t, x)$ and $u(t, x)$ be a solution of the vorticity
equations \eqref{Veqn}--\eqref{BSL} with initial data $\omega_0 \in
L^1 (\mathbb{R}^2)$. Let $p \in [1, \infty]$. Then for any index
$\beta \in Z^{+} \times Z^{+}$ and any given $R > 1$, the following
estimate is satisfied,
\begin{equation} \label{vtx}
    \begin{split}
    & \| \partial_x^{\beta}(e^{A(t - s)} v (s, \cdot))^R \|_p \\
    & \leq L_5 \Big(\frac{1}{R^{3 - \frac{2}{p}}   (t - s)^{\frac{1}{2}(|\beta|
-1)}   s^{2 - \frac{1}{p}}} + \frac{1}{R   (t - s)^{\frac{|\beta|}{2}}
 s^{\frac{3}{2} - \frac{1}{p}}}
+ \frac{ \| \omega_0^{R/8} \|}{(t - s)^{\frac{| \beta |}{2} }
s^{2 - \frac{1}{p}}} \Big),
    \end{split}
\end{equation}
where $ 0 < s < t$, and $L_5 = L_5 (| \beta|, p, \| \omega_0 \|)$ is a
positive constant.
\end{lemma}

\begin{proof}
Let $| \beta | = k \geq 0$. By Lemma \ref{Lm: sgspd}, for $t > 0$, we have
\begin{equation} \label{nonlterm}
    \begin{split}
&\| \partial_x^{\beta} (e^{A(t - s)} v(s, \cdot) )^R \|_p\\
&= \| \partial_x^{\beta} (e^{A(t - s)} (u(s, \cdot) \cdot \nabla) \omega (s, \cdot) )^R 
\|_p \\
& \leq L_3 (k, p)R^{- (3 - \frac{2}{p})} (t - s)^{- \frac{1}{2}(k - 1)} \| v(s,
 \cdot) \| + L_4 (k) (t - s)^{- \frac{k}{2}} \| v^{R/2}(s, \cdot) \|_p   ,
    \end{split}
\end{equation}
where by \eqref{Nw} and \eqref{Nu} we have
\begin{equation} \label{v1}
    \begin{split}
    \| v(s, \cdot) \| &= \int_{\mathbb{R}^2} | (u(s, x) \cdot \nabla )\omega(s, x) 
    | \,dx \\
    & \leq \| u_1(s, \cdot) \|_{\infty} \| \partial_{x_1} \omega (s, \cdot) \| 
    + \| u_2(s, \cdot) \|_{\infty} \| \partial_{x_2} \omega (s, \cdot) \| \\
    & \leq \left( N_u (0, 0, \infty) s^{- \frac{1}{2}} \| \omega_0 \|\right)
\left( 2N_{\omega}(0, 1, 1) s^{- (\frac{3}{2} - \frac{1}{p} )} \| \omega_0 \| \right)\\
& = N_{v,1} s^{-(2 - \frac{1}{p})},
    \end{split}
\end{equation}
in which $N_{v,1} = 2N_u (0, 0, \infty) N_{\omega}(0, 1, 1) \| \omega_0 \|^2$, and
\begin{equation} \label{v2}
    \begin{split}
& \|  v^{R/2}(s, \cdot) \|_p\\
&= \Big( \int_{\{| x | > R/2 \}}  | (u(s, x)\cdot \nabla)\omega(s, x)
 |^p \,dx \Big)^{1/p} \\
    & \leq \| u^{R/2} (s, \cdot) \|_{\infty} \left( \| \partial_{x_1}
\omega^{R/2}(s, \cdot) \|_p + \| \partial_{x_2}   \omega^{R/2}(s, \cdot) \|_p \right)
\\
& \leq 2N_{\omega} (0, 1, p) s^{- (\frac{3}{2} - \frac{1}{p} )} \| \omega_0 \|
\| u^{R/2}(s, \cdot) \|_{\infty} = N^{*} s^{-(\frac{3}{2} - \frac{1}{p})}
\| u^{R/2} (s,\cdot) \|_{\infty}   ,
    \end{split}
\end{equation}
in which $N^{*} = 2N_{\omega} (0, 1, p) \| \omega_0 \|$. 
Note that by Lemma \ref{Lm: uspdec} and \eqref{uinfest}, one has
\begin{equation} \label{ulmest}
    \| u^R (s, \cdot) \|_{\infty} \leq \frac{L_{1,\infty}}{R} \|\omega (s, \cdot)
 \| + L_{2,\infty} \| \omega^{R/2} (s, \cdot) \|^{1/2} \|
\omega^{R/2} (s, \cdot) \|_{\infty}^{1/2}   .
\end{equation}
Substituting \eqref{ulmest} with $R$ replaced by $R/2$ into \eqref{v2},
by Lemma \ref{Lm: wtheta} and \eqref{Nw}, we obtain
\begin{equation} \label{v3}
    \begin{split}
    &\| v^{R/2} (s, \cdot) \|_p \\
& \leq N^{*} s^{- (\frac{3}{2} - \frac{1}{p} )}
 \Big\{ \frac{2L_{1,\infty}}{R} \|\omega (s, \cdot) \|
 + L_{2,\infty} \| \omega^{R/4} (s, \cdot) \|^{1/2} \| \omega^{R/4} (s, \cdot)
 \|_{\infty}^{1/2}\Big\} \\
& \leq N^{*}  s^{- (\frac{3}{2} - \frac{1}{p} )}
 \Big\{ \frac{2L_{1,\infty}}{R} N_{\omega} (0, 0, 1) \| \omega_0 \|
 + L_{2,\infty} \Big[ \frac{4L_0(1)}{R} s ^{1/2} \| \omega_0 \|
 + \| \omega_0^{R/8} \| \Big]^{1/2}   \\
& \quad \times \Big[ \frac{4L_0 (\infty)}{R} s^{- 1/2} \| \omega_0 \|
 + s^{-1} \| \omega_0^{R/8} \| \Big]^{1/2} \Big\} \\
& \leq \frac{N^{*}}{s^{\frac{3}{2} - \frac{1}{p}}}
 \Big\{ 2L_{1,\infty} N_{\omega} (0, 0, 1)\frac{\|\omega_0 \|}{R} + L_{2,\infty}
  (4L_0(1)\|\omega_0\| + 1)^{1/2}  \\
& \quad \times (4L_0(\infty)\|\omega_0\| + 1)^{1/2} \left(R^{-1} s^{1/2}
  + \| \omega_0^{R/8} \| \right)^{1/2} \left(R^{-1} s^{-1/2}
  + s^{-1}\|\omega_0^{R/8} \| \right)^{1/2} \Big\} \\
& \leq \frac{N_{v,2}}{s^{\frac{3}{2} - \frac{1}{p}}}
 \Big(\frac{1}{R} + \frac{\|\omega_0^{R/8}\|}{s^{\frac{1}{2}}} \Big)
\end{split}
\end{equation}
where $N_{v,2}$ is a constant given by
\begin{align*}
    N_{v,2} &= N^{*}\big[ 2L_{1,\infty} N_{\omega} (0, 0, 1) \|\omega_0 \|\\
  &\quad + L_{2,\infty} (4L_0(1)\|\omega_0\| + 1)^{1/2} (4L_0(\infty)\|\omega_0\|
  + 1)^{1/2} \big].
\end{align*}
Finally we substitute \eqref{v1} and \eqref{v3} into \eqref{nonlterm}
to obtain
\begin{equation*}
    \begin{split}
&\| [\partial_x^{\beta} (e^{A(t - s)} (u(s, \cdot) \cdot \nabla) \omega (s, \cdot) ) ]^R 
 \|_p \\
& \leq \frac{L_3(k, p)   N_{v,1}}{R^{3 - \frac{2}{p}}
 (t - s)^{\frac{1}{2}(k - 1)}   s^{2 - \frac{1}{p}}}
 +   \frac{L_4(k)  N_{v,2}}{(t - s)^{\frac{k}{2}}   s^{\frac{3}{2}
 - \frac{1}{p}}} \Big(\frac{1}{R} + \frac{\|\omega_0^{R/8} \|}{s^{\frac{1}{2}}}
   \Big) \\
& \leq L_5 \Big(\frac{1}{R^{3 - \frac{2}{p}}   (t - s)^{\frac{1}{2}(k - 1)}
  s^{2 - \frac{1}{p}}} + \frac{1}{R   (t - s)^{\frac{k}{2}}
  s^{\frac{3}{2} - \frac{1}{p}}} + \frac{\| \omega_0^{R/8} \|}{(t - s)
  ^{\frac{k}{2}}   s^{2 - \frac{1}{p}}} \Big),
    \end{split}
\end{equation*}
where the constant $L_5 = L_5 (k, p, \|\omega_0 \|) = \max   \{L_3(k,
p) N_{v,1}, L_4(k) N_{v,2} \}$ and $k = | \beta |$. Therefore,
\eqref{vtx} is proved.
\end{proof}

\section{Exterior Decay Estimates of Temporal-Spatial Derivatives}

In this section we establish the exterior decay estimates of the
temporal-spatial derivatives of the vorticity solution
$\omega (t, x)$ of \eqref{Veqn}--\eqref{BSL}. These estimates of $L^p$-norms
of the derivatives are dominated by the combination of the spatial
decay rates $1/R$ and $\|\omega_0^{R/16}\|$ and by the temporal decay
rate $t ^{-(\alpha +\frac{1}{2}(|\beta| - 1))}$, as described in the
following two theorems. First we deal with the spatial
derivatives.

\begin{theorem} \label{Thm: wspdc}
Let $p \in [1, \infty]$. For any given $R > 1$ and any $\beta \in
Z^{+} \times Z^{+}$, the spatial derivative of the vorticity
solution $\omega (t, x)$ of \eqref{Veqn}--\eqref{BSL} with any
initial data  $\omega_0 \in L^1 (\mathbb{R}^2)$ has the following
decay property,
\begin{equation} \label{wspdc}
    \| \partial_{x}^{\beta} \omega^R (t, \cdot) \|_p
\leq \frac{B (| \beta |, p, \| \omega_0 \|)}{ t   ^{\frac{1}{2} ( | \beta | -1)}}
\Big( \frac{1}{R}  + \frac{1}{R   t ^{1 - \frac{1}{p}}}
 + \frac{\| \omega_0^{R/16} \|}{t   ^{\frac{3}{2} - \frac{1}{p}}} \Big),
\end{equation}
where $B(| \beta |, p, \| \omega_0 \|)$ is a positive constant depending on
$| \beta |$, $p$, and $\| \omega_0 \|$.
\end{theorem}

\begin{proof}
We have shown in Lemma \ref{Lm: wtheta} that \eqref{wspdc} holds for
$|\beta| = 0$. Now we prove that \eqref{wspdc} is valid for spatial
derivatives of $\omega(t, x)$ of any order $\beta$ by induction.

