text:
\section{Introduction}
This paper studies the stationary Navier--Stokes equations in the whole plane $\mathbb{R}^2$ driven by a force term
\begin{equation}  \label{NSE}
\left\{
\begin{aligned}
 & \Delta \mathbf{w} - (\mathbf{w} \cdot \nabla) \mathbf{w} - \nabla p = \mathbf{f}, \\
 & \nabla \cdot \mathbf{w} = 0, \\
 & \mathbf{w}(z) \to\mathbf{w}_\infty=\lambda \mathbf{e}_1 \ \ \text{as}\ \  |z| \to \infty
\end{aligned}
\right.
\end{equation}
where $\lambda\in\mathbb{R}_{\ge 0}$. We shall assume that the force is of compact support, i.e., $\mathrm{supp}\, \mathbf{f} \subseteq B_R$ for some $R > 0$.  We also assume a minimal regularity of $\mathbf{f}$, that is, $\mathbf{f} \in H^{-1}(B_{2R})$. By definition, $H^{-1}(B_{2R})$ is the dual space of $\dot{H}_0^1(B_{2R})$, see Section \ref{sec-not}.

When the total force $\mathcal{F} = \int_{\mathbb{R}^2} \mathbf{f} = 0$, one can find a rank-2 tensor $\mathbb{F} \in L^2(\mathbb{R}^2)$ such that $\mathbf{f} = \nabla \cdot \mathbb{F}$ (see Lemma \ref{lem-tensorF}). In this case, the existence of a family of $D$-solutions to \eqref{NSE}$_{1,2}$, parametrized by the average velocity in the unit disk $\Xint-_{B_1} \mathbf{w}$, is proved in \cite{GuilW2} based on a \emph{modified} invading domains method. Although this approach gives \emph{infinitely many} distinct solutions, the asymptotic behaviour of the solutions at spatial infinity is not clear. The solvability of \eqref{NSE} with arbitrary data $\mathbf{f}$ and $\lambda$ is an~important open problem.

To study the above problem, we consider Leray's original invading domains method, which was introduced in the celebrated 1933 paper \cite{Leray}. Namely, we first solve the Navier-Stokes equations in the disks $B_{R_k} (k =1,2,3,\cdots) $ with an increasing sequence of radii $R_k \to \infty$:
\begin{equation}  \label{NSE-k}
\left\{
\begin{aligned}
 & \Delta \mathbf{w}_k - (\mathbf{w}_k \cdot \nabla) \mathbf{w}_k - \nabla p = \mathbf{f} \quad \mathrm{in} \ B_{R_k},\\
 & \nabla \cdot \mathbf{w}_k = 0 \quad \mathrm{in} \ B_{R_k},\\
 & \mathbf{w}_k(z) = \mathbf{w}_\infty=\lambda \mathbf{e}_1 \ \mathrm{on} \ S_{R_k}.
\end{aligned}
\right.
\end{equation}
Existence of $D$-solutions $\mathbf{w}_k$ to the above system is a standard result. Heuristically, taking the limit $\mathbf{w}_k$ as $k \to \infty$ may produce a solution of \eqref{NSE}. However, several difficulties show up when one tries to adopt this approach.

\begin{enumerate}
  \item[(1)] One needs to prove a uniform bound (independent of $k$) for the Dirichlet integrals $D_k : = \int_{{B_{R_k}}} |\nabla \mathbf{w}_k|^2$.

In the case when $\mathcal{F} = 0$, and $\mathbf{f} = \nabla \cdot \mathbb{F}$ for some tensor $\mathbb{F} \in L^2(\mathbb{R}^2)$, we can test \eqref{NSE-k}$_{1}$ with $\mathbf{w}_k - \lambda \mathbf{e}_1$ and integrate by parts to get the energy equality
\begin{equation}
\int_{B_{R_k}} |\nabla \mathbf{w}_k|^2 = \int_{B_{R_k}} \mathbb{F} : \nabla \mathbf{w}_k
\end{equation}
where $\mathbb{F} : \nabla \mathbf{w}_k$ stands for $\sum_{i,j} \mathbb{F}_{ij} \partial_j w_{k,i}$. Using H\"older's inequality we get a nice bound on the Dirichlet integral
\begin{equation*}
\int_{B_{R_k}} |\nabla \mathbf{w}_k|^2 \le \int_{\mathbb{R}^2} |\mathbb{F}|^2.
\end{equation*}
However, for a general $\mathbf{f}$, such a direct estimate is unavailable.

\item[(2)] In addition to (1), one needs to prove the uniform boundedness of local norms of $\mathbf{w}_k$, \emph{e.g.}, $\|\mathbf{w}_k\|_{L^4(B_1)}$. Without such a bound, the weak limits of $\mathbf{w}_k$ are not well-defined.

\item[(3)] On the basis of (1) and (2), one may define $\mathbf{w}_L$ as the weak limit of some subsequence of $\mathbf{w}_k$. Then one still needs to study the asymptotic behaviour of $\mathbf{w}_L$ to answer the crucial question: do we have
\begin{equation}\label{des-eq}
\lim_{|z| \to \infty} \mathbf{w}_L(z) = \mathbf{w}_\infty ?
\end{equation}
\end{enumerate}

In the present paper we give complete solution for the first two problems~(1)--(2), and a partial solution to the third one. Namely, we prove that the uniform estimate $\|{\mathbf w}_k\|_{L^q(B_1)}+\|\nabla{\mathbf w}_k\|_{L^2(B_{R_k})}\le C$ is valid in general, therefore, the Leray solutions~${\mathbf w}_L$ are always well-defined (see Theorem~\ref{th-uniform-bound}). Also we justify the limiting equality~(\ref{des-eq}) if {\it at least one} of two additional assumption is fulfilled:
\begin{enumerate}
\item[(i)] the norm $\|{\mathbf f}\|_{H^{-1}(B_{2R})}$ is small enough with respect to~$\lambda$  (see Theorem~\ref{th-s1});

\item[(ii)] the drag force and the limiting velocity are zero: $\mathcal{F} = 0$, \,$\lambda=0$ (see Theorem~\ref{th-s2}).
\end{enumerate}

Finally, we prove the uniqueness of $D$-solutions to~(\ref{NSE})  under the assumption that the norm $\|{\mathbf f}\|_{H^{-1}(B_{2R})}$ is small enough with respect to~$\lambda$ (see
Theorem~\ref{th-uniq}). We emphasize here that the smallness conditions in Theorem~\ref{th-s1} and Theorem~\ref{th-uniq} are very different in nature.

The main tools for our research here are two general estimates of the difference between mean values of the velocity over two circles in terms of the Dirichlet integral between these two circles (see Theorems~\ref{th:estim1} and \ref{th:estim2}). These estimates are based on the technique of the~recent papers \cite{KPR20}, \cite{KR21new}. They are rather surprising since, as well known, it is not possible in general to control the uniform norm of a~function through its Dirichlet integral in large domains\footnote{For instance, the function $f(z) = \left(\ln(|z|)\right)^\frac13$ has a finite Dirichlet integral in $\mathbb{R}^2 \setminus B_2$ but diverges at infinity.}. Nevertheless, the~special structure of the Navier--Stokes system gives the possibility to derive such very general estimates, which turns out to be rather simple. Note, that the second (asymptotic) estimate is based on the  recent results from \cite{KR21new} concerning solutions to the Euler system in the~unit disk with constant vectors on the boundary (see Section~\ref{sec:3} of the present paper).

We emphasize, that all the estimates and conditions in  the main  theorems~\ref{th:estim1},\,\ref{th:estim2},\,\ref{th-uniform-bound}, \,\ref{th-s1}--\ref{th-s2},\,\ref{th-uniq} are scaling invariant.

Note, in conclusion, that the problem on solutions to steady Navier--Stokes system in exterior plane domain, due to the relative simplicity of its formulation, and the depth and elegance of the mathematical ideas hidden in it, has long attracted the attention of numerous researchers. In addition to the well-known classical works (\emph{e.g.},  \cite{Amick,F,FS2,FS,Fujita,GW1,GW,L,O,Sm}\,), there are also a number of recent papers (\emph{e.g.}, \cite{GaldiS,G,G2,JG,GuilW1,GuilW2,HW, KPR2, Nakat, RussoPG,S,Sazonov03, Y,Yam11,Yam18}).

\section{Preliminaries}

\subsection{Notations} \label{sec-not}
\emph{Throughout the paper, $\mathbf{w}_k$ will be the invading domain solutions to \eqref{NSE-k} for some sequence of radii $R_k \to \infty$.}

 We use $C$ to denote constants that are independent of $k, R,\lambda,\mathbf{f}$, etc. The exact values of $C$ may change from line to line.

We always assume that $\mathrm{supp} \, \mathbf{f} \subset  B_R$ and $\mathbf{f} \in H^{-1}(B_{2R})$. Definition and some properties of the $H^{-1}$ space are summarized below.

\subsection{Function spaces} \label{sec-fs}

We use standard notations for Sobolev spaces $H^1(\Omega)=W^{1,2}(\Omega)$. Let $\dot{H}_0^1(\Omega)$ be the completion of $C_c^\infty(\Omega)$ in $H^1(\Omega)$ as usual, with norm given by
\begin{equation}\label{sob1}
\|\varphi\|_{\dot{H}_0^1(\Omega)}^2 = \|\nabla \varphi\|^2_{L^2(\Omega)}.
\end{equation}
We recall the following elementary fact:
\begin{lemma}
For any function $\varphi\in H^1(B_R)$ and for every $q\in(1,\infty)$ the inequality
\begin{equation}\label{scale3}
\|\varphi\|_{L^q(B_R)}\le C \biggl(|\bar \varphi(R)| +\|\nabla \varphi\|_{L^2(B_R)}\biggr)R^{2/q}
\end{equation}
holds, where $\bar{\varphi}(R)$ is the average of $\varphi$ on the circle $S_R$, and $C$ is a constant depending only on $q$.
\end{lemma}

\begin{proof}
By scaling, the statement can be reduced to the case $R=1$. Then, the estimate follows from \cite[Theorem~1.1.16]{maz'ya}.
\end{proof}

The $H^{-1}$ norm of a (scalar valued) distribution $\psi$ in $B_{2R}$ is defined as
\begin{equation}
\|\mathbf{\psi}\|_{H^{-1}(B_{2R})} := \sup_{\varphi \in C_0^\infty(B_{2R}), \, \|\varphi\|_{\dot{H}_0^1(B_{2R}) } = 1 } \langle \psi, \varphi \rangle.
\end{equation}
Let $\sigma>0$. It is easy to check, that for the function $\varphi_\sigma(z):=\sigma^3\varphi(\sigma z)$ we have the following scaling identity:
\begin{equation}\label{scale1}
\|\varphi_\sigma\|_{H^{-1}(B_{2R/\sigma})}=\sigma\|\varphi\|_{H^{-1}(B_{2R})}.
\end{equation}

Let $\chi$ be a smooth cut-off function satisfying $\chi \equiv 1 \mathrm{\ on\ } B_R$ and $\mathrm{supp} \, \chi \subset B_{2R}$, and define the \emph{total force} $\mathcal{F} := \langle \mathbf{f}, \chi \rangle$. Since $\chi$ can be extended up to the function from $C^\infty_0(B_{2R})$ satisfying $|\nabla\chi|\le C\frac1R$, clearly, we have
\begin{equation}\label{scale2}
|\mathcal{F}| \le C \|\mathbf{f}\|_{H^{-1}(B_{2R})},
\end{equation}
where $C$ is some universal constant (does not depend on~$R$).
For simplicity, we will formally write $\mathcal{F} = \int_{\mathbb{R}^2} \mathbf{f}$ although it is possible that $\mathbf{f} \notin L^1_{\mathrm{loc}}$. Next, we state a result in the spirit of \cite[Lemma 3.6]{GuilW2}:
\begin{lemma} \label{lem-tensorF}
Under the above notations, if $\mathcal{F} = 0$, then there exists a tensor $\mathbb{F} \in L^2(\mathbb{R}^2)$ such that $\mathrm{supp}\, \mathbb{F} \subset B_{2R}$, $\mathbf{f} = \nabla \cdot \mathbb{F}$ on $\mathbb{R}^2$, and
$$\|\mathbb{F}\|_{L^2} \le C \|\mathbf{f}\|_{H^{-1}(B_{2R})}$$
with $C$ independent of $R$.
\end{lemma}
\begin{proof}
Let us define $\mathbb{F}$ to be $\nabla \mathbf{g}$ with $\mathbf{g} \in H^1({B_{2R}})$ solving
\begin{equation*}
\left\{
\begin{aligned}
 & \Delta \mathbf{g} = \mathbf{f},\quad \mathrm{in} \ \ B_{2R},  \\
 & \partial_n \mathbf{g} = 0, \quad \mathrm{on} \ \ \partial B_{2R}.
\end{aligned}
\right.
\end{equation*}
Such $\mathbf{g}$ exists uniquely up to adding constants since we assumed $\mathcal{F} = 0$. We extend $\mathbb{F}$ by $0$ outside $B_{2R}$. Then using $\partial_n \mathbf{g} = 0$ on $\partial B_{2R}$, one can check that $\nabla \cdot \mathbb{F} = \mathbf{f}$ holds in the sense of distributions not only on $B_{2R}$, but also on $\mathbb{R}^2$.
\end{proof}

The following change-of-domain lemma will be useful in Sections \ref{sec-limit} and \ref{sec-unique}.
\begin{lemma} \label{lem-change-of-domain}
Suppose that $\mathrm{supp} \, f \subset B_R$ and $f \in H^{-1}(B_{2R})$. Then for any $a \ge 2$,
\begin{equation}
	\|f\|_{H^{-1}(B_{aR})} \le C \left(1 + \left(\ln \frac{a}{2}\right)^\frac12 \right) \|f\|_{H^{-1}(B_{2R})}
\end{equation}
with $C$ independent of $a$ and $R$.
If in addition $\mathcal{F}=0$, then
\begin{equation*}
\|f\|_{H^{-1}(B_{aR})} \le C \|f\|_{H^{-1}(B_{2R})}.
\end{equation*}
\end{lemma}
\begin{proof}
Fix the cut-off function $\chi$ as above. For any $\varphi \in \dot{H}_0^1(B_{aR})$, we have $\varphi \chi \in \dot{H}_0^1(B_{2R})$ and $\langle f, \varphi \rangle = \langle f, \varphi \chi \rangle$. Notice that
\begin{align*}
\|\varphi \chi\|_{\dot{H}_0^1(B_{2R})} &\le  \|\chi \nabla \varphi\|_{L^2} + \|\varphi \nabla \chi\|_{L^2} \\
&\le \|\varphi\|_{\dot{H}_0^1(B_{aR})} + \frac{C}{R}\|\varphi\|_{L^2(B_{2R})} \\
&\overset{\eqref{scale3}}{\le}  \|\varphi\|_{\dot{H}_0^1(B_{aR})} + C  \bigg(|\bar \varphi(2R)| +\|\nabla \varphi\|_{L^2(B_{2R})} \bigg) \\
&\le C \|\varphi\|_{\dot{H}_0^1(B_{aR})} \left(1 + \left(\ln \frac{a}{2}\right)^\frac12 \right)
\end{align*}
In the last step we applied the estimate \eqref{eq-new2.9} below to control $|\bar \varphi(2R)| = |\bar \varphi(2R) - \bar \varphi(aR) |$.  Now, a standard duality argument proves the lemma.

If in addition $\mathcal{F} = 0$, then the corresponding claim follows from Lemma \ref{lem-tensorF}, and the fact that for any $a \ge 2$, we have
$$\|\nabla \cdot \mathbb{F}\|_{H^{-1}(B_{aR})} \le \|\mathbb{F}\|_{L^2}.$$
\end{proof}

\begin{lemma} \label{lem-new4}
For any $\phi \in H^1({\Omega}_{\rho_1, \rho_2})$, where $\Omega_{\rho_1, \rho_2} = \{z \in \mathbb{R}^2: 0 < \rho_1 <|z| < \rho_2\}$, we have
\begin{equation} \label{eq-new2.9}
|\bar{\phi}(\rho_2) - \bar{\phi}(\rho_1)| \le \frac{1}{\sqrt{2\pi}} \left(\int_{\Omega_{\rho_1, \rho_2} } |\nabla \phi|^2 \right)^\frac12 \cdot \left(\ln \frac{\rho_2}{\rho_1}\right)^\frac12
\end{equation}
\end{lemma}
For the proof of Lemma \ref{lem-new4}, see, \emph{e.g.}, \cite[Lemma 2.1]{KPR}.

\subsection{Properties of $D$-solutions}
We present two important lemmas for general $D$-solutions to the Navier--Stokes equations. They have been very useful in many previous studies on the Navier--Stokes exterior problem, see, for instance, \cite{GW, Amick, KPR, KPR20, KR21}. Lemma \ref{lem:pressure} was proved in \cite[Lemma 4.1]{GW}, and Lemma \ref{lem:angle} was proved in \cite[Theorem~4, page~399]{GW}.  Let $\mathbf{w}$ be a $D$-solution to the Navier--Stokes equations in some ring $\Omega_{r_1, r_2} = \{z \in \mathbb{R}^2: 0 < r_1 <|z| < r_2\}$, and $p$ be the corresponding pressure.

\begin{lemma}[\cite{GW}] \label{lem:pressure}
Denote by $\bar{p}(r)$ the average of $p$ over the circle $S_r$. Then for any $r_1 < \rho_1 \le \rho_2 < r_2$, we have
\begin{equation} \label{eq:pdifference}
|\bar{p}(\rho_2) - \bar{p}(\rho_1)| \le \frac{1}{4\pi} \int_{\Omega_{\rho_1, \rho_2}} |\nabla \mathbf{w}|^2  \ dxdy.
\end{equation}
\end{lemma}

Denote by $\bar{\mathbf{w}}(r)$ the average of $\mathbf{w}$ over the circle $S_r$. Note, that the direct estimates
for $\bar{\mathbf{w}}(r)$ are not so good:
\begin{equation} \label{eq2a}
|\bar{\mathbf{w}}(\rho_2) - \bar{\mathbf{w}}(\rho_1)| \le \frac{1}{\sqrt{2\pi}} \left(\int_{\Omega_{\rho_1, \rho_2} } |\nabla \mathbf{w}|^2 \right)^\frac12 \cdot \left(\ln \frac{\rho_2}{\rho_1}\right)^\frac12
\end{equation}
(see \eqref{eq-new2.9}). Nevertheless, the direction of $\bar{\mathbf{w}}(r)$ is still under
control of the Dirichlet integral:

\begin{lemma}[\cite{GW}] \label{lem:angle}
  Denote by $\bar{\mathbf{w}}$ the average of $\mathbf{w}$ over the circle $S_r$ and let $\varphi(r) \in [0,2\pi]$ (modulo $2\pi$) be the argument of the complex number associated with the vector $\bar{\mathbf{w}}(r) = (\bar{w}_1(r), \bar{w}_2(r))$, {\emph{i.e.}},
$\bar{\mathbf{w}}(r)=|\bar{\mathbf w}(r)|\,(\cos\varphi(r),\sin\varphi(r))$.  Assume also that $|\bar{\mathbf{w}}(r)| \ge \sigma > 0$ for some constant $\sigma$ and for all $r \in (r_1, r_2)$. Then for any $r_1 < \rho_1 \le \rho_2 < r_2$, we have
 \begin{equation} \label{eq:phidifference}
  |\varphi(\rho_2) - \varphi(\rho_1)| \le \frac{1}{4\pi \sigma^2} \int_{\Omega_{\rho_1, \rho_2}} \left( \frac{1}{r}|\nabla \omega| + |\nabla \mathbf{w}|^2 \right) \ dxdy.
 \end{equation}
 Here $\omega = \partial_2 w_1 - \partial_1 w_2$ is the vorticity of $\mathbf{w}$.
\end{lemma}

\section{The first basic estimate for the velocity}\label{sec-es1}
We have already mentioned, that the usual estimates for plane functions with finite Dirichlet integral are not efficient to compare the difference of mean values of the function over two circles whose radii are very different (see~(\ref{eq2a})\,). But the special structure of the plane
Navier--Stokes system allows to improve the issue. Here and in the next section we formulate and prove two important estimates for the velocity vector, which will be the main tool for future research in the paper.  Note, that they are valid not only for the solutions in the whole plane (considered above), but in much more general situations.

\begin{theorem}\label{th:estim1}
Let ${\mathbf w}$ be the $D$-solution to the Navier--Stokes system
 \begin{equation}  \label{NSE-gen}
\left\{\begin{aligned}
 & \Delta \mathbf{w} - (\mathbf{w} \cdot \nabla) \mathbf{w} - \nabla p = \mathbf{0}, \\
 & \nabla \cdot \mathbf{w} = 0
\end{aligned}
\right.
\end{equation}
 in the annulus type domain $\Omega_{r_1,r_2}=\{z\in{\mathbb R}^2:
 r_1\le|z|\le r_2\}$. Then
 \begin{equation} \label{eq:estim-m1}
 |\bar{\mathbf w}(r_1)-\bar{\mathbf w}(r_2)|\le C_*(1+\mu)\sqrt{D(r_1,r_2)},
 \end{equation}
 where
 \begin{equation} \label{eq:estim-m2}
\mu=\frac1{r_1{m}},\ \ \quad {m}:=\max\bigl\{|\bar{\mathbf w}(r_1)|,|\bar{\mathbf w}(r_2)|\bigr\},\qquad D(r_1,r_2):=\int\limits_{\Omega_{r_1,r_2}}|\nabla{\mathbf w}|^2
 \end{equation}
 and $C_*$ is some universal positive constant (does not depend on ${\mathbf w}, r_i$, etc.)
\end{theorem}

In order to prove the theorem, first of all we need to obtain the corresponding estimate for the {\it absolute value} of the velocity.

\begin{lemma}\label{lm:estim1}
Under conditions and notation of Theorem~\ref{th:estim1} the estimate
 \begin{equation} \label{eq:estim-ab1}
 \biggl| |\bar{\mathbf w}(r_2)|-|\bar{\mathbf w}(r_1)|\biggr|\le C_0(1+\mu)\sqrt{D(r_1,r_2)},
 \end{equation}
holds with some universal positive constant $C_0$  (not depending on ${\mathbf w}, r_i$, etc.)
\end{lemma}

\begin{proof}
If $|\bar{\mathbf w}(r_2)|=|\bar{\mathbf w}(r_1)|$, we have zero in the left hand side, and there is nothing to prove. For definiteness, assume that $$|\bar{\mathbf w}(r_2)|>|\bar{\mathbf w}(r_1)|$$ (in the case of opposite inequality the~arguments are the same). Let us make several simplifications. First of all, it is sufficient to consider the case
 \begin{equation} \label{f-add1}
{m}=|\bar{\mathbf w}(r_2)|=\max\limits_{r\in[r_1,r_2]}|\bar{\mathbf w}(r)|.
 \end{equation}
  Indeed, if the last assumption is not fulfilled, we can take
 $$r'_2\in[r_1,r_2]\ :\ |\bar{\mathbf w}(r'_2)|=\max\limits_{r\in[r_1,r_2]}|\bar{\mathbf w}(r)|,$$
 and then  consider the interval $[r_1,r'_2]$ instead of $[r_1,r_2]$.
 By construction,
 $$\!\!\!D(r_1,r'_2)<D(r_1,r_2),\quad|\bar{\mathbf w}(r_2)|-|\bar{\mathbf w}(r_1)|< |\bar{\mathbf w}(r'_2)|-|\bar{\mathbf w}(r_1)|, \quad\mu(r_1,r'_2)<\mu(r_1,r_2).$$
 So if we prove the required estimate (\ref{eq:estim-ab1}) for $r_1,r'_2$, then it implies the required estimate for initial pair $r_1,r_2$, etc. So below we assume that~(\ref{f-add1}) is fulfilled.

Further, it is sufficient to consider only the case when
  \begin{equation} \label{f-add2}
|\bar{\mathbf w}(r)|\ge\frac16{m}\qquad\forall r\in[r_1,r_2].
 \end{equation}
 Indeed, if the last assumption is not fulfilled, we can take
 $$r'_1=\max\biggl\{r\in[r_1,r_2]:|\bar{\mathbf w}(r)|\le\frac16{m}\biggr\},$$
 and then  consider the interval $[r'_1,r_2]$ instead of $[r_1,r_2]$.
 By construction,
  \begin{equation} \label{f-add3}
|\bar{\mathbf w}(r)|\ge\frac16{m}\qquad\forall r\in[r'_1,r_2],
 \end{equation}
 $$D(r'_1,r_2)<D(r_1,r_2),\qquad|\bar{\mathbf w}(r_2)|-|\bar{\mathbf w}(r_1)|\le {m}=\frac65\biggl(|\bar{\mathbf w}(r_2)|-|\bar{\mathbf w}(r'_1)|\biggr).$$
 So if we prove the required estimate (\ref{eq:estim-ab1}) for $r'_1,r_2$, then it implies the required estimate for initial pair $r_1,r_2$, etc. So below we assume that~(\ref{f-add2}) is fulfilled.

 Now take and fix some small number ${\varepsilon}_0\in(0,\frac1{100})$ (the exact value of ${\varepsilon}_0$ will be specified below). Obviously, it is sufficient to consider only the case when
 \begin{equation} \label{f-add4}
(1+\mu)\sqrt{D(r_1,r_2)}\le {\varepsilon}_0{m}.
 \end{equation}
 Indeed, if the opposite inequality were valid, then the required estimate~(\ref{eq:estim-ab1}) is fulfilled automatically with the fixed constant $C_0=\frac1{{\varepsilon}_0}$, and again there are nothing to prove. So below we assume that~(\ref{f-add4}) is fulfilled as well.