The variation-of-constant formula \eqref{voc} implies that
\begin{equation} \label{voc1}
    \partial_x^{\beta} \omega (t, \cdot) = \partial_x^{\beta} \left(e^{At/2} 
    \omega(t/2, \cdot)\right) - \int_{t/2}^t \partial_x^{\beta} \left(e^{A(t - s)} 
    (u(s, \cdot) \cdot \nabla) \omega (s, \cdot)\right) ds, \;\; t > 0.
\end{equation}

\textbf{Step 1.}
 Let $|\beta| = k \geq 1$. By Lemma \ref{Lm: sgspd} and \eqref{spddec},
 the linear part in \eqref{voc1} satisfies
\begin{equation} \label{spd1}
    \begin{split}
&\| \partial_{x}^{\beta} (e^{At/2} \omega(t/2, \cdot))^R \|_p \\
& \leq L_3 (k, p) R^{-(3 - \frac{2}{p})} t^{-\frac{1}{2} (k - 1)}
  \| \omega(t/2, \cdot) \|
  + L_4 (k) t^{-\frac{k}{2}} \| \omega (t/2, \cdot)^{R/2} \|_p, \quad
\text{for } p \in [1, \infty].
\end{split}
\end{equation}
Then using Lemma \ref{Lm: wtheta} and the estimates \eqref{Nw}
and \eqref{Nu}, we get from \eqref{spd1} that
\begin{equation} \label{linterm}
    \begin{split}
&\| \partial_{x}^{\beta} (e^{At/2} \omega(t/2, \cdot))^R \|_p \\
&\leq 2L_3 (k, p) N_{\omega}(0, 0, 1) R^{-(3 - \frac{2}{p})}
t^{- [\frac{1}{2} (k - 1) + (1 - \frac{1}{p}) ]} \| \omega_0 \| \\
&\quad + 2^{1 - \frac{1}{p}} L_4 (k)  t^{-\frac{k}{2}} L_0 (p, \|\omega_0\|)
\Big( \frac{1}{R} t^{- (\frac{1}{2} - \frac{1}{p})} \| \omega_0 \|
  + t^{- (1 - \frac{1}{p} )} \| \omega_0^{R/2} \| \Big) \\
& \leq \frac{L_6 (k, p, \| \omega_0 \|)}{t^{\frac{1}{2} (k - 1)}}
\Big( \frac{\| \omega_0 \|}{R   t^{1 - \frac{1}{p}}}
  + \frac{\| \omega_0^{R/2} \|}{t^{\frac{3}{2} - \frac{1}{p}}} \Big),
\end{split}
\end{equation}
where we used $R^{- 2(1 - 1/p)} \leq 1$ due to $R > 1$, and the
constant $L_6$ is given by
$$
    L_6 (k, p, \| \omega_0 \|) = 2 (L_3 (k, p) N_{\omega}(0, 0, 1)
  +  2^{1 - \frac{1}{p}} L_4 (k) L_0 (p, \|\omega_0\|)).
$$
Note that the estimate \eqref{linterm} for the linear part is valid for
all $\beta \in Z^{+} \times Z^{+}$.

The next two steps are devoted to the nonlinear integral term
in \eqref{voc1}.

\textbf{Step 2.} Since the nonlinear term in \eqref{Veqn}, which is
denoted by $v(t, x)$ in \eqref{vtxdef}, involves $\nabla \omega(t, x)$,
first we have to show that \eqref{wspdc} is valid for the gradient
$\nabla \omega(t, x)$ with $| \beta | = 1$ in this step. Indeed we show that
\begin{equation} \label{nbl}
    \| \nabla \omega^R (t, \cdot) \|_p \leq B_0 (p, \| \omega_0 \|)
\Big(\frac{1}{R   t^{1 - \frac{1}{p}}} + \frac{\|
\omega_0^{R/8} \|}{t^{\frac{3}{2} - \frac{1}{p}}} \Big),
\end{equation}
where $B_0 (p, \| \omega_0 \|)$ is a constant. According to \eqref{voc1},
$\nabla \omega(t, x)$ consists of the linear term and the nonlinear integral
term.

For the linear term, we apply the estimate \eqref{linterm} established in
Step 1 with $k = 1$ to get
\begin{equation} \label{lint}
    \| \nabla (e^{At/2} \omega(t/2, \cdot))^R \|_p \leq L_6 (1, p, \| \omega_0 \|)
 \Big( \frac{\| \omega_0 \|}{R   t^{1 - \frac{1}{p}}} + \frac{\| \omega_0^{R/2} \|}
{t^{\frac{3}{2} - \frac{1}{p}}} \Big).
\end{equation}
For the nonlinear term in \eqref{voc1} for the gradient $\nabla \omega (t, x)$, 
by the Jensen inequality and the convexity of the $L^p$ norm, we can apply 
Lemma \ref{Lm: nldec} to assert that for $k = 1$,
\begin{equation} \label{nonlint}
    \begin{split}
&\| \int_{t/2}^{t}   \nabla (e^{A(t - s)} (u(s, \cdot) \cdot \nabla)
 \omega (s, \cdot) )^R \,ds \|_p \\
& \leq L_5 (1, p, \| \omega_0 \|) \int_{t/2}^{t}
  \Big(\frac{1}{R^{3 - \frac{2}{p}}   s^{2 - \frac{1}{p}}}
  +  \frac{1}{R   (t - s)^{\frac{1}{2}}
  s^{\frac{3}{2} - \frac{1}{p}}} + \frac{\| \omega_0^{R/8} \|}{(t - s)
  ^{\frac{1}{2}}   s^{2 - \frac{1}{p}}} \Big) ds \\
& \leq L_7 (p, \| \omega_0 \|) \Big(\frac{1}{R   t^{1 - \frac{1}{p}}}
 + \frac{\| \omega_0^{R/8} \|}{t^{\frac{3}{2} - \frac{1}{p}}} \Big),
    \end{split}
\end{equation}
where
$$
    L_7 (p, \| \omega_0 \|) = 2^{3 - \frac{1}{p}} L_5 (1, p, \| \omega_0 \|).
$$
Collecting \eqref{lint} and \eqref{nonlint}, due to that
$\| \omega_0^{R/2} \| \leq \| \omega_0^{R/8} \|$, we obtain \eqref{nbl}
with the constant $B_0$ given by
$$
    B_0 (p, \| \omega_0 \|) = L_6(1, p, \| \omega_0 \|) \| \omega_0
\| + L_7 (p, \| \omega_0 \|).
$$
Therefore, \eqref{wspdc} is valid for $| \beta | = 1$, i.e. for the
gradient $\nabla \omega(t, x)$.

\textbf{Step 3.}
 Now we prove that \eqref{wspdc} is valid for spatial derivatives
of $\omega(t, x)$ of any order by induction. Assuming that \eqref{wspdc}
holds for all the spatial derivatives of $\omega(t, x)$ up to the order
$k$, and let $\beta \in Z^{+} \times Z^{+}$ has $| \beta | = k$.
We want to show that \eqref{wspdc} remains valid for
$\nabla \partial_x^{\beta} \omega(t, x)$.

By \eqref{voc1} and variable changes in the integral part, we can write
\begin{equation} \label{voc2}
    \nabla (\partial_x^{\beta} \omega^R (t, \cdot))
= \nabla [\partial_x^{\beta} (e^{At/2} \omega(t/2, \cdot))^R] - \int_{t/2}^t
 \nabla [e^{A(t - s)} \partial_x^{\beta}(u(s, \cdot) \cdot \nabla)
 \omega (s, \cdot))]^R \,ds,
\end{equation}
where $t > 0$. By \eqref{linterm} shown in Step 1, for the linear term
in \eqref{voc2} we have readily
\begin{equation} \label{linpt4}
    \| \nabla [\partial_x^{\beta} (e^{At/2} \omega(t/2, \cdot))^R] \|_p
 \leq \frac{L_6(k, p, \| \omega_0 \|)(\| \omega_0 \| + 1)}{t^{k/2}}
 \Big( \frac{1}{R   t^{1 - \frac{1}{p}}}
 + \frac{\| \omega_0^{R/2} \|}{t^{\frac{3}{2} - \frac{1}{p}}}\Big), \quad
 t > 0.
\end{equation}
Below we focus on estimating the nonlinear integral term in \eqref{voc2}.

Using the notation \eqref{vtxdef} and applying Lemma \ref{Lm: sgspd},
the $L^p$ norm of the integrand function in \eqref{voc2} admits the
estimate,
\begin{equation} \label{integrand}
    \begin{split}
&\| \nabla [e^{A(t - s)} \partial_x^{\beta}(u(s, \cdot) \cdot \nabla)
\omega (s, \cdot))]^R \|_p \\
&= \| \nabla (e^{A(t - s)} \partial_x^{\beta} v(s, \cdot))^R \|_p \\
 & \leq \frac{L_3 (k, p)}{R^{3 - \frac{2}{p}}} \| \partial_x^{\beta} v(s, \cdot) \| 
 + \frac{L_4 (k)}{(t - s)^{1/2}} \| \partial_x^{\beta} v^{R/2} (s, \cdot) \|_p.
    \end{split}
\end{equation}
By Leibniz formula in product differentiation and using \eqref{Nw} with 
$p = 1$ and \eqref{Nu} with $q = \infty$, we have
\begin{equation} \label{betav}
    \begin{split}
    \| \partial_x^{\beta} v(s, \cdot) \|
&= \| \sum_{\gamma \leq \beta}   \binom{\beta}{\gamma} \partial_x^{\gamma}
   u(s, \cdot) \cdot \partial_x^{\beta - \gamma} (\nabla \omega(s, \cdot)) \| \\
& \leq 2^{| \beta |} \sum_{\gamma \leq \beta}  \| \partial_x^{\gamma} u(s, \cdot)
  \|_{\infty}   \| \partial_x^{\beta - \gamma} \nabla \omega(s, \cdot) \| \\
 & \leq  2^{| \beta|}\sum_{\gamma \leq \beta}   \frac{N_u (0, | \gamma|, \infty)
  \|\omega_0\|}{s^{\frac{|\gamma|}{2} + \frac{1}{2}}} \cdot \frac{N_{\omega}
 (0, | \beta - \gamma|, 1) \|\omega_0\|}{s^{\frac{|\beta - \gamma|}{2}
  + \frac{1}{2}}} \\
&= \frac{L_8 (|\beta|, \|\omega_0 \|)}{s^{\frac{| \beta |}{2} + 1}},
    \end{split}
\end{equation}
where
$$
    L_8 (|\beta|, \|\omega_0\|) = 2^{| \beta|} \|\omega_0\|^2 \sum_{\gamma \leq \beta} 
      N_u (0, | \gamma|, \infty) N_{\omega} (0, | \beta - \gamma|, 1).
$$
Next we have
\begin{equation} \label{betavR}
    \begin{split}
  &\| \partial_x^{\beta} v^{R/2}(s, \cdot) \|_p \\
&= \| \sum_{\gamma \leq \beta}   \binom{\beta}{\gamma} \partial_x^{\gamma} 
 u^{R/2}(s, \cdot) \cdot \partial_x^{\beta - \gamma} (\nabla \omega^{R/2}(s, \cdot)) 
  \|_p \\
    & \leq \sum_{\gamma \leq \beta}  \binom{\beta}{\gamma} \| \partial_x^{\gamma} 
      u^{R/2}(s, \cdot) \|_{\infty}   \| \partial_x^{\beta - \gamma} \nabla 
      \omega^{R/2}(s, \cdot) \|_p \\
    & = \| u^{R/2}(s, \cdot) \|_{\infty}   \| \partial_x^{\beta} \nabla 
     \omega^{R/2}(s, \cdot) \|_p + \| \partial_x^{\beta} u^{R/2}(s, \cdot) \|_{\infty}  
      \| \nabla \omega^{R/2}(s, \cdot) \|_p \\
    & \quad + \sum_{0 < |\gamma| < |\beta|} \binom{\beta}{\gamma} \| 
     \partial_x^{\gamma} u(s, \cdot) \|_{\infty}   \| \partial_x^{\beta - \gamma} 
     \nabla \omega(s, \cdot) \|_p  = P_1 (s) + P_2 (s) + P_3 (s),
    \end{split}
\end{equation}
where we use $P_i (s), i = 1, 2, 3$, to denote the three parts in the entire 
sum of \eqref{betavR}.