Also we can assume without loss of generality that
$$r_2>100 r_1$$
(otherwise the required estimate (\ref{eq:estim-ab1}) follows from~(\ref{eq2a})\,).
Further proof splits into  several steps. Our general strategy is as follows: first of all, we obtain some uniform estimates for pressure, Bernoulli pressure $\Phi=p+\frac12|{\mathbf w}|^2$ and velocity in the suitable annulus type domain; then we deduce some improved estimates for the integral $\int|\nabla\omega|^2$, and finally we obtain the required estimate~(\ref{eq:estim-ab1}) using contradiction argument and level sets of Bernoulli pressure.

 \medskip

 \textsc{Step 1.} Using standard estimates (``self-improvement of regularity'') for stationary Stokes system, one can prove that for any $\rho\in[r_1,\frac15r_2]$
  \begin{equation} \label{eq-per1}
\biggl(\int\limits_{\frac32\rho\le r\le\frac92\rho} |\nabla^2{\mathbf w}|^\frac43+|\nabla p|^\frac43\biggr)^{\frac34}
\le C \,m\,\rho^\frac12 D(\rho,5\rho)^\frac12(1+\mu)
 \end{equation}
 (see, e.g., \cite[Lemma~8 and Appendix~II]{KR21} for the detailed calculations concerning this estimate\,).

 \medskip

 \textsc{Step 2.} Now we would like to obtain the uniform estimates for pressure. Using (\ref{eq-per1}), it is easy to see, that
there exist two ``good'' radii ${\tilde r}_1\in (\frac32r_1,2r_1)$, ${\tilde r}_2\in(\frac12r_2,\frac9{10}r_2)$ such that for $i=1,2$ and  for $D:=D(r_1,r_2)$ we have
\begin{equation} \label{per-eq3b}
\left\{
\begin{aligned}
&\int_{S_{{\tilde r}_i}} |\nabla \mathbf{w}|^2 ds\le \frac{C D}{{\tilde r}_i}, \\
&\int_{S_{{\tilde r}_i}} |\nabla \mathbf{w}| ds \le (2\pi {\tilde r}_i)^\frac12 \left( \int_{S_{{\tilde r}_i}} |\nabla \mathbf{w}|^2 ds \right)^\frac12  \le C D^\frac12, \\
& \int_{S_{{\tilde r}_i}} |\nabla p|^{\frac43} ds\le C \frac1{{{\tilde r}_i}^\frac13}\,m^\frac43\,D^\frac23(1+\mu)^\frac43, \\
&\int_{S_{{\tilde r}_i}} |\nabla p| ds \le (2\pi {\tilde r}_i)^\frac14 \left( \int_{S_{{\tilde r}_i}} |\nabla p|^\frac43 ds \right)^\frac34  \le C \,m\,D^\frac12(1+\mu).
\end{aligned}
\right.
\end{equation}
Without loss of generality we may assume that $\bar p({\tilde r}_1)=0$. Then by (\ref{eq:pdifference}) we have
\begin{equation} \label{eq-pr-mean}
|\bar p(r)|\le C D\le C{m}\sqrt{D}\qquad\mbox{ for any }r\in[r_1,r_2],
\end{equation}
thus the estimates~(\ref{per-eq3b}${}_4$) and the trivial inequality
$$|p(z)|\le\bar p(r)+\int\limits_{S_r}|\nabla p|\,ds\qquad\ \forall z\in S_r,$$
imply the corresponding pointwise bound for the pressure:
\begin{equation} \label{per-eq-3-5}
|p|\le C{m} (1+\mu) \sqrt{D}\mathrm{\ \ \ \ on\  } S_{{\tilde r}_i}, \ i=1,2.
\end{equation}

From the main estimate~(\ref{eq:pdifference}), using ``good circles'' technique, it is very easy to deduce from (\ref{eq-per1}) and  (\ref{per-eq-3-5}), that
\begin{equation} \label{per-eq-p1}
|p|\le C{m} (1+\mu) \sqrt{D}\mathrm{\ \ \ \ in\  \ } \Omega\bigl(6r_1,\frac16 r_2\bigr)
\end{equation}
(see Appendix~\ref{appendix-1} for details).

\medskip

 \textsc{Step 3.} Denote ${m}_0=|\bar{\mathbf w}(r_1)|$. Recall, that ${m}=|\bar{\mathbf w}(r_2)|$.
Then estimates (\ref{per-eq3b}${}_2$), (\ref{f-add4}), (\ref{per-eq-3-5}) imply the following bound for the Bernoulli pressure $\Phi=p+\frac12|{\mathbf w}|^2$\ \,:
\begin{equation} \label{per-eq-3-6}
\Phi\le \frac12{m}_0^2+C{m} (1+\mu) \sqrt{D}\mathrm{\ \ \ \ on \  } S_{{\tilde r}_1},
\end{equation}
\begin{equation} \label{per-eq-3-6-}
\Phi\le \frac12{m}^2+C{m} (1+\mu) \sqrt{D}\mathrm{\ \ \ \ on \  } S_{{\tilde r}_2},
\end{equation}
\begin{equation} \label{per-eq-3-7}
\Phi\ge \frac12{m}^2-C{m} (1+\mu) \sqrt{D}\mathrm{\ \ \ \ on \  } S_{{\tilde r}_2}
\end{equation}
for some universal constant~$C$ (does not depend on~${\mathbf w},D,{m},\mu$, etc.).
By one-sided maximum principle for the Bernoulli pressure (see, e.g., \cite{GW}\,),
\begin{equation} \label{per-eq-3-6+}
\Phi\le \frac12{m}^2+C{m} (1+\mu) \sqrt{D}\mathrm{\ \ \ \ in \  } \Omega\bigl({\tilde r}_1,{\tilde r}_2\bigr).
\end{equation}
In particular, by (\ref{per-eq-p1})
\begin{equation} \label{per-eq-be}
|{\mathbf w}|^2\le {m}^2+C{m} (1+\mu) \sqrt{D}\overset{(\ref{f-add4})}\le
C\,m^2 \mathrm{\ \ \ \ in \  } \Omega\bigl(6r_1,\frac16 r_2\bigr),
\end{equation}
in other words,
\begin{equation} \label{per-eq-be2}
|{\mathbf w}|\le C\,{m}\mathrm{\ \ \ \ in \  } \Omega\bigl(6r_1,\frac16 r_2\bigr).
\end{equation}
The last inequality give us the possibility to obtain the improved estimates for~$\nabla\omega$.

\medskip

 \textsc{Step 4.}  We prove that for any $\rho\in[6r_1,\frac1{30}r_2]$
 \begin{equation} \label{f-add5}
\int\limits_{2\rho\le r\le4\rho} |\nabla \omega|^2 \le C\frac1{\rho}{m}\,D(\rho,5\rho)(1+\mu).
 \end{equation}
Indeed, for $D_\rho:=D(\rho,5\rho)$ there exist two ``good'' radii $\rho_1\in (\rho,2\rho)$, $\rho_2\in(4\rho,5\rho)$ such that for $i=1,2$, we have
\begin{equation} \label{eq3b}
\int_{S_{\rho_i}} \omega^2 ds  \le  \frac{C D_\rho}{\rho},
\end{equation}
and
\begin{equation} \label{eq3c}
\left[ \partial_r \int_{S_{r}} \omega^2 ds \right]_{r=\rho_2}
 \le \frac{C D_\rho}{\rho^2},\qquad\qquad -\left[ \partial_r \int_{S_{r}} \omega^2 ds \right]_{r=\rho_1}\le \frac{C D_\rho}{\rho^2},
\end{equation}
The proof of~(\ref{eq3b})--(\ref{eq3c}) is based on some elementary real analysis and is given in the Appendix~\ref{appendix-2} for convenience.

Note that \eqref{per-eq-be2} imply
\begin{equation} \label{eq3de}
\max\limits_{z\in S_{\rho_i}} |{\mathbf w}(z)|\le C\,{m}.
\end{equation}
It is well-known that $\omega$ satisfies the~vorticity equation
\begin{equation} \label{eq:vorticity}
-\Delta \omega + \mathbf{w} \cdot \nabla \omega = 0.
\end{equation}
A standard energy estimate in the domain $\{ \rho_1 \le r \le \rho_2 \}$ for the above equation, together with the bounds \eqref{eq3b}--\eqref{eq3de}, gives
\begin{align*}
\int_{\rho_1 \le r \le \rho_2} |\nabla \omega|^2 &= \int_{S_{\rho_2}} \left(\omega \partial_r \omega  - \mathbf{w} \cdot \mathbf{e}_r \frac{\omega^2}{2}\right) ds \\
&\quad - \int_{S_{\rho_1}} \left(\omega \partial_r \omega  - \mathbf{w} \cdot \mathbf{e}_r \frac{\omega^2}{2}\right) ds \\
&\le C \frac{1}{\rho} D_\rho\biggl({m}+\frac1\rho\biggr)\le C \frac{1}{\rho}{m} D_\rho(1+\mu).
\end{align*}

\medskip

 \textsc{Step 5.} Applying (\ref{f-add5}) for $\rho=6r_1$, $12r_1$,\dots, etc., we obtain finally that
 \begin{equation} \label{f-add8}
\int\limits_{12r_1\le r\le\frac18r_2} r|\nabla \omega|^2 \le C{m} D(1+\mu).
 \end{equation}

 \medskip

 \textsc{Step 6.}  We need a pair of ``good circles'' with the same properties as in Step~2--3, but now inside the annulus $\Omega(12r_1,\frac18r_2)$ (in order to use estimates~(\ref{f-add8})\,). So, repeating the arguments of these steps, we obtain, that there exists $\tilde\rho_1\in\bigl(12r_1,24 r_1\bigr)$, \
 $\tilde\rho_2\in\bigl(\frac1{16}r_2,\frac18r_2\bigr)$ such that
 \begin{equation} \label{eq-3-6}
\Phi\le \frac12{m}_0^2+C{m} (1+\mu) \sqrt{D}\mathrm{\ \ \ \ on \  } S_{{\tilde \rho}_1},
\end{equation}
\begin{equation} \label{eq-3-7}
\Phi\ge \frac12{m}^2-C{m} (1+\mu) \sqrt{D}\mathrm{\ \ \ \ on \  } S_{{\tilde \rho}_2}
\end{equation}
for some universal constant~$C$ (does not depend on~${\mathbf w},D,{m},\mu$, etc.).
 Therefore,
\begin{equation} \label{eq-3-8}
\min\limits_{z\in S_{\tilde\rho_2}} \Phi(z)\ge \max\limits_{z\in S_{\tilde\rho_1}} \Phi(z)+\frac12\biggl({m}^2-{m}_0^2-C{m} (1+\mu) \sqrt{D}\biggr)
\end{equation}

Let $\varphi(r)$ be the direction of the vector $\bar{\mathbf{w}}(r) = (\bar{w}_1(r), \bar{w}_2(r))$, {\emph{i.e.}},
$\bar{\mathbf{w}}(r)=|\bar{\mathbf w}(r)|\,(\cos\varphi(r),\sin\varphi(r)).$  Without loss of generality we may assume, that  $\varphi(\tilde\rho_2)=0$.
Then from the inequality~(\ref{f-add8}) we have
\begin{equation} \label{eq-3-8'}
\int\limits_{12r_1\le r\le\frac18r_2} \frac1r|\nabla \omega| \le \biggl(C{m} D(1+\mu)\biggr)^{\frac12}\sqrt{\frac\pi{6 r_1}}\le C {m} (1+\mu) \sqrt{D},
 \end{equation}
consequently, from the formula~(\ref{eq:phidifference}) and assumptions~(\ref {f-add2}), (\ref {f-add4}) we obtain that
 \begin{equation} \label{eq-3-9}
|\varphi(r)|\le C'{\varepsilon}_0\qquad\forall r\in[\tilde\rho_1,\tilde\rho_2]
\end{equation}
 with some universal constant~$C'$ (does not depend on~${\mathbf w},D,{m},\mu$, etc.).

Without loss of generality we may assume that all the constants $C$ in the inequalities (\ref{eq-3-8})--(\ref{eq-3-8'}) are the same. Now we are ready for the key step of the proof.

\medskip

\textsc{Step 7.} We claim that the inequality
\begin{equation} \label{eq-3-10}
{m}-{m}_0\le 5C(1+\mu)\sqrt{D}
\end{equation}
holds for sufficiently small~${\varepsilon}_0$, where $C$ is the same as in~(\ref{eq-3-8})--(\ref{eq-3-8'}).

Indeed, suppose this claim fails, than
\begin{equation} \label{eq-3-11}
{m}-{m}_0> 5C(1+\mu)\sqrt{D},
\end{equation}
in particular, from~(\ref{eq-3-8}) we have
\begin{equation} \label{eq-3-12}
\min\limits_{z\in S_{\tilde\rho_2}} \Phi(z)> \max\limits_{z\in S_{\tilde\rho_1}} \Phi(z)+2C{m} (1+\mu) \sqrt{D}.
\end{equation}
Now, we are in a position to apply the methods of \cite{KPR20} based on level set structures of $\Phi$ to obtain a contradiction when $\varepsilon_0$ is sufficiently small. For a reader's convenience, we recall the main ideas of the proof in~\cite{KPR20} adapted for the present paper.

From (\ref{eq-3-12}) it follows immediately, that there are two closed regular level sets $S'_{k}$ and $S''_{k}$ of $\Phi$
such that:

\begin{enumerate}
\item[(i)] \ $\Phi|_{S'}\equiv t'$, \ \ \ \ \ $\Phi|_{S''}\equiv t''$, \ \ \ \ $t''-t'>2C{m} (1+\mu) \sqrt{D}$;
\item[(ii)] \qquad $S'$, $S''$ \ are smooth closed curves (homeomorphic to the circle) surrounding the origin, both of them lie between circles $S_{\tilde\rho_1}$ and $S_{\tilde\rho_2}$.
\item[(iii)] \qquad the vorticity $\omega(z)$ does not change sign between the curves $S'$, $S''$; for definiteness, we can assume without loss of generality that
\begin{equation}\label{eq-3-13}
\omega(z)>0\qquad\mbox{ for all $z$ between  $S'$ and $S''$}
\end{equation}
\end{enumerate}
(for the last assertion, see \cite[Step~6]{KPR20}\,).

Using (\ref{eq-3-9}), it is not difficult to prove for ${\varepsilon}_0$ small enough, that there exists a unit vector $\tilde{\mathbf e}=(\cos\tilde\theta,\sin\tilde\theta)$ such that the segment
$L=\{\rho\tilde{\mathbf e}:\rho\in[\tilde\rho_1,\tilde\rho_2]\}$ satisfies the following properties:
\begin{enumerate}
\item[(iv)] \ ${\mathbf w}^\bot(z)\cdot\tilde{\mathbf e}<0$ for any $z\in L$, where we denote $(a,b)^\bot=(-b,a)$;
\item[(v)] \ $\int\limits_L|\nabla\omega|\,ds< 2C{m} (1+\mu) \sqrt{D}$.
\end{enumerate}
Now take two points $A\in L\cap S'$ \ and \ $B\in L\cap S''$ such that the line segment $[A,B]$ lies between the curves $S'$ and $S''$.
Recall, that the gradient of the Bernoulli pressure satisfies the identity
$$\nabla\Phi\equiv-\nabla^\bot\omega+\omega\cdot{\mathbf w}^\bot.$$
Then we have
\begin{equation}
\label{en25}
\begin{array}{c}
t''-t'=\Phi(B)-\Phi(A)=\int\limits_{[A,B]}\nabla\Phi\cdot{\tilde{\mathbf e}}\,ds\\[12pt]
=-\int\limits_{[A,B]}\nabla^\bot\omega\cdot{\tilde{\mathbf e}}\,ds+
\int\limits_{[A,B]}\omega{\mathbf w}^\bot\cdot{\tilde{\mathbf e}}\,dr:=I+II.
\end{array}
\end{equation}
Estimate the~terms~$I$ and $II$ separately. From the above property (v) we have
\begin{equation}
\label{en26}
I< 2C{m} (1+\mu) \sqrt{D}.
\end{equation}
On the other hand, from (iii)--(iv) we obtain
\begin{equation}
\label{en27}
II<0.
\end{equation}
Therefore,
$$t''-t'\le2C{m} (1+\mu) \sqrt{D},$$
a contradiction with~(i).

Of course, the above items (i)--(v) are only short description. In case of interest, a reader can find a detailed justification for all these steps in~\cite{KPR20}.

\medskip

The proof of  Lemma~\ref{lm:estim1} is finished completely.
\end{proof}

\medskip

{\it Proof of Theorem~\ref{th:estim1}}.  \ We start from the same simplifications as in the proof of Lemma~\ref{lm:estim1}. For definiteness, assume that $|{\mathbf w}(r_2)|\ge|{\mathbf w}(r_1)|$.
Because of the triangle inequality, it is sufficient to consider the case
 \begin{equation} \label{fg-add1}
{m}=|\bar{\mathbf w}(r_2)|=\max\limits_{r\in[r_1,r_2]}|\bar{\mathbf w}(r)|.
 \end{equation}
Also, it is sufficient to consider only the situation when
  \begin{equation} \label{fg-add2}
|\bar{\mathbf w}(r)|\ge\frac16{m}\qquad\forall r\in[r_1,r_2]
 \end{equation}
  \begin{equation} \label{fg-add4}
\sqrt{D(r_1,r_2)}=\sqrt{D}\le \frac1{100}{m}.
 \end{equation}
(see the explanation in the proof of Lemma~\ref{lm:estim1}). Further we have to repeat all the arguments of the proof of Lemma~\ref{lm:estim1} up to inequality~(\ref{eq-3-8'}).
Then from the formulas~(\ref{eq:phidifference}), (\ref{eq-3-8'}), and assumptions~(\ref {fg-add2}), (\ref {fg-add4}) we obtain that
 \begin{equation} \label{g-eq-3-9}
|\varphi(r)|\le C\frac1{{m}^2}\biggl(D+{m}(1+\mu)\sqrt{D}\biggr)\le C'\frac1{{m}}(1+\mu)\sqrt{D}\qquad\forall r\in[\tilde\rho_1,\tilde\rho_2].
\end{equation}
The last formula and (\ref{eq:estim-ab1}) imply easily
\begin{equation} \label{geq:estim-m1}
 |\bar{\mathbf w}(\tilde\rho_1)-\bar{\mathbf w}(\tilde\rho_2)|\le C(1+\mu)\sqrt{D}.
 \end{equation}
Then, by virtue of elementary estimate~(\ref{eq2a})   and inclusions $\tilde\rho_1\in(12r_1,24r_1)$, \ $\tilde\rho_2\in\bigl(\frac1{16} r_2,\frac18 r_2\bigr)$, we obtain the required inequality~(\ref{geq:estim-m1}).

The proof of Theorem~\ref{th:estim1} is finished completely. $\qed$

\section{The second basic (asymptotic) estimate for the velocity}\label{sec-es2}

For the limiting case (when $r_{i}$ are very large and velocity is ``almost constant'' on the boundary circles) the general estimate of Theorem~\ref{th:estim1} can be improved essentially.

\begin{theorem}\label{th:estim2}
Let ${\mathbf w}_k$ be a sequence of $D$-solutions to the Navier--Stokes system
 \begin{equation}  \label{NSE-gen-k}
\left\{\begin{aligned}
 & \Delta \mathbf{w}_k - (\mathbf{w}_k \cdot \nabla) \mathbf{w}_k - \nabla p_k = \mathbf{0}, \\
 & \nabla \cdot \mathbf{w}_k = 0
\end{aligned}
\right.
\end{equation}
 in the annulus type domains $\Omega_{r_{1k},r_{2k}}$. Suppose, in addition, that
\begin{equation} \label{eq:r-ass}
r_{1k}\to+\infty,\qquad\frac{r_{2k}}{r_{1k}}\to+\infty,
 \end{equation}
 and  there exist two vectors ${\mathbf w}_0$, ${\mathbf w}_\infty\in{\mathbb R}^n$ such that
 \begin{equation} \label{e-eq:estim-a1}
 \max_{z\in S_{r_{1k}}}|{\mathbf w}_k(z)-{\mathbf w}_0|\to0, \qquad\max_{z\in S_{r_{2k}}}|{\mathbf w}_k(z)-{\mathbf w}_\infty|\to0\qquad\mbox{\rm as }k\to\infty.
 \end{equation}
 Then
 \begin{equation} \label{2e-eq:estim-a2}
|{\mathbf w}_0-{\mathbf w}_\infty|\le C_{**}\frac{D_*}{{m}},
 \end{equation}
 where ${m}:=\max\bigl\{|{\mathbf w}_0|,|{\mathbf w}_\infty|\bigr\}$, $D_* =\varliminf\limits_{k\to\infty}\int_{\Omega_{r_{1k},r_{2k}}}|\nabla{\mathbf w}_k|^2$,
 and $C_{**}$ is some universal positive constant (does not depend on ${\mathbf w}_k$, etc.)
\end{theorem}

To prove this theorem, we have to use some fine properties of solutions to Euler system from~\cite{KR21new}, considered in the next section~\ref{sec:3}.

\subsection{On solutions to Euler equations}
\label{sec:3}

In this section we consider some properties of weak solutions ${\mathbf v}\in W_{\mathrm loc}^{1,2}(\Omega)$ to the Euler system
\begin{equation}
\label{Eul}\left\{\begin{array}{rcl} \big({\bf
v}\cdot\nabla\big){\bf v}+\nabla p & = & 0 \qquad \ \ \
\hbox{\rm in }\;\;B_1,\\[4pt]
\hbox{\rm div}\,{\bf v} & = & 0\qquad \ \ \ \hbox{\rm in }\;\;B_1,
\\[4pt]
{\bf v} &  = & {\mathbf e}\ \
 \qquad\  \hbox{\rm on }\;\;S_1=\partial\Omega,
\end{array}\right.
\end{equation}
where $S_1=\{z\in{\mathbb R}^2:|z|=1\}$ is the unit circle and ${\mathbf e}=(1,0)$ is the unit vector. Here $\Omega$ is the unit disk or its complement, \emph{i.e.},
\begin{equation}
\label{om1}\Omega=B_1
\end{equation}
or
\begin{equation}
\label{om2}\Omega={\mathbb R}^2\setminus B_1.
\end{equation}
Suppose that
\begin{equation}
\label{de}\int\limits_{\Omega}|\nabla{\mathbf v}|^2<{\varepsilon}^2
\end{equation}
for some constant ${\varepsilon}\in(0,1)$. Then from the first equation~(\ref{Eul}${}_1$) one can assume that $|p|\sim{\varepsilon}$.
Nevertheless, surprisingly much better estimate holds.

\begin{theorem} \label{th:euler_est}
Let ${\mathbf v}\in W^{1,2}_{\mathrm{loc}}(\Omega)$ and $p\in W^{1,1}_{\mathrm{loc}}(\Omega)$ satisfy the Euler equations~{\rm (\ref{Eul}${}_{1-2}$)} for almost all~$z\in \Omega$. Suppose also that the~estimate~{\rm(\ref{de})} and the boundary condition~{\rm(\ref{Eul}${}_{3}$)} are fulfilled. Then $p\in C(\bar{\Omega})$, moreover,
\begin{equation}
\label{de-e}\sup\limits_{z_1,z_2\in \bar \Omega}|p(z_1)-p(z_2)|\le C\,{\varepsilon}^2,
\end{equation}
where $C$ is some universal constant (does not depend on ${\varepsilon},{\mathbf v},p$\,).
\end{theorem}

\begin{proof}
For the case $\Omega=B_1$ the result was proved in~\cite{KR21new}. So let us consider the case of exterior domain~$\Omega={\mathbb R}^2\setminus B_1$.