By using \eqref{uinfest}, \eqref{Nw} and \eqref{Nu}, we have
\begin{equation} \label{p1}
    \begin{split}
    P_1 (s)
&= \| u^{R/2}(s, \cdot) \|_{\infty}   \| \partial_x^{\beta} \nabla 
  \omega^{R/2}(s, \cdot) \|_p \\
& \leq \frac{N_{\omega} (0, |\beta| + 1, p) \|\omega_0\|}{s^{\frac{|\beta|}{2}
 + \frac{3}{2} - \frac{1}{p}}} \\
&\quad\times \Big( \frac{L_{1, \infty}}{R} \| \omega(s, \cdot) \|
  + L_{2, \infty} \| \omega^{R/4}(s, \cdot) \|^{1/2} \| \omega^{R/4} (s, \cdot)
  \|_{\infty}^{1/2} \Big)
\end{split}
\end{equation}
where by Lemma \ref{Lm: wtheta},
\begin{align*}
    &\| \omega^{R/4}(s, \cdot) \|^{1/2} \| \omega^{R/4} (s, \cdot) \|_{\infty}^{1/2} \\
    & \leq \Big(L_0(1, \|\omega_0\|)
\Big(\frac{2s^{1/2}}{R} \|\omega_0\| + \|\omega_0^{R/8}\|\Big)
 L_0(\infty, \|\omega_0\|)\\
&\quad\times \Big(\frac{2s^{-1/2}}{R} \|\omega_0\| + s^{-1} \|\omega_0^{R/8}\|
\Big) \Big)^{1/2} \\
    & \leq L_9 (\|\omega_0\|)\Big( \frac{1}{R} + s^{-1/2} \| \omega_0^{R/8}
 \| \Big),
\end{align*}
in which
$$
    L_9 (\|\omega_0\|) = (2L_0(1, \|\omega_0\|) \| \omega_0 \| + 1)^{1/2}
(2L_0(\infty, \|\omega_0\|) \| \omega_0 \| + 1)^{1/2}.
$$
From \eqref{p1} it follows that
\begin{equation} \label{p12}
    P_1(s) \leq \frac{B_1 (|\beta|, p, \|\omega_0\|)}{s^{\frac{|\beta|}{2}
+ \frac{3}{2} - \frac{1}{p}}} \Big( \frac{1}{R} + s^{-1/2} \| \omega_0^{R/8}
\| \Big),
\end{equation}
where
$$
    B_1 (|\beta|, p, \|\omega_0\|) = N_{\omega} (0, |\beta| + 1, p) 
    \|\omega_0\| \left\{L_{1,\infty} N_{\omega} (0, 0, 1) \|\omega_0 \| 
    + L_{2,\infty} L_9 (\|\omega_0\|)\right\}.
$$
Next by \eqref{Nu} and \eqref{nbl} for $\nabla \omega(t, x)$ proved in Step 2, 
we have
\begin{equation} \label{p2}
    \begin{split}
    P_2 (s)
& = \| \partial_x^{\beta} u^{R/2}(s, \cdot) \|_{\infty}   \| \nabla \omega^{R/2}(s,
 \cdot) \|_p \\
& \leq \frac{N_u (0, |\beta|, \infty) \| \omega_0 \|}{s^{\frac{1}{2} (| \beta |
  + 1)}}   B_0 (p, \| \omega_0 \|) \Big( \frac{1}{R   s^{1 - \frac{1}{p}}}
 + \frac{\| \omega_0^{R/16} \|}{s^{\frac{3}{2} - \frac{1}{p}}} \Big) \\
& \leq \frac{B_2 (|\beta|, p, \|\omega_0\|)}{s^{\frac{|\beta|}{2}
 + \frac{3}{2} - \frac{1}{p}}} \Big( \frac{1}{R} + s^{-1/2}
\| \omega_0^{R/16} \| \Big),
\end{split}
\end{equation}
where the constant $B_2 (|\beta|, p, \|\omega_0\|)$ is given by
$$
    B_2 (|\beta|, p, \|\omega_0\|) = N_u (0, |\beta|, \infty) \| \omega_0 \|  
    B_0 (p, \| \omega_0 \|).
$$
Moreover, by the assumption of induction in this step that \eqref{wspdc} holds 
for all the spatial derivatives of $\omega(t, x)$ up to the order $k$, and 
here $|\beta| = k$, we have
\begin{equation} \label{p3}
    \begin{split}
P_3 (s) &=  \sum_{0 < |\gamma| < |\beta|} \binom{\beta}{\gamma} 
 \| \partial_x^{\gamma} u(s, \cdot) \|_{\infty}   \| \partial_x^{\beta - \gamma} 
  \nabla \beta(s, \cdot) \|_p \\
    & \leq  \sum_{0 < |\gamma| < |\beta|} \binom{\beta}{\gamma}
  \frac{N_u (0, |\gamma|, \infty) \|\omega_0\|}{s^{\frac{1}{2}(|\gamma| + 1)}}
   \frac{B(|\beta - \gamma|, p, \|\omega_0\|)}{s^{\frac{1}{2}(|\beta - \gamma|
 + 1 - 1)}}\\
&\quad\times \Big(\frac{1}{R} + \frac{1}{R  s^{1 - \frac{1}{p}}}
 + \frac{\|\omega_0^{R/16} \|}{s^{\frac{3}{2} - \frac{1}{p}}} \Big) \\
& \leq \frac{B_3 (|\beta|, p, \|\omega_0 \|)}{s^{\frac{1}{2}(|\beta| + 1)}}
 \Big(\frac{1}{R} + \frac{1}{R  s^{1 - \frac{1}{p}}}
  + \frac{\|\omega_0^{R/16} \|}{s^{\frac{3}{2} - \frac{1}{p}}} \Big),
    \end{split}
\end{equation}
where the constant $B_3 (|\beta|, p, \|\omega_0 \|)$ is given by
$$
    B_3 (|\beta|, p, \|\omega_0 \|) =  \sum_{0 < |\gamma| < |\beta|} 
    \binom{\beta}{\gamma} N_u (0, |\gamma|, \infty)   \|\omega_0\|  B(|\beta - \gamma|,
     p, \|\omega_0\|).
$$
Now substituting \eqref{p12}. \eqref{p2} and \eqref{p3} into
\eqref{betavR}, we get
\begin{equation} \label{bvRpn}
    \begin{split}
    \|\partial_x^{\beta} v^{R/2} (s, \cdot) \|_p
&\leq P_1 (s) + P_2 (s) + P_3 (s) \\
& \leq \frac{B_1 + B_2 + B_3}{s^{\frac{|\beta|}{2} + \frac{3}{2}
  - \frac{1}{p}}} \left( \frac{1}{R} + \frac{\|\omega_0^{R/16} \|}
  {s ^{\frac{1}{2}}} \right) +  \frac{B_3}{R  s ^{\frac{1}{2}(|\beta| + 1)}}  ,
    \end{split}
\end{equation}