By well-known fact concerning $D$-solutions to Euler and
Navier--Stokes system (see, \emph{e.g.}, \cite[Lemma~4.1]{GW}), the
averages of the pressure $\bar p(r)$ are uniformly bounded and have some limit at infinity, without loss of generality we may assume that this limit iz zero:
\begin{equation} \label{co-3}
\bar p(r)\to0\qquad\mbox{ as }r\to\infty.
\end{equation}
Moreover, since
$\int\limits_{\Omega}|\nabla{\mathbf v}|^2dx<\infty$, there exists an
increasing sequence $r_i\to +\infty$ such that
\begin{equation} \label{co-4}
\int\limits_{S_{r_i}}|\nabla{\mathbf v}|\,ds\le \frac{{\varepsilon}_i}{\sqrt{\ln r_i}} \qquad\mbox{ with }{\varepsilon}_i\to0\quad\mbox{ as
}i\to\infty
\end{equation}
and
\begin{equation} \label{co-4'}
\sup\limits_{z\in S_{r_i}} \bigl|{\mathbf v}(z)\bigr|\le {\varepsilon}_i\sqrt{\ln
r_i}
\end{equation}
(see \cite[Lemmas~2.1--2.2]{GW})). From
(\ref{co-3})--(\ref{co-4'}) and from the equation~(\ref{Eul}$_1$)
it follows that
\begin{equation} \label{co-5}
\sup\limits_{z\in S_{r_i}} \bigl|p(z)\bigr|\to0\qquad\mbox{ as }i\to\infty.
\end{equation}
Indeed,
$$
|p(r_i,\theta)-\bar p(r_i)|\le \int\limits_{S_{r_i}}|\nabla p|\,ds
\leq \int\limits_{S_{r_i}}|{\mathbf v}|\cdot|\nabla {\mathbf v}|\,ds\leq {\varepsilon}_i\sqrt{\ln
r_i} \int\limits_{S_{r_i}}|\nabla {\mathbf v}|\,ds\le {\varepsilon}^2_i.
$$

Taking divergence on the first equation in \eqref{Eul} gives
\begin{equation}\label{r2.2}
\Delta p = - \nabla {\mathbf v}\cdot (\nabla{\mathbf v})^\intercal
\end{equation}
We can extend ${\mathbf v}$ outside $\Omega$ by the constant vector ${\mathbf e}$ so that it is globally defined and divergence free in $\mathbb{R}^2$. By the classical div-curl lemma (see, \emph{e.g.}, \cite{CLMS}), \,$\nabla{\mathbf v}\cdot (\nabla{\mathbf v})^\intercal$ belongs to the Hardy space $\mathcal{H}^1(\mathbb{R}^2)$.
Put
$$
G(x)=-\frac{1}{2\pi}\int\limits_{\Omega} \log|x-y|(\nabla{\mathbf
v}\cdot\nabla{\mathbf v}^\top)(y) dy.
$$
By Calder\'on--Zygmund theorem for Hardy's
spaces \cite{St}, \ $G\in D^{2,1}({\mathbb R}^2)\cap D^{1,2}({\mathbb R}^2)$. By
classical facts from the theory of Sobolev spaces (see, \emph{e.g.},
\cite{maz'ya}\,), the last inclusion implies that~$G$ is
continuous and convergent to zero at infinity, in particular,
\begin{equation} \label{co-6--}
\|\nabla G\|_{L^2({\mathbb R}^2)}+\|\nabla^2 G\|_{L^1({\mathbb R}^2)}\le C{\varepsilon}^2,
\end{equation}
\begin{equation} \label{co-6}
\sup\limits_{z\in{\mathbb R}^2}|G(z)|< C{\varepsilon}^2,
\end{equation}
\begin{equation} \label{Gto0}
G(z) \to 0, \quad \mathrm{as} \ z \to \infty.
\end{equation}
Consider the decomposition
\begin{equation} \label{decomp0}
p=p_*+G.
\end{equation}
By construction, $\Delta p_*=0$
in $\Omega$, \emph{i.e.}, $p_*$ is a harmonic function, and
by~(\ref{co-5}) we have
\begin{equation} \label{co-7}
\sup\limits_{z\in S_{r_i}}|p_*(z)|\to0\qquad\mbox{ as }i\to\infty.
\end{equation}
Of course, it implies
\begin{equation} \label{co-7-11}
|p_*(z)|\to0\qquad\mbox{ as }|z|\to\infty.
\end{equation}
Since $p = G + p_*$, we have $p(z) \to 0$ as $z \to \infty$. In \cite[Lemma~9]{KR21new}, it was proved that $p$ satisfies the Neumann boundary conditions (in weak sense) on~$S_1$:
$$\partial_n p=0\qquad\mbox{ on }S_1.$$
Therefore, $p$ can be solved from \eqref{r2.2} using the Green function for the Poisson problem in $\Omega$  with Neumann boundary conditions:
\begin{equation}
p(x) = -\frac{1}{2\pi} \int_\Omega \left( \log |x - y| + \log \left|\frac{x}{|x|^2} - y\right| - \log|y| \right) (\nabla{\mathbf
v}\cdot\nabla{\mathbf v}^\top)(y) dy
\end{equation}
It is important to observe that in the above formula, the integral domain $\Omega$ can be replaced by $\mathbb{R}^2$ , since $\nabla{\mathbf
v}\cdot\nabla{\mathbf v}^\top = 0$ in $B_1$ due to our extension of $\mathbf{v}$ by constant. Finally, by the same reasons for the validity of \eqref{co-6--}-\eqref{Gto0}, we obtain the required bound~(\ref{de-e}).
\end{proof}

The proof of the central Theorem~\ref{th:euler_est} is finished. In \cite[\S2]{KR21new} it was shown, that the established estimates imply

\begin{corollary} \label{lem:pE}
Let the assumptions of Theorem~\ref{th:euler_est} be fulfilled. Then there exists a set  $\mathscr{F}_0\subset(\frac12,\frac32)$ of positive measure (having one-sided density~$1$ at~$1$\,) \ such that
\begin{equation} \label{eq:pr-gr7}
S_r\subset\Omega \qquad\mbox{\rm \ and \ \ }\max_{z\in S_r}\biggl(|{\mathbf v}(z)-{\mathbf e}| + \bigl|p(z) - \Xint-_{S_{r}} p\bigr|\biggr) \le C {\varepsilon}^2\qquad\qquad\forall r\in\mathscr{F}_0,
\end{equation}where $C$ is some universal constant (does not depend on ${\varepsilon},{\mathbf v},p$, etc.\,).
\end{corollary}

Sometimes solutions to Euler system can be obtained as a~limit of solutions to Navier--Stokes equations when viscosity coefficient tends to zero. We can use this fact in
the following useful statement.

\begin{theorem}\label{th:estim-eul-lim}
Let the assumptions of Theorem~\ref{th:estim2} be fulfilled.  Suppose, in addition, that $\bar p_k(r_{1k})=0$. Then there exists $\delta\in\bigl(0,\frac12)$ such that
 \begin{equation}\label{eq:fin-Eul-1}
{m}_0\cdot\bigl|{\mathbf w}_k(z)-{\mathbf w}_0\bigr| + \bigl|p_k(z)\bigr| \le C D_*+{\varepsilon}_k\qquad\ \forall z\in S_{(1+\delta)r_{1k}},
\end{equation}
 \begin{equation}\label{eq:fin-Eul-2}
{m}_\infty\cdot\bigl|{\mathbf w}_k(z)-{\mathbf w}_\infty \bigr| + \bigl|p_k(z)\bigr| \le C D_*+{\varepsilon}_k\qquad\ \forall z\in S_{(1-\delta)r_{2k}},
\end{equation}
where ${m}_\infty=|{\mathbf w}_\infty|$, \ ${m}_0=|{\mathbf w}_0|$, $D_*=\varliminf\limits_{k\to\infty}\int_{\Omega_{r_{1k},r_{2k}}}|\nabla{\mathbf w}_k|^2$, \ ${\varepsilon}_k\to0$ as $k\to\infty$,
 and $C$ is some universal positive constant (does not depend on ${\mathbf w}_k,\,r_{ik}$, etc.)
\end{theorem}

\begin{proof}
The assertion of the last theorem was deduced (by corresponding scaling and limiting procedure) from Theorem~\ref{th:euler_est} and Corollary~\ref{lem:pE} in
\cite[Section~4]{KR21new} (see, \emph{e.g.}, the formula~(4.10) in \cite{KR21new} and commentaries).
\end{proof}

\subsection{Proof of Theorem \ref{th:estim2}}\label{sec-es-pr}

Let the conditions of Theorem~\ref{th:estim2} be fulfilled.
As before, we have to make several simplifications. For definiteness, assume that
${m}=|{\mathbf w}_\infty|\ge|{\mathbf w}_0|$. Denote ${m}_0=|{\mathbf w}_0|$.
Now take and fix some small number ${\varepsilon}_0\in(1,\frac1{100})$ (the exact value of ${\varepsilon}_0$ will be specified below). Obviously, it is sufficient to consider only the case when
 \begin{equation} \label{2-f-add4}
D_*<{\varepsilon}_0{m}^2.
 \end{equation}
 Indeed, if the opposite inequality valid, then the required estimate~(\ref{2e-eq:estim-a2}) is fulfilled automatically with the fixed constant $C=\frac2{{\varepsilon}_0}$, and there are nothing to prove. So below we assume that~(\ref{2-f-add4}) is fulfilled as well, therefore,
  \begin{equation} \label{2e-f-add4}
D(r_{1k},r_{2k})<{\varepsilon}_0{m}^2
 \end{equation}
 for $k$ large enough. Applying Theorem~\ref{th:estim1} (with $\mu$ arbitrary small as $k\to\infty$ because of assumption $r_{1k}\to+\infty$\,), we obtain
   \begin{equation} \label{2e-f-add2}
\frac12{m}<|\bar{\mathbf w}_k(r)|<\frac32{m}\qquad\forall r\in[r_{1k},r_{2k}].
 \end{equation}
Take $\delta\in(0,\frac12)$ from Theorem~\ref{th:estim-eul-lim}
and denote $\rho_{1k}=(1-\delta)r_{1k}$,  \ $\rho_{2k}=(1-\delta)r_{2k}$.
 Repeating Steps~1--6 of the proof of Lemma~\ref{lm:estim1} (with evident modifications and arbitrary small~$\mu$ as $k\to+\infty$), we obtain that
 \begin{equation} \label{2e-eq-3-8'}
\int\limits_{\rho_{1k}\le r\le\rho_{2k}} \frac1r|\nabla \omega_k| \le {\varepsilon}_k\to0\qquad\mbox{ as \ }k\to+\infty,
 \end{equation}
consequently, from the formula~(\ref{eq:phidifference}) and assumptions~(\ref {2e-f-add2}), (\ref {2e-f-add4}) we obtain that
 \begin{equation} \label{2e-eq-3-9}
|\varphi_k(r)|\le \frac1{4\pi{m}^2} \bigl(D_*+{\varepsilon}_k\bigr)\le \frac1{4\pi}{\varepsilon}_0\qquad\forall r\in[\rho_{1k},\rho_{2k}]
\end{equation}
 for all sufficiently large~$k$.

 Recall, that by Theorem~\ref{th:estim-eul-lim},
 \begin{equation}\label{2e-eq:fin-Eul-1}
{m}_0\bigl| {\mathbf w}_k(z)-{\mathbf w}_0\bigr| + \bigl|p_k(z)\bigr| \le C D_*+{\varepsilon}_k\qquad\ \forall z\in S_{\rho_{1k}},
\end{equation}
 \begin{equation}\label{2e-eq:fin-Eul-2}
{m}\bigl|{\mathbf w}_k(z)-{\mathbf w}_\infty\bigr| + \bigl|p_k(z)\bigr| \le C D_*+{\varepsilon}_k\qquad\ \forall z\in S_{\rho_{2k}}.
\end{equation}
Consequently, using ${m}_0\ge\frac12{m}$, we obtain:
\begin{equation} \label{2e-eq-3-8}
\min\limits_{z\in S_{\rho_{2k}}} \Phi_k(z)\ge \max\limits_{z\in S_{\rho_{1k}}} \Phi_k(z)+\frac12\biggl({m}^2-{m}_0^2-12CD_*-{\varepsilon}_k\biggr),
\end{equation}
where $C$ is the same as in~(\ref{2e-eq:fin-Eul-1})--(\ref{2e-eq:fin-Eul-2}).

Below we consider the case $D_*>0$ (the case $D_*=0$ can be considered similarly, with some evident simplifications). Now we claim that the inequality
\begin{equation} \label{2e-eq-3-10}
{m}-{m}_0\le \frac{10}{m} CD_*
\end{equation}
holds for sufficiently small~${\varepsilon}_0$, where $C$ is the same as in~(\ref{2e-eq-3-8}).
Indeed, suppose that the opposite inequality is valid, then
 from~(\ref{2e-eq-3-8}) we have
\begin{equation} \label{2e-eq-3-12}
\min\limits_{z\in S_{\rho_{2k}}} \Phi_k(z)> \max\limits_{z\in S_{\rho_{1k}}} \Phi_k(z)+CD_*.\end{equation}
Now we can apply the methods of \cite{KPR20} (based on level set structures of $\Phi_k$) in the same way as on the Step~7 of the proof of previous Lemma~\ref{lm:estim1}, and to obtain the desired contradiction. So the~claim~(\ref{2e-eq-3-10}) is proved.