Finally we substitute \eqref{betav} and \eqref{bvRpn} into \eqref{integrand}
and then we can estimate the nonlinear integral term in \eqref{voc2}
as follows, noting that $|\beta| = k \geq 0$ and $R > 1$,
\begin{equation} \label{nonlpt}
    \begin{split}
    &\| \int_{t/2}^t \nabla [e^{A(t - s)} \partial_x^{\beta}(u(s, \cdot) \cdot \nabla) 
    \omega (s, \cdot))]^R \,ds   \|_p \\
    & \leq \int_{t/2}^t \| \nabla [e^{A(t - s)} \partial_x^{\beta}(u(s, \cdot) 
    \cdot \nabla) \omega (s, \cdot))]^R \|_p \,ds \\
    &  \leq \int_{t/2}^t \left( \frac{L_3 (|\beta|, p)}{R^{3 - \frac{2}{p}}} \| 
    \partial_x^{\beta} v(s, \cdot) \| + \frac{L_4 (|\beta|)}{(t - s)^{1/2}} \| 
    \partial_x^{\beta} v^{R/2} (s, \cdot) \|_p \right) ds \\
    & \leq \int_{t/2}^t \frac{L_3 (|\beta|, p)}{R^{3 - \frac{2}{p}}} \cdot 
    \frac{L_8 (|\beta|, \|\omega \|_0)}{s^{\frac{| \beta |}{2} + 1}} \,ds \\
    & \quad + \int_{t/2}^t  \frac{L_4 (|\beta|)}{(t - s)^{1/2}}
\Big[ \frac{B_1 + B_2 + B_3}{s^{\frac{|\beta|}{2} + \frac{3}{2}
  - \frac{1}{p}}}  \Big( \frac{1}{R} + \frac{\|\omega_0^{R/16} \|}
  {s^{\frac{1}{2}}} \Big) +  \frac{B_3}{R  s^{\frac{1}{2}(|\beta| + 1)}}\Big] ds \\
    & \leq L_3 (|\beta|, p)   \frac{L_8 (|\beta|, \|\omega \|_0)}{R   
    (t/2)^{\frac{|\beta|}{2}}} + L_4(|\beta|)   \frac{2B_3   (t/2)^{1/2}}{R  
    (t/2)^{\frac{1}{2}(|\beta| + 1)}} \\
 &\quad  + L_4 (|\beta|)   \frac{2(B_1 + B_2 + B_3)(t/2)^{1/2}}{(t/2)
  ^{\frac{|\beta|}{2} + \frac{3}{2} - \frac{1}{p}}}
 \Big( \frac{1}{R} + \frac{\|\omega_0^{R/16} \|}{(t/2)^{\frac{1}{2}}} \Big) \\
    & \leq \frac{B_4(|\beta| + 1, p, \|\omega_0 \|)}{t  ^{\frac{1}{2}[(|\beta|
  + 1) -1]}} \Big(\frac{1}{R} + \frac{1}{R   t ^{1 - \frac{1}{p}}}
 + \frac{\| \omega_0^{R/16} \|}{t   ^{\frac{3}{2} - \frac{1}{p}}} \Big),
    \end{split}
\end{equation}
where
\begin{align*}
   B_4(|\beta| + 1, p, \|\omega_0 \|)
    & = \max \big\{2^{\frac{|\beta|}{2}}L_3 (|\beta|, p) L_8 (|\beta|, \|\omega \|_0)
+ 2^{\frac{|\beta|}{2} + 1} L_4(|\beta|)B_3,  \\
&\quad  2^{\frac{|\beta|}{2} + 2 -\frac{1}{p}} L_4(|\beta|) (B_1 + B_2 + B_3)
\big\}.
\end{align*}
We assemble \eqref{voc2}, \eqref{linpt4} (noting that $|\beta| = k$ and
$\|\omega_0^{R/2} \| \leq \|\omega_0^{R/16}\|$) and \eqref{nonlpt} to obtain
\begin{equation} \label{asb}
    \|\nabla \partial_x^{\beta} \omega^R (t, x) \|_p \leq \frac{B (| \beta | + 1,
 p, \| \omega_0 \|)}{ t ^{\frac{|\beta|}{2}}} \Big( \frac{1}{R}
+ \frac{1}{R   t ^{1 - \frac{1}{p}}}
+ \frac{\| \omega_0^{R/16} \|}{t   ^{\frac{3}{2} - \frac{1}{p}}} \Big),
\end{equation}
where
\begin{equation*}
    B (| \beta | + 1, p, \| \omega_0 \|) = \max   \{L_6 (|\beta|, p,
  \|\omega_0\|) (\|\omega_0\| + 1), \; B_4(|\beta| + 1, p, \|\omega_0 \|) \}.
\end{equation*}
The inequality \eqref{asb} shows that \eqref{wspdc} is valid for
$|\beta| = k + 1$. Thus by induction we have proved that \eqref{wspdc}
is valid for all order $\beta \in Z^{+} \times Z^{+}$.
\end{proof}

Based on Theorem \ref{Thm: wspdc}, we now prove the following result
on the exterior decay estimates of all the temporal-spatial derivatives
of the vorticity.

\begin{theorem} \label{Thm: tsdrdc}
    For any given $R > 1$, $\alpha \in Z^{+}$ and $\beta \in Z^{+} \times Z^{+}$,
the temporal-spatial derivative of the vorticity solution $\omega(t, x)$
of \eqref{Veqn}--\eqref{BSL} with initial data
$\omega_0 \in L^1(\mathbb{R}^2)$ has the following decay property,
\begin{equation} \label{tspdc}
    \| \partial_t^{\alpha} \partial_{x}^{\beta} \omega^R (t, \cdot) \|_p
\leq \frac{K (\alpha, | \beta |, p, \| \omega_0 \|)}{ t   ^{\alpha + \frac{1}{2}
( | \beta | -1)}} \Big( \frac{1}{R}  + \frac{1}{R   t ^{1 - \frac{1}{p}}}
+ \frac{\| \omega_0^{R/16} \|}{t   ^{\frac{3}{2} - \frac{1}{p}}} \Big),
\end{equation}
where $K (\alpha, | \beta |, p, \| \omega_0 \|)$ is a positive constant depending
on $\alpha, |\beta|, p$, and $\|\omega_0\|$.
\end{theorem}

\begin{proof}
We shall prove \eqref{tspdc} by induction with respect to the order
$\alpha$ of temporal derivatives of $\partial_x^{\beta} \omega(t, x)$.
In Theorem \ref{Thm: wspdc}, we have shown that \eqref{tspdc} is
valid for $\alpha = 0$ and any $\beta \in Z^{+} \times Z^{+}$.
Now assuming that \eqref{tspdc} is valid for an $\alpha \geq 0$ and any
$\beta \in Z^{+} \times Z^{+}$, we prove that \eqref{tspdc} holds for
$\alpha + 1$ and any $\beta$.

By the $C^{\infty}$ regularity of the vorticity and velocity in
\eqref{Veqn}--\eqref{BSL}, one has
\begin{equation} \label{tsdw}
    \partial_t^{\alpha + 1} \partial_{x}^{\beta} \omega = \partial_t^{\alpha} 
    \partial_{x}^{\beta} \Delta \omega - \partial_t^{\alpha} \partial_{x}^{\beta} 
    ((u\cdot \nabla) \omega)
\end{equation}
and, for any given $R > 1$,
\begin{equation} \label{tsest}
    \| \partial_t^{\alpha + 1} \partial_{x}^{\beta} \omega^R(t, \cdot) \|_p
\leq \| \partial_t^{\alpha} \partial_{x}^{\beta} \Delta \omega^R(t, \cdot) \|_p
 + \| \partial_t^{\alpha} \partial_{x}^{\beta} ((u^R \cdot \nabla) \omega^R (t, \cdot))
\|_p   .
\end{equation}
By the assumption of induction, for the first term on the right-hand side
of \eqref{tsest} we have
\begin{equation} \label{tlin}
    \| \partial_t^{\alpha} \partial_{x}^{\beta} \Delta \omega^R (t, \cdot) \|_p
\leq \frac{K (\alpha, | \beta | + 2, p, \| \omega_0 \|)}{ t   ^{\alpha
 + \frac{1}{2} ( | \beta | + 1)}}
\Big( \frac{1}{R}  + \frac{1}{R   t ^{1 - \frac{1}{p}}}
+ \frac{\| \omega_0^{R/16} \|}{t   ^{\frac{3}{2} - \frac{1}{p}}} \Big).
\end{equation}
For the second term on the righr-hand side of \eqref{tsest},
from \eqref{Nu} and the assumption of induction we get
\begin{equation} \label{tnl}
    \begin{split}
    &\| \partial_t^{\alpha} \partial_{x}^{\beta} ((u^R(t, \cdot) \cdot \nabla) 
    \omega^R (t, \cdot)) \|_p \\
    & \leq \sum_{\eta \leq \alpha} \sum_{\gamma \leq \beta} \binom{\alpha}{\eta} 
    \binom{\beta}{\gamma} \| \partial_t^{\eta} \partial_{x}^{\gamma} u^R(t, \cdot) 
    \|_{\infty} \| \partial_t^{\alpha - \eta} \partial_{x}^{\beta - \gamma} 
    \nabla \omega^R (t, \cdot)  \|_p \\
    & \leq \sum_{\eta \leq \alpha} \sum_{\gamma \leq \beta} \binom{\alpha}{\eta}
  \binom{\beta}{\gamma} \frac{N_u (\eta, |\gamma|, \infty) \|\omega_0\|}
 {t  ^{\eta + \frac{|\gamma|}{2} + \frac{1}{2}}} \cdot \frac{K(\alpha
 - \eta, |\beta - \gamma| + 1, p, \|\omega_0\|)}{t  ^{\alpha - \eta
 + \frac{1}{2} |\beta - \gamma|}}  \\
    & \quad\times \Big( \frac{1}{R}  + \frac{1}{R   t ^{1 - \frac{1}{p}}}
+ \frac{\| \omega_0^{R/16} \|}{t   ^{\frac{3}{2} - \frac{1}{p}}} \Big) \\
    & \leq \frac{L_{10} (\alpha, |\beta|, p, \|\omega_0\|)}{t  ^{\alpha 
    + \frac{1}{2}
 (|\beta| + 1)}} \Big( \frac{1}{R}  + \frac{1}{R   t ^{1 - \frac{1}{p}}}
+ \frac{\| \omega_0^{R/16} \|}{t   ^{\frac{3}{2} - \frac{1}{p}}} \Big),
    \end{split}
\end{equation}
where
\begin{align*}
&L_{10} (\alpha, |\beta|, p, \|\omega_0\|) \\
&= \sum_{\eta \leq \alpha} \sum_{\gamma \leq \beta} \binom{\alpha}{\eta}
\binom{\beta}{\gamma} N_u (\eta, |\gamma|, \infty) \|\omega_0\| K(\alpha
- \eta, |\beta - \gamma| + 1, p, \|\omega_0\|).
\end{align*}
Substituting \eqref{tlin} and \eqref{tnl} into \eqref{tsest}, we get the
 following estimate,
\begin{equation} \label{assemb}
    \| \partial_t^{\alpha + 1} \partial_{x}^{\beta} \omega^R(t, \cdot) \|_p
\leq \frac{K(\alpha + 1, |\beta|, p, \|\omega_0\|)}{t ^{\alpha + 1 + \frac{1}{2}
(|\beta| - 1)}}  \Big( \frac{1}{R}  + \frac{1}{R   t ^{1 - \frac{1}{p}}}
+ \frac{\| \omega_0^{R/16} \|}{t   ^{\frac{3}{2} - \frac{1}{p}}} \Big),
\end{equation}
with
\begin{equation} \label{constK}
    K(\alpha + 1, |\beta|, p, \|\omega_0\|) = K (\alpha, | \beta | + 2, p,
     \| \omega_0 \|)
+ L_{10} (\alpha, |\beta|, p, \|\omega_0\|).
\end{equation}
It follows from \eqref{assemb} and \eqref{constK} that the estimate
\eqref{tspdc} is valid\ for $\alpha + 1$ and any $\beta \in Z^{+} \times Z^{+}$.
Thus by induction this theorem is proved.
\end{proof}

\section{Attraction of Oseen Vortex and Velocity Solutions}

As an application of the obtained derivative estimates of the 2D
vorticity and velocity solutions, we shall sketch the proof of the
following theorem, which shows that the temporal-spatial
derivatives of the 2D vorticity and velocity solutions on
$\mathbb{R}^2$ behave as the derivatives of the same order of the
corresponding Oseen solutions asymptotically in the sense of $L^p
(p \geq 1)$ and $L^q (q \geq 2)$ convergence, respectively.

\begin{theorem} \label{Thm3}
Let $\omega(t, x)$ be the vorticity solution of
\eqref{Veqn}--\eqref{BSL} with initial vorticity $\omega_0 \in
L^1(\mathbb{R}^2)$. Let $\omega^{*} (t, x)$ be the Oseen vortex
\eqref{OsV} with $\kappa = \kappa (\omega_0)$ given in \eqref{kw0}. Then
for any $\alpha \in Z^{+}$ and any $\beta \in Z^{+} \times Z^{+}$, it
hold that
\begin{equation} \label{wconv}
    \lim_{t \to \infty} t ^{\alpha + \frac{|\beta|}{2}
+ 1 - \frac{1}{p}} \| \partial_{t}^{\alpha} \partial_{x}^{\beta} \omega (t, \cdot )
- \partial_{t}^{\alpha} \partial_{x}^{\beta} \omega^{*} (t, \cdot ) \|_{p} = 0,
 \quad p \in [1, \infty],
\end{equation}
and
\begin{equation} \label{uconv}
    \lim_{t \to \infty} t ^{\alpha + \frac{|\beta|}{2} + \frac{1}{2}
- \frac{1}{q}} \| \partial_{t}^{\alpha} \partial_{x}^{\beta} u (t, \cdot )
- \partial_{t}^{\alpha} \partial_{x}^{\beta} u^{*} (t, \cdot ) \|_{q} = 0,
\quad q \in (2, \infty],
\end{equation}
where $u(t, x)$ and $u^{*}(t, x)$ are the velocity solutions given
by \eqref{BSL} corresponding to $\omega(t, x)$ and the Oseen vortex
$\omega^{*}(t, x)$, respectively.
\end{theorem}

The proof of theorem \ref{Thm3} uses the shown estimates and takes the
approach of the parabolic similarity \cite{CP96, GW05, CL96,  WB98}
and the compactness processing in $L^p$ spaces.

Let $(\omega, u)$ be a vorticity-velocity solution of
\eqref{Veqn}--\eqref{BSL} with any initial vorticity $\omega_0 \in
L^1 (\mathbb{R}^2)$. For any scalar constant $\lambda > 0$, we define
\begin{equation} \label{simfn}
    \begin{gathered}
    \omega_{\lambda} (t, x) = \lambda^2 \omega (\lambda^2 t, \lambda x), \\
    u_{\lambda} (t, x) = \lambda u (\lambda^2 t, \lambda x), \\
    \omega_{0, \lambda} (x) = \lambda^2 \omega_0 (\lambda x)  \; 
    ( = \omega_{\lambda} (0, x) \; ).
    \end{gathered}
\end{equation}
For any $\lambda > 0, t > 0, x \in \mathbb{R}^2$, it is easy to check
that $(\omega_\lambda, u_\lambda)$ satisfies the vorticity equation
\eqref{Veqn}--\eqref{BSL} and that
\begin{equation} \label{sclp}
    \begin{gathered}
 \| \omega_{0, \lambda} \| = \| \omega_0 \|, \quad
 \kappa (\omega_{0, \lambda} ) = \kappa (\omega_0),\\
 \omega_\lambda^{*} (t, x) = \omega^{*} (t, x), \quad
 u_\lambda^{*} (t, x) = u^{*} (t, x).
    \end{gathered}
\end{equation}
Since
\begin{equation*}
    \partial_t^\alpha \partial_x^\beta \omega_\lambda (t, x) 
    - \partial_t^\alpha \partial_x^\beta \omega^{*} (t, x) = \lambda  ^{2 + 2\alpha
     + |\beta|} [ \partial_t^\alpha \partial_x^\beta \omega(\lambda^2 t, \lambda x) 
     - \partial_t^\alpha \partial_x^\beta \omega^{*}(\lambda^2 t, \lambda x) ],
\end{equation*}
we find that
\begin{equation} \label{drsclpn}
    \| \partial_t^\alpha \partial_x^\beta \omega_\lambda (1, \cdot) 
    - \partial_t^\alpha \partial_x^\beta \omega^{*} (1, \cdot) \|_p 
    = \lambda ^{2 + 2\alpha + |\beta| - \frac{2}{p}} \| \partial_t^\alpha 
    \partial_x^\beta \omega_\lambda (\lambda^2, \cdot) - \partial_t^\alpha 
    \partial_x^\beta \omega^{*} (\lambda^2, \cdot) \|_p.
\end{equation}
Taking $t = \lambda^2$, we have the \emph{equivalence} that \eqref{wconv} holds 
for $p \in [1, \infty]$ if and only if

\begin{equation} \label{eqvl}
    \lim_{\lambda \to \infty} \| \partial_t^\alpha \partial_x^\beta 
    \omega_\lambda (1, \cdot) - \partial_t^\alpha \partial_x^\beta \omega^{*} (1, \cdot)
     \|_p = 0, \quad \text{for} \; p \in [1, \infty].
\end{equation}
Without confusion $\partial_t^\alpha \partial_x^\beta w(1, x)$ means
$(\partial_t^\alpha \partial_x^\beta w(t, x))|_{t = 1}$, for 
$w = \omega_\lambda, \omega^{*}, u_\lambda$, or $u^{*}$.

Below we prove \eqref{eqvl}, which is equivalent to \eqref{wconv}.
The proof involves the following lemma, which crucially relies on
the exterior decay estimates of temporal-spatial derivatives of
the vorticity solutions shown in Theorem \ref{Thm: tsdrdc} and the
Riesz criterion \cite[Theorem 2.21]{rA75} of compactness of
subsets in $L^p (\mathbb{R}^n)$.

\begin{lemma} \label{Lm: compac}
The set of functions $\{\omega_\lambda (1, x)\}_{\lambda \geq 1}$ defined in
\eqref{simfn} is a precompact set in the Sobolev space $W^{|\beta|,
p}(\mathbb{R}^2)$, for any $p \in [1, \infty]$ and any integer
$|\beta | \geq 0$.
\end{lemma}

\begin{proof} First let $p \in [1, \infty)$.
Let $\omega_0 \in L^1 (\mathbb{R}^2)$ be arbitrarily given. In order to
prove this lemma, it suffices to show that for any given integer
$|\beta| = k$,
\begin{equation} \label{critern}
    \big\{\partial_x^{\gamma} \omega_\lambda (1, x)\big\}_{\lambda \geq 1}
\text{is precompact in $L^p (\mathbb{R}^2)$, for all 0 }
 \leq \gamma \leq \beta.
\end{equation}
By the Riesz criterion of compactness, we just check the following
three conditions. Let $\gamma \leq \beta$.

\textbf{(a)} $\{\partial_x^{\gamma} \omega_\lambda (1, x)\}_{\lambda \geq 1}$ is
a bounded subset in $L^p (\mathbb{R}^2)$. In fact, \eqref{Nw} and
\eqref{sclp} yield
$$
    \|\partial_x^{\gamma} \omega_\lambda (1, \cdot) \|_p \leq N_\omega (0, |\gamma|, p)
    \|\omega_{0,\lambda} \| = N_\omega (0, |\gamma|, p) \|\omega_0 \| 
    \leq \max_{\gamma \leq \beta} \{N_\omega (0, |\gamma|, p)\} \|\omega_0\|,
$$
for all $\lambda \geq 1$, which confirms this assertion.

\textbf{(b)} The uniform convergence to zero at infinity for 
$\{\partial_x^{\gamma} \omega_\lambda (1, x)\}_{\lambda \geq 1}$ is attested 
by applying Theorem \ref{Thm: wspdc} and \eqref{wspdc} to 
$\omega_\lambda (t, \cdot)$ with $t = 1$ and choosing $R > 1$ to be 
sufficiently large such that
\begin{equation} \label{ucgl}
    B(|\gamma|, p, \|\omega_0\|) \Big( \frac{2}{R} + \| \omega_0^{R/16} \| \Big)
< \varepsilon.
\end{equation}

\textbf{(c)} The equicontinuity of  $\{\partial_x^{\gamma} \omega_\lambda (1,
x)\}_{\lambda \geq 1}$ in $L^p (\mathbb{R}^2)$ is shown below. By the
formula \eqref{voc1} on the time interval $[\frac{1}{2}, 1]$ for
the vorticity solutions $\omega_\lambda (t, x)$ and $\omega_\lambda (t, x - y)$,
where $y$ is a parameter, we need to estimate
\begin{equation} \label{oml}
    \begin{split}
    \| \partial_x^\gamma \omega_\lambda (1, \cdot - y) - \partial_x^\gamma 
    \omega_\lambda (1, \cdot)\|_p &\leq \| \partial_x^\gamma e^{A(1/2)}
    (\omega_\lambda (1/2, \cdot - y) - \omega_\lambda (1/2, \cdot)) \|_p \\
    &\quad + \int_{1/2}^1 \| \partial_x^\gamma e^{A(1 - s)}(v_\lambda (s, \cdot - y) 
    - v_\lambda (s, \cdot)) \|_p \,ds,
    \end{split}
\end{equation}
where, as in \eqref{vtxdef}, $v_\lambda(t, x)  = (u_\lambda (t, x) \cdot
\nabla)\omega_\lambda (t, x)$ and $v_\lambda(t,  x - y)  = (u_\lambda (t, x - y)
\cdot \nabla)\omega_\lambda (t, x - y)$. We shall estimate the two terms
in \eqref{oml} respectively.

\textbf{(i)} First we estimate the linear term on the right-hand side
of \eqref{oml}. Let
$$
    E(x) = E(x_1. x_2) = \frac{1}{2\pi \nu} \exp (- \frac{| x |^2}{2\nu}),
\quad x \in \mathbb{R}^2.
$$
According to \eqref{dsg} and by variable changes, we have
\begin{equation} \label{linpt}
    \begin{split}
&\|\partial_x^\gamma  e^{A(1/2)} (\omega_\lambda (1/2, \cdot - y)
 - \omega_\lambda (1/2, \cdot)) \|_p \\
& = \big\| \frac{1}{2\pi \nu} \int_{\mathbb{R}^2} \partial_x^\gamma
\exp \big(- \frac{|x - \xi|^2}{2\nu}\big) (\omega_\lambda (1/2, \xi - y)
 - \omega_\lambda (1/2, \xi)) \,d\xi \big\|_p \\
 & (\text{by the variable change} \; \zeta = \xi - x) \\
& = \big\| \frac{1}{2\pi \nu} \int_{\mathbb{R}^2} \exp
 \big(- \frac{|\zeta|^2}{2\nu}\big) \partial_\zeta^\gamma (\omega_\lambda (1/2, \zeta
  + x - y) - \omega_\lambda (1/2, \zeta + x)) \,d\zeta \big\|_p   \\
    &  (\text{by the variable changes} \; \eta = \zeta + x - y  \text{ and }  
    \eta = \zeta + x, \; \text{respectively}) \\
    & = \big\| \frac{1}{2\pi \nu} \int_{\mathbb{R}^2}
\big[ \exp \big(- \frac{|x - y - \eta|^2}{2\nu}\big)
 - \exp \big(- \frac{|x - \eta|^2}{2\nu}\big) \big] \partial_\eta^\gamma
 \omega_\lambda (1/2, \eta) \,d\eta \big\|_p \\
& = \| (E(x - y) - E(x)) \ast (\partial_x^\gamma   \omega_\lambda (1/2, x)) \|_p.
    \end{split}
\end{equation}
By the convolution property \eqref{convln}, \eqref{Nw}, and \eqref{sclp},
it follows that
\begin{align*}
     &\|(E(x - y) - E(x)) \ast (\partial_x^\gamma   \omega_\lambda (1/2, x)) \|_p \\
     & \leq \| E(x - y) - E(x) \|   N_\omega (0, |\gamma|, p)   
     2^{ |\gamma|/2} \|\omega_0\|.
\end{align*}
Let $\varepsilon$ be arbitrarily given. By the continuity of integrable functions
with respect to the $L^1$-norm \cite[Chapter VIII, Proposition 1.4]{aT88},  
here for the function $E(x)$ there exists a uniform 
$\delta_1 (\varepsilon) > 0$ independent of $\lambda \geq 1$, such that 
if $| y | < \delta_1$, then
\begin{equation} \label{linptest}
    \begin{split}
    \| \partial_x^\gamma & e^{A(1/2)} (\omega_\lambda (1/2, \cdot - y) 
     - \omega_\lambda (1/2, \cdot)) \|_p \\
    & \leq  \| E(x - y) - E(x) \|   N_\omega (0, |\gamma|, p)   
    2^{ |\gamma|/2} \|\omega_0\| < \frac{\varepsilon}{2}  .
    \end{split}
\end{equation}