The formulas (\ref{2e-eq:fin-Eul-1})--(\ref{2e-eq:fin-Eul-2}), (\ref{2e-eq-3-10}) imply that
\begin{equation} \label{2e-eq-3-13}
\bigl| \bar{\mathbf w}_k(\rho_{1k})-{\mathbf w}_0\bigr| \le C\frac1{m} D_*\end{equation}
\begin{equation} \label{2e-eq-3-14}
\bigl| \bar{\mathbf w}_k(\rho_{2k})-{\mathbf w}_\infty\bigr| \le C\frac1{m} D_*.\end{equation}
\begin{equation} \label{2e-eq-3-15}
\biggl| |\bar{\mathbf w}_k(\rho_{2k})|-|\bar{\mathbf w}_k(\rho_{1k})|\biggr| \le C\frac1{m} D_*\end{equation}
with some universal constant~$C$.
We need the following elementary fact: for any pair of vectors ${\mathbf a}=a{\mathbf e}_{\mathbf a}$,\ \ ${\mathbf b}=b{\mathbf e}_{\mathbf b}$, where ${\mathbf e}_{\mathbf a}$ and ${\mathbf e}_{\mathbf b}$ are the corresponding unit vectors, one has:
$$\bigl|{\mathbf a}-{\mathbf b}\bigr|\le \bigl|a-b\bigr|+b\bigl|{\mathbf e}_{\mathbf a}-{\mathbf e}_{\mathbf b}\bigr|.$$
This elementary formula together with established estimates~(\ref{2e-eq-3-15}), (\ref{2e-eq-3-9}) imply
\begin{equation} \label{2e-eq-3-16}
\bigl| \bar{\mathbf w}_k(\rho_{2k})-\bar{\mathbf w}_k(\rho_{1k})\bigr| \le C\frac1{m} D_*.\end{equation}
Thus by (\ref{2e-eq-3-13})--(\ref{2e-eq-3-14}) the proof of Theorem~\ref{th:estim2} is finished completely. $\qed$

\section{Uniform bounds of the invading domain solutions $\mathbf{w}_k$}\label{sec-unif}

The main result of this section is the following $k$-uniform bounds.

\begin{theorem} \label{th-uniform-bound}
Suppose that $R > 0$,
$\mathrm{supp \, } \mathbf{f} \subset B_R$, and $A = \|\mathbf{f}\|_{H^{-1}(B_{2R})}<\infty $, and consider $R_k \ge 2R$. Then the following uniform estimates for the invading domain solutions $\mathbf{w}_k$ to \eqref{NSE-k} are valid:
\begin{equation} \label{eq31}
D_k:=\int_{{B_{R_k}}} |\nabla \mathbf{w}_k|^2 \le C_1A\biggl(A+\lambda+\frac{A^{\frac13}}{R^{\frac23}}\biggr),
\end{equation}
\begin{equation} \label{eq31-1}
\max_{R \le r \le R_k} |\bar{\mathbf{w}}_k(r)| \le C_2\biggl(A+\lambda+\frac{A^{\frac13}}{R^{\frac23}}\biggr),
\end{equation}
where $C_i$ are some universal constants {\rm(}do not depend on ${\mathbf w}_k$ and parameters $A,R,\lambda$, etc.{\rm)}.
\end{theorem}

\begin{remark}
\label{scale-i}
The considered problem~(\ref{NSE-k}) has the following scaling property: if $\mathbf{w}_k$ is a solution to~(\ref{NSE-k}) with parameters $(A,\lambda,R)$, then for any $\tau>0$ the mapping ${\mathbf w}_{\tau k}(z):=\tau{\mathbf w}_k(\tau z)$ is a solution to~(\ref{NSE-k}) with the corresponding parameters
$(\tau A,\,\tau\lambda,\,\tau^{-1}R)$. Note, that basic estimates in theorems~\ref{th:estim1},\,\ref{th:estim2},\,\ref{th-uniform-bound},\,\ref{th-s1}--\ref{th-s2},\,\ref{th-uniq} are scaling invariant.
\end{remark}

\begin{proof}
Let $\chi$ be a smooth cut-off function satisfying $\chi \equiv 1 \mathrm{\ on\ } B_R$, $\chi = 0$ outside $ B_{2R}$, and $|\nabla\chi|\le \frac{C}R$. By the system \eqref{NSE-k} and the assumptions on $\mathbf{f}$, we have the following direct estimate of the Dirichlet integral
\begin{align} \label{unif-eq-ee}
D_k = \int_{{B_{R_k}}}|\nabla \mathbf{w}_k|^2 &= \int_{B_{R_k}} \mathbf{f} \cdot (\mathbf{w}_k - \mathbf{w}_\infty)  \nonumber \\
&= \int_{B_{R_k}} \chi \mathbf{f} \cdot (\mathbf{w}_k - \mathbf{w}_\infty) \nonumber \\
&\le \|\mathbf{f}\|_{H^{-1}(B_{2R})} \|\chi(\mathbf{w}_k - \mathbf{w}_\infty)\|_{H_0^1(B_{2R})}  \nonumber \\
&\le C\,A \biggl(\|\nabla\mathbf{w}_k\|_{L^2(B_{2R})} +\frac1R\|\mathbf{w}_k - \mathbf{w}_\infty\|_{L^2(B_{2R})}\biggr)  \nonumber \\
&\le C\,A \bigl( D_k^\frac12 +  |\bar{\mathbf{w}}_k(2R)| + \lambda\bigr),
\end{align}
here we used (\ref{scale3}) for the last inequality
(the exact values of $C$ may change from line to line).
Using \eqref{unif-eq-ee} it is easy to deduce
\begin{equation} \label{unif-eq-e-uk}
D_k \le C\,A (A+|\bar{\mathbf{w}}_k(2R)| + \lambda).
\end{equation}
Therefore,
\begin{equation} \label{unif-eq-e}
D_k \le C\,A (A+\Lambda_k+ \lambda),
\end{equation}
where we denote
$\Lambda_k =\max_{R \le r \le R_k} |\bar{\mathbf{w}}_k(r)|$. Let $\Lambda_k=|\bar{\mathbf{w}}_k(r_k)|$ with some $r_k\in[R,R_k]$.
Applying Theorem \ref{th:estim1} in the annulus type domain $\Omega_{r_k, R_k}$ and inserting \eqref{unif-eq-e}, we obtain
\begin{align} \label{unif-eq-1e}
|\Lambda_k - \lambda| &\le C_*(1+ \frac{1}{R\Lambda_k})\sqrt{D_k} \nonumber \\
&\le C (1+ \frac{1}{R\Lambda_k})\sqrt{A (A + \Lambda_k + \lambda)}
\end{align}
Now we have to consider two different cases:

\medskip
{\sc Case 1.}  $\Lambda_k\le 2(A+\lambda)$. Then the required estimates~(\ref{eq31})--(\ref{eq31-1}) follow immediately.

\medskip
{\sc Case 2.} Now suppose that $\Lambda_k> 2(A+\lambda)$
Then from (\ref{unif-eq-1e}) we obtain:
\begin{equation} \label{unif-eq-w-d3}
 \Lambda_k\le C (1+ \frac{1}{R\Lambda_k})\sqrt{A \Lambda_k}
\end{equation}
If $\frac1{R\Lambda_k}\le 2$, then, similarly to the Case~1, we have $\Lambda_k\le C\,A$, and the required estimate~(\ref{eq31}) follow easily from~(\ref{unif-eq-e}). So we can assume in addition that~$\frac1{R\Lambda_k}>2$. Then
\begin{equation} \label{unif-eq-w-d4}
 \Lambda_k\le C \frac{1}{R\Lambda_k}\sqrt{A \Lambda_k} \le C \frac{1}{R}\sqrt{\frac{A}{\Lambda_k}}.
\end{equation}
Therefore
\begin{equation} \label{unif-eq-w-d5}
 \Lambda_k\le C \frac{A^{\frac13}}{R^{\frac23}},
\end{equation}
and finally the required estimates~(\ref{eq31})--(\ref{eq31-1}) are~valid in this case as~well.
\end{proof}

\begin{corollary} \label{cor-convergence}
Under the hypothesis of Theorem \ref{th-uniform-bound}, for any sequence $R_k \to \infty$, there exists a subsequence of the invading domain solutions $\mathbf{w}_k$ to \eqref{NSE-k} which converges weakly  to a $D$-solution $\mathbf{w}_L$ to the Navier--Stokes equations \eqref{NSE}$_{1,2}$. Moreover, the convergence is strong in the $C^m_{\mathrm loc}$-topology in $\mathbb{R}^2 \setminus \bar{B}_R$  for any $m \ge 0$, and also strong in $L^2(B_{2R})$.
\end{corollary}

\begin{proof}
Theorem \ref{th-uniform-bound} implies that
\begin{equation} \label{eq-710}
\|\mathbf{w}_k\|_{H^1(B_\rho)} \le C(A, R, \lambda, \rho)
\end{equation}
uniformly in $k$, for any $\rho>0$.  Hence, we can extract a subsequence of $\mathbf{w}_k$ which converges weakly to some function $\mathbf{w}_L \in H^1_{\mathrm{loc}}(\mathbb{R}^2)$ satisfying the equations \eqref{NSE}$_{1,2}$ and
\begin{equation}
\int_{\mathbb{R}^2} |\nabla \mathbf{w}_L|^2 \le \varliminf\limits_{k\to \infty} \int_{B_{R_k}} |\nabla \mathbf{w}_k|^2 \le C(A, R, \lambda)
\end{equation}
By the standard local regularity theory for stationary Navier--Stokes equations (see, \emph{e.g.}, \cite[Section 2.3]{KR21}), we can obtain uniform local $C^m$ bounds for $\mathbf{w}_k$ in $\mathbb{R}^2 \setminus \bar{B}_R$ (outside the support of $\mathbf{f}$) for any $m > 0$. This together with Arzela-Ascoli theorem implies the local strong $C^m$ convergence in $\mathbb{R}^2 \setminus \bar{B}_R$. Finally, \eqref{eq-710} and the compact embedding $H^1(B_{2R}) \subset L^2(B_{2R})$ implies the strong convergence in $L^2(B_{2R})$.
\end{proof}

\section{The limiting velocity of $\mathbf{w}_L$} \label{sec-limit}

Let $\mathbf{w}_k$ be the invading domain solutions that we studied in the last section. By passing to a subsequence, we assume that $\mathbf{w}_k$ converges weakly to $\mathbf{w}_L$ in the sense described in Corollary \ref{cor-convergence}. In this section, we study the fundamental question: \emph{does $\mathbf{w}_L$ achieve the prescribed limiting velocity $\mathbf{w}_\infty = \lambda \mathbf{e}_1$ at spatial infinity?}

Since $\mathbf{w}_L$ is a $D$-solution to the Navier--Stokes equations (without force) in the exterior domain $\mathbb{R}^2 \setminus B_{2R}$, it has some finite uniform limit at spatial infinity (this important fact is established through the papers \cite{GW, Amick, KPR}). Let us denote $\mathbf{w}_0 = \lim_{r\to \infty} \mathbf{w}_L$.  We state a simple but crucial identity for the tail Dirichlet integral
\begin{equation} \label{eq-81}
D_* = \lim_{r \to \infty} \overline{\lim_{k \to \infty}}  \int_{\Omega_k \cap \{|z| \ge  r\}} |\nabla \mathbf{w}_k|^2 dxdy.
\end{equation}

\begin{lemma} \label{lem-D*}
If $\mathbf{w}_0 \neq 0$, then $D_* = - \mathcal{F} \cdot (\mathbf{w}_\infty - \mathbf{w}_0)$.
\end{lemma}
\begin{proof}
Recall the following energy equalities for the stationary Navier--Stokes solutions $\mathbf{w}_k$ and $\mathbf{w}_L$ respectively,
\begin{equation} \label{eq-82}
D_k = \int_{B_{R_k}} |\nabla \mathbf{w}_k|^2 = \int_{B_{R_k}} \mathbf{f} \cdot (\mathbf{w}_k - \mathbf{w}_\infty) = \int_{B_{R_k}} \mathbf{f} \cdot \mathbf{w}_k  -  \mathcal{F} \cdot \mathbf{w}_\infty,
\end{equation}
\begin{equation} \label{eq-83}
D_L = \int_{\mathbb{R}^2} |\nabla \mathbf{w}_L|^2 = \int_{\mathbb{R}^2} \mathbf{f} \cdot (\mathbf{w}_L - \mathbf{w}_0) = \int_{\mathbb{R}^2} \mathbf{f} \cdot \mathbf{w}_L  -  \mathcal{F} \cdot \mathbf{w}_0.
\end{equation}
The last line relies on the asymptotic behaviour of $\mathbf{w}_L$ at spatial infinity when $\mathbf{w}_0 \neq 0$. By the result of \cite{S}, $\mathbf{w}_L$ is \emph{physically reasonable} in the sense of \cite[page 350]{Sm}. In \cite[Theorem 5]{Sm}  it is proved that $\mathbf{w}_L$ exhibits the wake region behaviour, and has the polynomial convergence rates at spatial infinity indicated by the Oseen fundamental tensor (with some logarithmic correction). Thus \eqref{eq-83} can be obtained with multiplying \eqref{NSE}$_1$ by $\mathbf{w}_L - \mathbf{w}_0$ and integrating on $B_\rho$, then sending $\rho \to \infty$. Observe that $\int \mathbf{f} \cdot \mathbf{w}_k \to \int \mathbf{f} \cdot \mathbf{w}_L$ since $\mathbf{w}_k \rightharpoonup \mathbf{w}_L$ in $H^1(B_{2R})$. It remains to conclude using the simple fact that
\begin{align}
D_* &= \lim_{r \to \infty}  \overline{\lim_{k \to \infty}}  \left\{ D_k -\int_{\Omega_k \cap \{|z| \le  r\}} |\nabla \mathbf{w}_k|^2 dxdy \right\} \\
&=  \overline{\lim_{k \to \infty}}  D_k - D_L.
\end{align}
\end{proof}

Next, we justify the equality $\mathbf{w}_0 = \mathbf{w}_\infty$ in two different scenarios.

\subsection{Scenario I : $\lambda$ is much larger than force} \label{sec-81}

\begin{theorem}  \label{th-s1}
There exists an universal constant ${\varepsilon}>0$ such that,
under the hypothesis of Theorem \ref{th-uniform-bound}, if
 \begin{equation} \label{eq-s1-1}
A\le \frac{{\varepsilon}^2}{\ln^\frac12  \left(\frac{1}{\lambda R} + 2\right)}\lambda,
 \end{equation}
then we have
\begin{enumerate}
\item $\int_{B_{R_k}} |\nabla \mathbf{w}_k|^2 \le C_1 {\varepsilon}^2 \lambda^2$, for $R_k \ge 2\hat{R} := 2\max\{R, \frac{1}{\lambda}\}$,
\item $\max_{\hat{R} \le r \le R_k}|\bar{\mathbf{w}}_k(r)| \le C_2\lambda$,
\item The limiting solution $\mathbf{w}_L$ satisfies the condition $\mathbf{w}_0 = \mathbf{w}_\infty = \lambda \mathbf{e}_1$.
\end{enumerate}
Here again $C_i$ are some universal constants (do not depend on~${\mathbf w}_k$, $A$, $\lambda$, etc.). If, in addition, the total force $\mathcal{F} = 0$, then the factor $\ln^\frac12  \left(2+\frac{1}{\lambda R}\right)$ in \eqref{eq-s1-1} can be removed.
\end{theorem}

\begin{proof}

To prove Claims~1--2, we consider two separate cases:

\smallskip
{\bf Case~I: $\lambda R \ge 1$.} In this case, $ \ln^\frac12  \left(\frac{1}{\lambda R} + 2\right)  \sim 1$. The term $\frac{A^{\frac13}}{R^{\frac23}}$ in \eqref{eq31}--\eqref{eq31-1}  can be estimated as
\begin{align}\label{est-sc1-1}
\frac{A^{\frac13}}{R^{\frac23}}=\frac{A^{\frac13}\lambda^{\frac23}}{R^{\frac23}\lambda^{\frac23}}&\le  A^{\frac13}\lambda^{\frac23}  \overset{\eqref{eq-s1-1}}{\le} {\varepsilon}^\frac23 \lambda
\end{align}
Then Claims~1--2 follows immediately from \eqref{eq-s1-1} and Theorem~\ref{th-uniform-bound}.

\smallskip
{\bf Case~II: $\lambda R < 1$.} By Lemma \ref{lem-change-of-domain}, we have
\begin{equation}
\|\mathbf{f}\|_{H^{-1}(B_\frac{2}{\lambda})} \le C \left( 1 + \ln ^\frac12 \left(\frac{1}{\lambda R}\right) \right) \|\mathbf{f}\|_{H^{-1}(B_{2R})}
\end{equation}
It is easy to check that, in this case, $ 1 + \ln ^\frac12 \left(\frac{1}{\lambda R}\right)  \sim \ln^\frac12  \left(\frac{1}{\lambda R} + 2\right)$. Since $\frac{1}{\lambda} > R$, $\mathbf{f}$ is of course compactly supported in $B_{\frac{1}{\lambda}}$. Hence, we can reduce to Case I with the new parameters $$R' = \frac{1}{\lambda}$$ and
$$A' \le C \ln^\frac12 \left(\frac{1}{\lambda R} + 2\right) A.$$
Hence, Claims~1--2 are again valid in this case.

Note that when $\mathcal{F} = 0$, the logarithmic factors can be removed due to the last statement in Lemma \ref{lem-change-of-domain}.

\smallskip
To prove Claim~3, notice that we are now in a position similar to \cite{KR21new}. More precisely,  if ${\varepsilon}<{\varepsilon}_0$ with ${\varepsilon}_0$ small enough, then we have

\smallskip

(a) $D_k = \int |\nabla \mathbf{w}_k|^2  \le {\varepsilon} \lambda^2$,

\smallskip

(b) The total force $|\mathcal{F}| = \left| \int_{\mathbb{R}^2} \mathbf{f} \right| \le {\varepsilon} \lambda$.

\smallskip

These are the key ingredients to invoke the proof of \cite[Theorem 1]{KR21new} (see Section 6 there). Here we can simplify the proof essentially using our second basic estimate in Theorem~\ref{th:estim2}. Namely, using Corollary~\ref{cor-convergence}, we can find $r_{1k} \to \infty$ such that  \eqref{eq:r-ass} and \eqref{e-eq:estim-a1} are valid. By Theorem \ref{th:estim2} we have
 \begin{equation} \label{eq-86}
|{\mathbf w}_0-{\mathbf w}_\infty|\le C_{**}\frac{D_*}{\lambda},
 \end{equation}
where $D_*$ is the tail Dirichlet energy defined in \eqref{eq-81} which controls the limit $\varliminf\limits_{k\to\infty}\int_{\Omega_{r_{1k},r_{2k}}}|\nabla{\mathbf w}_k|^2$. By Lemma \ref{lem-D*} and (a), (b), we have
\begin{equation} \label{eq-87}
D_* \le \begin{cases}
|\mathcal{F}| |\mathbf{w}_\infty - \mathbf{w}_0| \le {\varepsilon} \lambda |\mathbf{w}_\infty - \mathbf{w}_0| , \quad \mathrm{if} \ \mathbf{w}_0 \neq 0, \\
 \varliminf\limits_{k\to \infty} D_k \le {\varepsilon} \lambda^2, \quad \mathrm{if} \ \mathbf{w}_0 = 0.
\end{cases}
\end{equation}
By \eqref{eq-86} -- \eqref{eq-87} we obtain $|\mathbf{w}_\infty - \mathbf{w}_0| \le C_{**} {\varepsilon}|\mathbf{w}_\infty - \mathbf{w}_0|$ which, by taking ${\varepsilon}$ small, immediately implies $\mathbf{w}_\infty = \mathbf{w}_0$.

\end{proof}

\subsection{Scenario II :  zero total force, and $\lambda = 0$}

\begin{theorem}\label{th-s2}
Under the hypothesis of Theorem \ref{th-uniform-bound}, assume in addition that the total force vanishes,   \emph{i.e.}, $\mathcal{F} = \int_{\mathbb{R}^2} \mathbf{f} = \mathbf{0}$. Moreover, we consider zero prescribed limiting velocity, \emph{i.e.}, $\mathbf{w}_\infty = 0$. Then the limiting solution $\mathbf{w}_L$ satisfies the condition $\mathbf{w}_0 = \mathbf{w}_\infty = 0$.
\end{theorem}

\begin{proof}
We assume that $\mathbf{w}_0 \neq 0$ and  obtain a contradiction. By Lemma \ref{lem-D*} we have
\begin{equation} \label{eq-88}
D_* = - \mathcal{F} \cdot (0 - \mathbf{w}_0) = 0
\end{equation}
Using Corollary \ref{cor-convergence}, we can find $r_{1k} \to \infty$ such that  \eqref{eq:r-ass} and \eqref{e-eq:estim-a1} are valid. By Theorem \ref{th:estim2} we have
 \begin{equation} \label{eq-89}
|{\mathbf w}_0-{\mathbf w}_\infty|\le C_{**}\frac{D_*}{|\mathbf{w}_0|},
 \end{equation}
where $D_*$ is the tail Dirichlet energy defined in \eqref{eq-81} which controls the limit $\varliminf\limits_{k\to\infty}\int_{\Omega_{r_{1k},r_{2k}}}|\nabla{\mathbf w}_k|^2$. \eqref{eq-88} -- \eqref{eq-89} immediately imply that $\mathbf{w}_0 = \mathbf{w}_\infty$, a contradiction with the initial assumption.
\end{proof}

\section{A uniqueness theorem in the case of small force} \label{sec-unique}

\begin{theorem} \label{th-uniq} {
There exists a~universal constant ${\varepsilon}>0$ such that,
under the hypothesis of Theorem \ref{th-uniform-bound}, if
 \begin{equation} \label{eq-uni-1}
  A< \frac{{\varepsilon}^2}{(1+\lambda R)^3\ln^\frac12  \left(2+\frac{1}{\lambda R}\right)}\lambda,
 \end{equation}
then the problem \eqref{NSE} is uniquely solvable in the class of $D$-solutions. If, in addition, the total force $\mathcal{F} = 0$, then the factor $\ln^\frac12  \left(2+\frac{1}{\lambda R}\right)$ in \eqref{eq-uni-1} can be removed.}
\end{theorem}

\begin{remark}
 In fact, the essence of the proof below can be best understood by setting $R=1, \lambda =1$.
\end{remark}

\begin{proof}
The existence of a $D$-solution $\mathbf{w}_L$ to \eqref{NSE} when $f$ is sufficiently small was already proved in Theorem \ref{th-s1} based on the invading domains method. We just need to show uniqueness. Using scaling, we may assume, without loss of generality, that $\lambda=1$, \emph{i.e.}, $\mathbf{w}_\infty = \mathbf{e}_1$. In particular, the assumption (\ref{eq-uni-1}) will now read
 \begin{equation} \label{eq-uni-0}
  A< \frac{{\varepsilon}^2}{(1+R)^3\ln^\frac12  \left(2+\frac{1}{R}\right)}. \end{equation}
 By the results of \cite{S, Sm}, $\mathbf{w}$ is physically reasonable in the sense of \cite[page 350]{Sm}. In \cite[Theorem 5]{Sm}, it is proved that $\mathbf{w}$ exhibits the wake region behaviour, and has the polynomial convergence rates at spatial infinity indicated by the Oseen fundamental tensor (with some logarithmic correction). Hence, the following energy equality is valid in whole space,
\begin{equation}
D = \int_{\mathbb{R}^2} |\nabla \mathbf{w}|^2 = \int \mathbf{f} \cdot (\mathbf{w} - \mathbf{e}_1).
\end{equation}
Using the ideas of proving Theorems~\ref{th-uniform-bound}, \ref{th-s1}, we obtain the same type of estimate, that is,
\begin{equation} \label{eq5D}
D \le C A\lambda\le C\frac{{\varepsilon}^2}{(1+R)^3\ln^\frac12  \left(2+\frac{1}{R}\right)}.
\end{equation}
Below we have to consider two different cases:

\medskip
\centerline{\bf Case~I: $R\ge1$.}
Then by Claim~2 of Theorem~\ref{th-s1} we obtain
\begin{equation} \label{eq5supw-}
\sup_{r \ge R} |\bar{\mathbf w}(r)| \le C.
\end{equation}
More precisely, using the first basic estimate of Theorem~\ref{th:estim1}, from (\ref{eq5D}) we obtain immediately
\begin{equation} \label{eq5supw}
\sup_{r \ge R} |\bar{\mathbf w}(r)-{\mathbf e}_1| \le C\frac{{\varepsilon}}{(1+R)^{3/2}}.
\end{equation}
As a by-product in the proof of Theorem~\ref{th:estim1}, from the estimate~(\ref{f-add8}) we also have
\begin{equation} \label{eq54}
\int_{r \ge \frac32 R } r |\nabla \omega|^2  \le C\frac{{\varepsilon}^2}{(1+R)^3},
\end{equation}
and consequently,
 \begin{equation} \label{eq54-}
\int_{r\ge \frac32 R }|\nabla \omega|^2  \le C\frac{{\varepsilon}^2}{(1+R)^{4}}.
\end{equation}
Next, the proof will be divided into a few steps.

{
\medskip
\textsc{Step 1.}  For simplicity, we denote
\begin{equation}
{\tilde\varepsilon}=R^{-3/2}{\varepsilon},
\end{equation}
so that
\begin{equation}\label{equniq-0}
D+\int_{r \ge \frac32 R } r |\nabla \omega|^2  \le C{\tilde\varepsilon}^2,
\end{equation}
\medskip
\begin{equation}\label{equniq-00}
\sup_{r \ge R} |\bar{\mathbf w}(r)-{\mathbf e}_1| \le C{\tilde\varepsilon}.
\end{equation}
By \eqref{equniq-0}--\eqref{equniq-00}, we can find a good circle $S_{r_*}, r_* \in [\frac32R, 2R]$, on which
\begin{equation}\label{equniq-1}
\max\limits_{z\in S_{r_*}}|{\mathbf w}(z)-{\mathbf e}_1|\le C{\tilde\varepsilon}
\end{equation}
\begin{equation} \label{equniq-1-1}
\max\limits_{z\in S_{r_*}}|p(z) - \bar{p}(r_*)| \le C {\tilde\varepsilon}.
\end{equation}
Without loss of generality, let us assume that $p \to 0$ as $r \to \infty$. Due to Lemma \ref{lem:pressure}, we have
\begin{equation}
|\bar{p}(r_*)|\le C{\tilde\varepsilon}^2.
\end{equation}
Using the standard decomposition of pressure (see, \emph{e.g.},~(\ref{decomp0}) or  \cite[Corollary 16]{KR21new}), we deduce that
\begin{equation}\label{equniq-2}
|p(z)|\le C{\tilde\varepsilon}\qquad\mbox{ for }|z|\ge 2R.
\end{equation}
Following \cite[Section 3]{KR21}\footnote{Amick's special $\omega$-level sets as described in \cite[Lemma 13]{KR21} are still available in the whole plane setting outside the support of $\mathbf{f}$.}, we can find a sequence of good radii $r_k \in [2^kR, 2^{k+1}R), \\ k=1,2, 3, \cdots$ such that
\begin{equation}
|\mathbf{w} - \mathbf{e}_1| \le C {\tilde\varepsilon},
\end{equation}
on each circle $S_{r_k}$. Locating these ``good circles" is a necessary preparation for Step~2.