\textbf{(ii)} Now we estimate the nonlinear integral term in \eqref{oml}.
Let
$$
F(x, s) = \frac{1}{4\pi \nu (1 - s)} \exp \Big( -   \frac{| x
|^2}{4\nu (1 - s)} \Big), \quad 0 < s < 1.
$$
By variable changes similar to what we did in deriving \eqref{linpt}, we have
\begin{equation} \label{nlpt}
    \begin{split}
  &\int_{1/2}^1 \| \partial_x^\gamma  e^{A(1 - s)} (v_\lambda (s, \cdot - y)
   - v_\lambda (s, \cdot)) \|_p \,ds \\
  & \leq \int_{1 - d}^1  ( \|F(\cdot - y, s) \| + \|F(\cdot, s) \|)\, \|\partial_x^\gamma 
    v_\lambda (s, \cdot) \|_p \,ds \\
  &\quad  + \int_{1/2}^{1- d} \| F(\cdot - y, s) - F(\cdot, s) \|\, \|
   \partial_x^\gamma   v_\lambda (s, \cdot) \|_p \,ds
    \end{split}
\end{equation}
where $0 < d < 1/2$. By \eqref{Nw}, \eqref{Nu}, \eqref{sclp}, and
since $s \geq 1/2$ in \eqref{nlpt},  we get
\begin{align*}
    &\| \partial_x^\gamma  v_\lambda (s, x) \|_p \\
& \leq \sum_{\mu \leq \gamma} \binom{\gamma}{\mu} \| (\partial_x^\mu 
 u_\lambda (s, x))   \partial_x^{\gamma - \mu} (\nabla \omega_\lambda (s, x)) \|_p \\
& \leq \sum_{\mu \leq \gamma} \binom{\gamma}{\mu} N_u (0, |\mu|, \infty) 
 s^{- (\frac{|\mu|}{2} + \frac{1}{2})} \|\omega_{0,\lambda}\|   N_{\omega} 
 (0, |\gamma - \mu| + 1, p) s^{-\frac{|\gamma - \mu| + 1}{2}} 
  \|\omega_{0,\lambda}\| \\
& \leq \sum_{\mu \leq \gamma} \binom{\gamma}{\mu} 2^{\frac{|\gamma|}{2}
 + 1} N_u (0, |\mu|, \infty) N_{\omega} (0, |\gamma - \mu|
 + 1, p) \|\omega_0\|^2 \\
&= D(|\gamma|, p, \|\omega_0\|),
\end{align*}
where $D(|\gamma|, p, \|\omega_0\|)$ is a uniform constant independent of
$\lambda \geq 1$. For the arbitrarily given $\varepsilon > 0$ mentioned in (i),
there exists a $d(\varepsilon) \in (0, 1/2)$ such that
$d(\varepsilon) D(|\gamma|, p, \|\omega_0\|) < \varepsilon/8$, which yields
\begin{equation} \label{dest}
     \int_{1 - d}^1  ( \|F(\cdot - y, s) \| + \|F(\cdot, s) \|)   \|\partial_x^\gamma 
      v_\lambda (s, \cdot) \|_p \,ds \leq 2d(\varepsilon) D(|\gamma|, p, \|\omega_0\|)
      < \frac{\varepsilon}{4}  ,
\end{equation}
due to  $\|F(\cdot - y, s) \| = \|F(\cdot, s) \| = 1$. Moreover, we have
\begin{align*}
&\| F(x - y, s)  - F(x, s) \| \\
& = \int_{\mathbb{R}^2} \frac{1}{4\pi \nu (1 - s)}
\Big| \exp \Big( -   \frac{|x - y|^2}{4\nu (1 - s)} \Big)
- \exp \Big( -   \frac{|x - y|^2}{4\nu (1 - s)} \Big) \Big| dx \\
& = \int_{\mathbb{R}^2} \frac{1}{\pi} \left| \exp (- |\widetilde{x} - \widetilde{y}|^2) 
- \exp (- |\widetilde{x}|^2 ) \right| d\widetilde{x} ,
\end{align*}
where
$$
    \widetilde{x} = \frac{x}{\sqrt{4\nu (1 - s)}} \quad \text{and}
\quad \widetilde{y} = \frac{y}{\sqrt{4\nu (1 - s)}}  .
$$
By the continuity of the integrable function $\exp(- |\widetilde{x} |^2)$
with respect to the $L^1$-norm, for the arbitrarily given $\varepsilon > 0$ 
there exists a uniform $\delta_2 (\varepsilon) > 0$ independent of 
$\lambda \geq 1$,
such that if
\begin{equation} \label{dlt2}
     \frac{| y |}{\sqrt{4\nu (1 - s)}}  = | \widetilde{y} | < \delta_2 (\varepsilon), 
     \quad \text{for any} \; s \in [ 1/2, 1 - d(\varepsilon)],
\end{equation}
then
$$
    \| F(x - y, s) - F(x, s) \| < \frac{\varepsilon}{2 D(|\gamma|, p, 
    \|\omega_0\|)},
$$
so that
\begin{equation} \label{hest}
 \int_{1/2}^{1- d} \| F(\cdot - y, s) - F(\cdot, s) \| \, \| \partial_x^\gamma
v_\lambda (s, \cdot) \|_p \,ds < \frac{\varepsilon}{4}.
\end{equation}
Note that \eqref{dlt2} holds if $| y | <  \sqrt{4\nu d(\varepsilon)} 
\delta_2 (\varepsilon)$.
>From \eqref{nlpt}, \eqref{dest}, and \eqref{hest}, we have shown that if
 $| y | <  \sqrt{4\nu d(\varepsilon)} \delta_2 (\varepsilon)$, then
\begin{equation} \label{nlptest}
    \int_{1/2}^1 \| \partial_x^\gamma e^{A(1 - s)}(v_\lambda (s, \cdot - y)
     - v_\lambda (s, \cdot)) \|_p < \frac{\varepsilon}{4} + \frac{\varepsilon}{4} 
     = \frac{\varepsilon}{2}\,.
\end{equation}
In turn by \eqref{oml}, \eqref{linptest} and \eqref{nlptest},
there exists a uniform constant
$$
    \delta = \delta (\varepsilon) = \min \{\delta_1 (\varepsilon), 
    \sqrt{4\nu d(\varepsilon)}
 \delta_2 (\varepsilon)\},
$$
which is independent of $\lambda \geq 1$, such that if $| y | < \delta$, then
\begin{equation} \label{eqconwl}
    \| \partial_x^\gamma \omega_\lambda (1, \cdot - y)
     - \partial_x^\gamma \omega_\lambda (1, \cdot)\|_p < \frac{\varepsilon}{2} 
     + \frac{\varepsilon}{2} = \varepsilon.
\end{equation}
Therefore, the equicontinuity of $\{\partial_x^\gamma   \omega_\lambda (1,
x)\}_{\lambda \geq 1}$ in $L^p (\mathbb{R}^2)$ is proved.

By the Riesz criterion, we have proved the statement
\eqref{critern} and consequently $\{\omega_\lambda (1, x) \}_{\lambda \geq
1}$ is a precompact set in the space $W^{|\beta|, p} (\mathbb{R}^2)$
for all $\beta \in Z^{+} \times Z^{+}$ and for $p \in [1, \infty)$.
For $p = \infty$, by using Ascoli-Arzel\`{a} Theorem, we can also
prove that $\{\omega_\lambda (1, x) \}_{\lambda \geq 1}$ is a precompact set
in the space $W^{|\beta|, \infty} (\mathbb{R}^2)$ for all $\beta \in
Z^{+} \times Z^{+}$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{Thm3}]
By the result shown in Lemma \ref{Lm: compac}, for $p \in [1,
\infty]$, and for any $\beta \in Z^{+} \times Z^{+}, \{\omega_\lambda (1,
x) \}_{\lambda \geq 1}$ is precompact in $W^{|\beta|, p}
(\mathbb{R}^2)$. Thus there exist a sequence $\{\omega_{\lambda_\ell} (1,
x) \}_{\ell = 1}^{\infty}$ of $\{\omega_\lambda (1, x) \}_{\lambda \geq 1}$,
where $\lambda_\ell < \lambda_{\ell + 1} \to \infty$ as $\ell \to \infty$,
and a function $h(x) \in W^{|\beta|, p} (\mathbb{R}^2)$ such that
\begin{equation} \label{lmh}
    \lim_{\ell \to \infty} \omega_{\lambda_\ell} (1, \cdot) = h(\cdot)
\quad \text{in }  W^{|\beta|, p} (\mathbb{R}^2).
\end{equation}
It has been shown (as mentioned in Section 1) that
\begin{equation*}
    \lim_{t \to \infty} t  ^{1 - \frac{1}{p}} \| \omega (t, \cdot) 
    - \omega^{*} (t, \cdot) \|_p = 0,
\end{equation*}
which implies
\begin{equation} \label{wlim}
    \lim_{\ell \to \infty} \| \omega_{\lambda_\ell} (1, \cdot) 
    - \omega^{*} (1, \cdot) \|_p = 0
\end{equation}
due to the equivalence of \eqref{wconv} and \eqref{eqvl} here for
$\alpha = 0, |\beta| = 0$. By the uniqueness of limit in $L^p
(\mathbb{R}^2)$,  \eqref{lmh} and \eqref{wlim} imply that
\begin{equation} \label{hlim}
    h(x) = \omega^{*} (1, x) \quad \text{in }  L^p (\mathbb{R}^2)
\text{ and also in } W^{|\beta|, p} (\mathbb{R}^2).
\end{equation}
 From \eqref{lmh} and \eqref{hlim} it follows that
\begin{equation} \label{wconvx}
    \lim_{\ell \to \infty} \| \partial_x^\beta \omega_{\lambda_\ell} (1, \cdot)
- \partial_x^\beta \omega^{*} (1, \cdot) \|_p = 0.
\end{equation}
Since any sequence itself of $\{\omega_\lambda (1, \cdot) \}_{\lambda \geq
1}$ contains a subsequence convergent to the same limit $\omega^{*}
(1, \cdot)$ in $W^{|\beta|, p} (\mathbb{R}^2)$, we have proved that
\eqref{eqvl} holds for $\alpha = 0$, namely,
\begin{equation} \label{wconvgl}
    \lim_{\lambda \to \infty} \| \partial_x^\beta \omega_\lambda (1, \cdot)
 - \partial_x^\beta \omega^{*} (1, \cdot) \|_p = 0  \text{ for any }
 \beta \in Z^{+} \times Z^{+}, \; p \in [1, \infty].
\end{equation}
Note that \eqref{wconvgl} is stronger than the sequence limit \eqref{wconvx}.

Now we take the bootstrap approach to prove \eqref{eqvl} and equivalently
 \eqref{wconv} for any $\alpha \geq 1$, based on the result \eqref{wconvgl}. 
 Suppose that  \eqref{eqvl} is valid for an $\alpha \geq 0$ with any 
 $\beta \in Z^{+} \times Z^{+}$ and $p \in [1, \infty]$. From \eqref{Veqn} 
 we have
\begin{gather*}
    \partial_t^{\alpha + 1} \partial_x^\beta \omega_\lambda 
    = \partial_t^{\alpha} \partial_x^\beta
 \Delta \omega_\lambda - \partial_t^{\alpha} \partial_x^\beta (u_\lambda \cdot \nabla )
 \omega_\lambda,
\\
    \partial_t^{\alpha + 1} \partial_x^\beta \omega^{*} = \partial_t^{\alpha} 
    \partial_x^\beta \Delta \omega^{*} - \partial_t^{\alpha} 
    \partial_x^\beta (u^{*} \cdot \nabla )\omega^{*}.
\end{gather*}
Consequently,
\begin{equation} \label{txab}
    \begin{split}
    &\| \partial_t^{\alpha + 1}  \partial_x^\beta \omega_\lambda(1, \cdot)
  - \partial_t^{\alpha + 1} \partial_x^\beta \omega^{*} (1, \cdot) \|_p \\
    & \leq \| \partial_t^{\alpha} \partial_x^\beta (\Delta \omega_\lambda(1, \cdot)
  - \Delta \omega^{*} (1, \cdot)) \|_p + \| \partial_t^{\alpha} 
  \partial_x^\beta (v_\lambda(1, \cdot)
  - v^{*} (1, \cdot)) \|_p
    \end{split}
\end{equation}
where $v_\lambda (t, x) = (u_\lambda (t, x) \cdot \nabla)\omega_\lambda (t, x)$ 
and $v^{*} (t, x) = (u^{*} (t, x) \cdot \nabla)\omega^{*} (t, x)$. 
By the assumption of induction, we readily have
\begin{equation} \label{Tm1}
    \lim_{\lambda \to \infty} \|\partial_t^\alpha \partial_x^\beta 
    (\Delta \omega_\lambda (1, \cdot)
- \Delta \omega^{*} (1, \cdot)) \|_p = 0.
\end{equation}
Moreover,
\begin{equation} \label{Tm2}
    \begin{split}
     &\| \partial_t^{\alpha} \partial_x^\beta (v_\lambda(1, \cdot) - v^{*} (1, \cdot)) 
     \|_p \\
     & \leq \sum_{\theta \leq \alpha}   \sum_{\gamma \leq \beta}   
     \binom{\alpha}{\theta} \binom{\beta}{\gamma} ( \| \partial_t^\theta 
     \partial_x^\gamma (u_\lambda (1, \cdot) 
     - u^{*} (1, \cdot)) \|_{\infty} \| \partial_t^{\alpha - \theta} 
     \partial_x^{\beta - \gamma} \nabla \omega_\lambda (1, \cdot) \|_p \\
     &\quad  +  \| \partial_t^\theta \partial_x^\gamma u^{*} (1, \cdot) \|_{\infty} \|
      \partial_t^{\alpha - \theta} \partial_x^{\beta - \gamma} \nabla 
      (\omega_\lambda (1, \cdot) - \omega^{*} (1, \cdot)) \|_p ).
    \end{split}
\end{equation}
Since $\| \partial_t^\theta \partial_x^\gamma u^{*} (1, \cdot) \|_\infty$ is a
constant and by the assumption of induction we obtain
\begin{equation*}
    \lim_{\lambda \to \infty} \|\partial_t^{\alpha - \theta} \partial_x^{\beta
 - \gamma} \nabla (\Delta \omega_\lambda (1, \cdot) - \Delta \omega^{*} (1, \cdot)) \|_p
  = 0\,.
\end{equation*}
It follows that
\begin{equation} \label{Tm2sec}
    \lim_{\lambda \to \infty}   \sum_{\theta \leq \alpha} \sum_{\gamma
\leq \beta} \binom{\alpha}{\theta} \binom{\beta}{\gamma} \| \partial_t^\theta
\partial_x^\gamma u^{*} (1, \cdot) \|_{\infty} \| \partial_t^{\alpha - \theta}
\partial_x^{\beta - \gamma} \nabla (\omega_\lambda (1, \cdot) - \omega^{*} (1, \cdot))
 \|_p = 0\,.
\end{equation}
For another part in \eqref{Tm2}, note that \eqref{Nw} and \eqref{sclp} yield
\begin{equation} \label{wbdd}
    \| \partial_t^{\alpha - \theta} \partial_x^{\beta - \gamma} \nabla 
    \omega_\lambda (1, \cdot) \|_p \leq N_\omega (\alpha - \theta, |\beta - \gamma| +1,
     p) \|\omega_0\|.
\end{equation}
The rest issue is the convergence of
$\| \partial_t^\theta \partial_x^\gamma (u_\lambda (1, \cdot) - u^{*}
 (1, \cdot)) \|_{\infty}$.
By the Sobolev imbedding theorem, there is a uniform constant
$C_0 > 0$ such that $\| \varphi \|_\infty \leq C_0 \| \varphi
\|_{H^2}$, where $H^2$ stands for the Sobolev space $H^2
(\mathbb{R}^2)$. By the Biot-Savart law, we get
\begin{equation} \label{upn}
    \begin{split}
    &\|  \partial_t^\theta \partial_x^\gamma (u_\lambda (1, \cdot) 
    - u^{*} (1, \cdot))  \|_{\infty} \\[5pt]
&\leq C_0 \| \partial_t^\theta \partial_x^\gamma (u_\lambda (1, \cdot) - u^{*} (1, 
\cdot)) \|_{H^2} \\[5pt]
    &  \leq C_0 \sum_{|\rho| \leq 2} \| \partial_t^\theta 
    \partial_x^{\gamma + \rho} (u_\lambda (1, \cdot) - u^{*} (1, \cdot)) \|_2  \\
    & = \frac{C_0}{2\pi} \sum_{|\rho| \leq 2}
 \big\|   \int_{\mathbb{R}^2}  \partial_t^\theta \partial_x^{\gamma + \rho}
 \frac{(x - y)^{\bot}}{|x - y|^2}   (\omega_\lambda (1, y) - \omega^{*} (1, y))
 \,dy   \big\|_2 \\
&\text{(by a variable changes as in \eqref{linpt})} \\
& = \frac{C_0}{2\pi} \sum_{|\rho| \leq 2} \big\|   \int_{\mathbb{R}^2}
  \frac{(x - y)^{\bot}}{|x - y|^2}    \partial_t^\theta \partial_y^{\gamma
  + \rho}  (\omega_\lambda (1, y) - \omega^{*} (1, y)) \,dy   \big\|_2 \\
& \leq \frac{C_0}{2\pi} \sum_{|\rho| \leq 2} \big\|   \int_{\mathbb{R}^2}
  \frac{1}{|x - y|}   \left| \partial_t^\theta \partial_y^{\gamma + \rho}
  (\omega_\lambda (1, y) - \omega^{*} (1, y)) \right| dy   \big\|_2 \\
& \leq \frac{C_0}{2\pi} \sum_{|\rho| \leq 2} S_{2, 1, 2}   \|
  \partial_t^\theta \partial_x^{\gamma + \rho}  (\omega_\lambda (1, x) - \omega^{*} (1, x))
  \| \longrightarrow 0, \quad \text{as }  \lambda \to \infty,
\end{split}
\end{equation}
by the assumption of induction and the Hardy-Littlewood-Sobolev inequality 
\eqref{HLS} here with $n = 2, a = 2, p = 1, q = 2$. Thus we can combine 
\eqref{wbdd} and \eqref{upn} to get
\begin{equation} \label{Tm2fir}
    \lim_{\lambda \to \infty}   \sum_{\theta \leq \alpha} \sum_{\gamma \leq \beta}
\binom{\alpha}{\theta} \binom{\beta}{\gamma} \| \partial_t^\theta \partial_x^\gamma
(u_\lambda (1, \cdot) - u^{*} (1, \cdot)) \|_{\infty} \| \partial_t^{\alpha - \theta}
 \partial_x^{\beta - \gamma} \nabla \omega_\lambda (1, \cdot) \|_p = 0.
\end{equation}
Finally, substituting \eqref{Tm2sec} and \eqref{Tm2fir} into \eqref{Tm2},
we obtain
\begin{equation} \label{Tm2res}
    \lim_{\lambda \to \infty}   \| \partial_t^{\alpha} \partial_x^\beta 
    (v_\lambda(1, \cdot) - v^{*} (1, \cdot)) \|_p  = 0\,.
\end{equation}
Then \eqref{txab}, \eqref{Tm1} and \eqref{Tm2res} altogether imply that
\eqref{eqvl} holds for $\alpha + 1$; i.e.,
$$
    \lim_{\lambda \to \infty}   \| \partial_t^{\alpha + 1} \partial_x^{\beta} 
    (\omega_\lambda (1, \cdot) - \omega^{*} (1, \cdot)) \|_p = 0, \quad \beta \in Z^{+} 
    \times Z^{+},\; p \in [1, \infty],
$$
and equivalently \eqref{wconv} holds for $\alpha + 1$ and any
$ \beta \in Z^{+} \times Z^{+}, \; p \in [1, \infty]$.
This bootstrap induction shows that the convergence statement
\eqref{wconv} is valid.

By the Biot-Savart law and utilizing the corresponding equivalence
of \eqref{uconv} and the following statement,
$$
    \lim_{\lambda \to \infty}   \| \partial_t^{\alpha} 
    \partial_x^{\beta} (u_\lambda (1, \cdot)
 - u^{*} (1, \cdot)) \|_q = 0, \quad \alpha \geq 0, \; \beta \in Z^{+} \times Z^{+}, \;
  q \in (2, \infty],
$$
we can also prove that \eqref{uconv} is valid. The detail is omitted.
\end{proof}

\textbf{Remark.}  For the three-dimensional Navier-Stokes equation
\cite[Chapter 6]{SY02}, the global existence and regularity of
strong  solutions is an extremely challenging open problem, that
is defined by the Clay Mathematics Institute as one of the century
problems in mathematics for the new millennium. Nevertheless, the
study of 3D vorticity plays an important role. In this regard, one
of the major differences between 2D and 3D vorticity behavior
arises from the fact that the 3D vorticity is governed by a
different evolutionary equation:
\begin{equation} \label{3dvorty}
    \frac{\partial \omega}{\partial t} = \nu \Delta \omega - (u \cdot \nabla)\omega
+ (\omega \cdot \nabla)u, \quad t > 0, \; x \in \mathbb{R}^3,
\end{equation}
where the additional nonlinear term $(\omega \cdot \nabla)u$ known
as the vorticity stretching amplifies the vorticity strength by the
interaction with the gradient of velocity field. Therefore, the dynamics,
regularity, and asymptotics of the three-dimensional vorticity evolution
are much more complicated than the two-dimensional case.

The extension of Oseen vortices in the three-dimensional case is
Burgers vortices. A Burgers vortex is an exact solution of the
3D Navier-Stokes equation that is a superposition of a background
irrotational strain field/flow and a 2D vortex, just like the Oseen vortex,
in the plane perpendicular to the strain axis. In recent years some
results from the ongoing researches are emerging in regard to the local
stability and asymptotic structures of the 3D Burgers vortices.
In \cite{GW06} it has been shown that the whole family of axisymmetric
and non-axisymmetric Burgers vortex solutions parametrized by the total
circulation numbers of the 3D Navier-Stokes equation is locally stable
for small Reynolds numbers. Based on several numerical simulation results
of turbulent fluid flows shown in \cite{SR04} and others, very recently
in \cite{yM09} the existence of Burgers vortices as the swirling,
tube-like structures in 3D turbulence is rigorously proved for high
Reynolds numbers with asymmetric parameters less than one half. However,
the stability of the 3D Burgers vortices for high Reynolds numbers is
still not known even though some numerical studies indicate some kind
of tendency of persistence possibly toward stability.

\begin{thebibliography}{99}

\bibitem{rA75}
    R. S. Adams,
    \emph{Sobolev Spaces},
    Academic Press, New York, 1975.
\bibitem{ABD05}
    M. Arnold, Y. Bakhtin, E. Dinaburg,
    \emph{Regularity of solutions to vorticity Navier-Stokes systems on 
    $\mathbb{R}^2$},
    Comm. Math. Phys. \textbf{258} (2005), 339--348.
\bibitem{BJ05}
    H. O. Bae and B. J. Jin,
    \emph{Upper and lower bounds of temporal and spatial decays for the 
    Navier-Stokes equations},
    J. Diff. Eqns. \textbf{209} (2005), 365--391.
\bibitem{BdV07}
    H. Beir\~{a}o da Veiga,
    \emph{Vorticity and regularity for solutions of initial-boundary value 
    problems for the Navier-Stokes equations},
    Hyperbolic problems and regularity questions, 39--47, 
    Trends Math., Birkh\"{a}user, Basel, 2007.
\bibitem{BA94}
    M. Ben-Artzi,
    \emph{Global solutions of two-dimensional Navier-Stokes and Euler equations},
    Arch. Rational Mech. Anal. \textbf{128} (1994), 329--358.
\bibitem{BC93}
    A. L.  Bertozzi and P. Constantin,
    \emph{Global regularity for vortex patches},
    Comm. Math. Phys. \textbf{152} (1993), 19--28.
\bibitem{lB04}
    L. Brandolese,
    \emph{Space-time decay of Navier-Stokes flows invariant under rotations},
    Math. Ann. \textbf{329} (2004), 685--706.
\bibitem{hB87}
    H. Brezis,
    \emph{Analyse Fonctionnelle: Th\'{e}orie    et Applications} (second edition),
    Masson, Paris, 1987.
\bibitem{CL96}
    E. A. Carlen and M. Loss,
    \emph{Optimal smoothing and decay estimates for viscously damped 
    conservation laws, with applications to the 2-D Navier-Stokes equation},
    Duke Math. J. \textbf{81} (1996), 135--157.
\bibitem{CP96}
    M. Cannone and F. Planchon,
    \emph{Self-similar solutions for Navier-Stokes equations in} $\mathbb{R}^3 $,
    Comm. Partial Diff. Eqns. \textbf{21} (1996), 179--193.
\bibitem{aC94}
    A. Carpio,
    \emph{Asymptotic behavior for the vorticity equations in dimensions two and 
    three},
    Comm. Partial Diff. Eqns. \textbf{19} (1994), 827--872.
\bibitem{DS84}
    E. B. Davies and B. Simon,
    \emph{Ultracontractivity and the heat kernel for Schr\"{o}dinger semigroups},
    J. Funct. Anal. \textbf{59} (1984), 335--395.
\bibitem{FS96}
    R, Farwig and H. Sohr,
    \emph{Global estimates in weighted spaces of weak solutions of the 
    Navier-Stokes equations in exterior domains},
    Arch. Math. \textbf{67} (1996), 319--330.
\bibitem{bF04}
    B. Franke,
    \emph{About the decay of vorticity in a two-dimenional incompressible 
    viscous fluid},
    Semigroup Forum, \textbf{69} (2004), 303--316.
\bibitem{gG94}
    G. Galdi,
    \emph{An Introduction to the Mathematical Theory of the Navier-Stokes 
    Equations II},
    Springer, Berlin, 1994.
\bibitem{GW05}
    T. Gallay and C. E. Wayne,
    \emph{Global stability of vortex solutions of the two-dimensional
     Navier-Stokes equation},
    Comm. Math. Phys.   \textbf{255} (2005), 97--129.
\bibitem{GW06}
    T. Gallay and C. E. Wayne,
    \emph{Three-dimensional stability of Burgers vortices: The low Reynolds 
    number case},
    Physica D: Nonlinear Phenomena, \textbf{213} (2006), 164--180.
\bibitem{GK88}
    Y. Giga and T. Kambe,
    \emph{Large time behavior of the vorticity of two-dimensional viscous 
    flow and its applications to vortex formation},
    Comm. Math. Phys. \textbf{117} (1988), 549--568
\bibitem{GS03}
    Y. Giga and O. Sawada,
    \emph{On regularizing decay rate estimates for solutions to the 
    Navier-Stokes initial value problem},
    Nonlinear Analysis and Applications: to V. Lakshmikantham on his 80th 
    birthday,     Kluwer Acad. Publ., Dordecht, 2003, Vol. 1-2, 549--562.
\bibitem{lG76}
    L. Gross,
    \emph{Logarithmic Sobolev inequalities},
    Amer. J. Math. \textbf{97} (1976), 1061--1083.
\bibitem{lH90}
    L. H\"{o}rmander,
    \emph{The Analysis of Linear Partial Differential Operators I}, 
    (second Edition),  Springer-Verlag, Berlin, 1990.
\bibitem{cH01}
    C. Huang,
    \emph{Singular integral system approach to regularity of 3D vortex patches},
    Indiana Univ. Math. J. \textbf{50} (2001), 509--552.
\bibitem{cH06}
    C. Huang,
    \emph{Global smooth nonconstant vortex patches in bounded domains},
    Far East J. Math. Sci. \textbf{22} (2006), 227--238.
\bibitem{tK94}
    T. Kato,
    \emph{The Navier-Stokes equation for an incompressible fluid in 
    $\mathbb{R}^2$ with a measure as the initial vorticity},
    Diff. Integral Eqns. \textbf{7} (1994), 949--966.
\bibitem{KR05}
    H. Kwean and J. Roh,
    \emph{The global attractor of the 2D $g$-Navier-Stokes equations on 
    some unbounded domains},
    Comm. Korean Math. Soc. \textbf{20} (2005), 731--749.
\bibitem{yM09}
    Y. Maekawa,
    \emph{On the existence of Burgers Vortices for high Reynolds numbers},
    J. Math. Anal. Appl. \textbf{349} (2009), 181--200.
\bibitem{aM86}
    A. J. Majda,
    \emph{Vorticity and the mathematical theory of incompressive fluid flow},
    Comm. Pure Appl. Math. \textbf{39} (1986), 187--220.
\bibitem{MB02}
    A. J. Majda and A.L. Bertozzi,
    \emph{Vorticity and Incompressible Flow},
    Cambridge Univ Press, Cambridge, 2002.
\bibitem{MP94}
    C. Marchioro and M. Pulvirenti,
    \emph{Mathematical Theory of Incompressible Nonviscous Fluids},
    Springer, New York, 1994.
\bibitem{Met91}
    W. H. Matthaeus, W. T. Stribling, D. Martinez, S. Oughton, and D. Montgomery,
    \emph{Selective decay and coherent vortices in two-dimensional 
    incompressible turbulence},
    Phys. Rev. Lett. \textbf{66} (1991), 2731--2734.
\bibitem{cO11}
    C. W. Oseen,
    \emph{\"{U}ber Wirbelbewegung in einer reibenden Fl\"{u}ssigkeit},
    Arkiv Mat. Astr. Fys. \textbf{7} (1911), 1--13.
\bibitem{RGE02}
    L. Rossi and J. Graham-Eagle,
    \emph{On the existence of two-dimensional, localized, rotating, 
    self-similar vortical structures},
    SIAM J. Appl. Math. \textbf{62} (2002), 2112--2128.
\bibitem{SS00}
    M. E. Schonbek and T. Schonbek,
    \emph{On the boundedness and decay of the moments of solutions of 
    the Navier-Stokes equations},
    Adv. Diff. Eqns. \textbf{5} (2000), 861--898.
\bibitem{SY02}
        G. R. Sell and Y. You,
        \emph{Dynamics of Evolutionary Equations},
        Springer, New York, 2002.
\bibitem{SR04}
    P. J. Schmid and M. Rossi,
    \emph{Three-dimensional stability of a Burgers vortex},
    J. Fluid Mech. \textbf{500} (2004), 103--112.
\bibitem{sT99}
    S. Takahashi,
    \emph{A weighted equations approach to decay rate estimates for the 
    Navier-Stokes equations},
    Nonlinear Analysis \textbf{27} (1999), 751--789.
\bibitem{aT88}
    A. Torchinsky,
    \emph{Real Variables},
    Addison-Wesley, Redwood City, 1988.
\bibitem{WB98}
    Th. P. Witelski and A. J. Bernoff,
    \emph{Self-similar asymptotics for linear and nonlinear diffusion equations},
    Studies in Appl. Math. \textbf{100} (1998), 153--193.
\end{thebibliography}

\end{document}