\medskip
\textsc{Step 2.} Recall Amick's auxiliary function $\gamma = \frac{|{\mathbf w}|^2}{2} + p - \omega \psi$. Here $\psi$ is the stream function satisfying $\nabla \psi = {\mathbf w}^\perp = -w_2 \mathbf{e}_1 + w_1 \mathbf{e}_2$. Let $\psi(2R, 0) = 0$ to be definite. It is known that $\gamma$ satisfies the two-sided maximum principle and converges to $\frac12$ at infinity, see \cite{Amick}. Using \eqref{equniq-1}--\eqref{equniq-1-1}, (\ref{equniq-2}), and the maximum principle for Bernoulli pressure $\Phi = \frac{|{\mathbf w}|^2}{2} + p$, we have
\begin{equation}
|\mathbf{w}(z)| \le C,
\end{equation}
for all $|z| \ge r_*$.
Similarly, using (\ref{equniq-0})--(\ref{equniq-1-1}) and the two-sided maximum principle for $\gamma$  we can deduce that
\begin{equation}
|\gamma(z) - \frac{1}{2}| \le C R^\frac12 {\tilde\varepsilon}
\end{equation}
for all $|z| \ge 2R$. The $R^\frac12$ factor shows up here because, on a typical circle $S_{r_*}$, $R<r_*<2R$, the term $\omega \psi$ in the definition of $\gamma$ is bounded by $R^{-\frac12} {\tilde\varepsilon} \cdot R = R^\frac12 {\tilde\varepsilon}$. Following \cite[Appendix~I]{KR21}, and using the~information~from Step~1
we obtain the key pointwise estimate
\begin{equation} \label{amick-prel}
|\mathbf{w}(z) - \mathbf{e}_1| \le C \, R^\frac12 \, {\tilde\varepsilon}=C\, R^{-1}\, {\varepsilon},
\end{equation}
for all $|z| \ge 2R$.
Also, from~(\ref{scale3}) and (\ref{equniq-0})--(\ref{equniq-00}) we have
\begin{equation} \label{ballest}
\|\mathbf{w} - \mathbf{e}_1\|_{L^q(B_{3R})}\le C \,R^{\frac2q-\frac{3}{2}} \, {\varepsilon},
\end{equation}
in particular,
\begin{equation} \label{ballest-1}
\|\mathbf{w} - \mathbf{e}_1\|_{L^5(B_{3R})}\le C \,R^{-\frac{11}{10}} \, {\varepsilon}.
\end{equation}
}

\medskip
\textsc{Step 3.} Following \cite[Section 5]{KR21}\footnote{The rescaling procedure in \cite{KR21} is not needed here since $\lambda$ is fixed to be 1. However we need to deal with the $R$-dependence in the estimates here while $R=1$ in \cite{KR21}. The big denominator $(1+R)^3$ in (\ref{eq5D}) is useful to absorb all the $R$-dependence that shows up.}, and setting ${\varepsilon}$ sufficiently small, one can obtain the pointwise decay estimate
\begin{equation} \label{eq5amick}
|(\mathbf{w} - \mathbf{e}_1)_i(z)| \le C {\varepsilon} h_i(z), \quad i = 1,2,
\end{equation}
for $|z| \ge 2R$, where $h_i(z)$ is the majorant function given by \footnote{In \cite{FS, KR21}, $h_i(z)$  is defined as $\log \frac{2}{|z|}$ inside the unit disk. Such a logarithmic singularity plays an crucial role there as it accounts for the delicate boundary effects in the flow around obstacle problem. For the whole plane problem, we only use the definition of $h$ outside $B_1$.}
\begin{align}
 |z|>1 &:  \quad \begin{cases}
                   h_1(z) = |z|^{-\frac12},\\
                   h_2(z) = |z|^{-\frac35}.
                  \end{cases}
\end{align}

\textsc{Step 4.}
Let
\begin{equation}
I_i(z)(\mathbf{u}, \mathbf{v})= \int_{\mathbb{R}^2} \partial_l E_{ij}(z-z') u_j(z') v_l(z') \, dz'_1 dz'_2:=I^R_i(z)(\mathbf{u}, \mathbf{v})+I^\infty_i(z)(\mathbf{u}, \mathbf{v}),
\end{equation}
where
\begin{equation}
I^R_i(z)(\mathbf{u}, \mathbf{v}) := \int_{B_{2R}} \partial_l E_{ij}(z-z') u_j(z') v_l(z') \, dz'_1 dz'_2,
\end{equation}
\begin{equation}
I^\infty_i(z)(\mathbf{u}, \mathbf{v}) := \int_{\mathbb{R}^2\setminus B_{2R}} \partial_l E_{ij}(z-z') u_j(z') v_l(z') \, dz'_1 dz'_2.
\end{equation}
Here $\mathbf{E}$ is the fundamental Oseen tensor, i.e, the fundamental solution to the Oseen system:
\begin{equation}
\left\{
\begin{aligned}
 & \Delta E_{ij} - \partial_1 E_{ij} - \partial_i e_j = \delta_{ij} \delta_0, \quad i,j = 1, 2, \\
 & \sum_{i=1,2} \partial_i E_{ij} = 0, \quad j =1,2,
\end{aligned}
\right.
\end{equation}
where $\delta_0$ is the standard $\delta$-function supported at the origin. For more information on $\mathbf{E}$, see for instance \cite[Section 2.4]{KR21} or \cite[Section 2]{GuilW1}.

Recall, that $\partial_l E_{ij}\in L^q({\mathbb R}^2)$ for any $q\in(3/2,2)$, in particular,
\begin{equation}\label{bile1}
\partial_l E_{ij}\in L^{5/3}({\mathbb R}^2)
\end{equation}
(for the list of integrability properties for $E_{ij}$, see, \emph{e.g.}, in~\cite{S}). Define the auxiliary norm
\begin{equation}
\|\mathbf{u}\|_Y : = \sup_{|z| \ge 2 R; i=1,2} \frac{|u_i(z)|}{h_i(z)}.
\end{equation}

We will need a crucial bilinear estimate (see \cite[Lemmas~3.1--3.2]{FS}): there exists an absolute constant $C$ such that
\begin{equation} \label{eq5bilinear}
\sup_{|z| \ge 1; i=1,2} \frac{|I^\infty_i(z)(\mathbf{u}, \mathbf{v})|}{h_i(z)}\le C\,\|\mathbf{u}\|_Y
\|\mathbf{v}\|_Y.
\end{equation}
In particular,
\begin{equation} \label{bile2-} \left| \mathbf I^\infty(z)(\mathbf{u}, \mathbf{v}) \right| \le
                   C\,\|\mathbf{u}\|_Y
\|\mathbf{v}\|_Y|z|^{-\frac12},\qquad |z|\ge1.
\end{equation}
Further, from (\ref{bile1}) and the elementary estimate $\|\mathbf{u}\|_{L^5({\mathbb R}^2\setminus B_{2R})}\le C\,R^{-\frac1{10}}\|\mathbf{u}\|_Y$ we have:
\begin{equation} \label{bile2}\left| \mathbf I^\infty(z)(\mathbf{u}, \mathbf{v}) \right|\le
                   C\,\|\mathbf{u}\|_Y
\|\mathbf{v}\|_YR^{-\frac15},\ \qquad |z|\le1
\end{equation}
(really, the last pointwise estimate is valid for all $z\in{\mathbb R}^2$, but we will use it in the unit ball only).
Therefore, from~(\ref{bile2-})--(\ref{bile2}) we conclude:
\begin{equation} \label{bile3}
\|\mathbf I^\infty\|_{L^5(B_{2R})}\le C\,\|\mathbf{u}\|_Y
\|\mathbf{v}\|_Y.
\end{equation}
\begin{equation} \label{bile4}
\|\mathbf I^\infty\|_{Y}\le C\,\|\mathbf{u}\|_Y
\|\mathbf{v}\|_Y.
\end{equation}
Similarly, from (\ref{bile1})  and H\"older's inequality we have:
\begin{equation} \label{bile5}
|\mathbf I^R(z)(\mathbf{u}, \mathbf{v})|\le
                   C\,\|\mathbf{u}\|_{L^5(B_{2R})}
                   \|\mathbf{v}\|_{L^5(B_{2R})},\qquad\ \forall z\in{\mathbb R}^2.
\end{equation}
In particular,
\begin{equation} \label{bile6}
\|\mathbf I^R\|_{L^5(B_{2R})}\le C\,\|\mathbf{u}\|_{L^5(B_{2R})}
\|\mathbf{v}\|_{L^5(B_{2R})}R^{\frac25}.
\end{equation}
It is well-known that
$|\nabla\mathbf E(z)|\le C\,\frac{1}{|z|}$  for $|z|\ge1$ (see, \emph{e.g.},~\cite[\S2]{Sm}\,). This implies
\begin{equation} \label{bile7}
|\mathbf I^R(z)(\mathbf{u}, \mathbf{v})|\le
                   C\,\|\mathbf{u}\|_{L^5(B_{2R})}
                   \|\mathbf{v}\|_{L^5(B_{2R})}\frac{R^\frac65}{|z|},\qquad\ \forall z\in{\mathbb R}^2\setminus B_{2R}.
\end{equation}
Therefore,
\begin{equation} \label{bile9}
\|\mathbf I^R(\mathbf{u}, \mathbf{v})\|_{Y}\le C\,\|\mathbf{u}\|_{ L^5(B_{2R})}
\|\mathbf{v}\|_{L^5(B_{2R})}R^{ \frac45}.
\end{equation}
Finally, we can summarize all previous estimates in the following form:
\begin{equation}\label{sum-bil1}
\|\mathbf I(\mathbf{u}, \mathbf{v})\|_{X}\le C\,\|\mathbf{u}\|_X \biggl(
\|\mathbf{v}\|_Y +\|\mathbf{v}\|_{L^5(B_{2R})} R^{\frac45}\biggr),
\end{equation}
and also
 \begin{equation}\label{sum-bil2}
\|\mathbf I(\mathbf{u}, \mathbf{v})\|_{X}\le C\,\|\mathbf{v}\|_X \biggl(
\|\mathbf{u}\|_Y +\|\mathbf{u}\|_{L^5(B_{2R})} R^{\frac45}\biggr),
\end{equation}
where we denote
\begin{equation}
\|\mathbf{u}\|_X : =\|\mathbf{u}\|_Y+ \|\mathbf{u}\|_{L^5(B_{2R})}.
\end{equation}

\textsc{Step 5.} The local uniqueness argument we are going to invoke next is similar to that in \cite[Section 7]{FS}. Suppose $\mathbf{w}^{(1)}, \mathbf{w}^{(2)}$ are two $D$-solution to \eqref{NSE}. Denote $\mathbf{u}^{(k)} := \mathbf{w}^{(k)} - \mathbf{e}_1$, $k =1,2$,
then by Steps~2--3 we have
\begin{equation} \label{eq5Xsmall}
\|\mathbf{u}^{(k)}\|_X\le C {\varepsilon}, \quad k = 1,2,
\end{equation}
\begin{equation} \label{eq5Xsmall--}
\|\mathbf{u}^{(k)}\|_{L^5(B_{2R})}\le C {\varepsilon} R^{-\frac{11}{10}}, \quad k = 1,2.
\end{equation}
Since $\mathbf{u}^{(k)}$ satisfies the system
\begin{equation}
\left\{
\begin{aligned}
&\Delta \mathbf{u}^{(k)}  - \partial_1 \mathbf{u}^{(k)} - \nabla p^{(k)} = \mathbf{u}^{(k)} \cdot \nabla \mathbf{u}^{(k)} + \mathbf{f},  \\
& \nabla \cdot \mathbf{u}^{(k)} = 0,
\end{aligned}
\right.
\end{equation}
and the decay estimate \eqref{eq5Xsmall}, we obtain the following standard representation formula,
$$
\mathbf{u}^{(k)}(z) = \int_{\mathbb{R}^2} \mathbf{E}(z-z') \cdot \left\{ (\mathbf{u}^{(k)} \cdot \nabla) \mathbf{u}^{(k)} + \mathbf{f} \right\}(z') \, dz'_1 dz'_2, \quad  k=1,2.
$$
From this representation we deduce
\begin{align} \label{eq5udiff}
(u_i^{(1)} - u_i^{(2)})(z) &= \int E_{ij}(z-z') (\mathbf{u}^{(1)} \cdot \nabla \mathbf{u}^{(1)} - \mathbf{u}^{(2)} \cdot \nabla \mathbf{u}^{(2)})_j(z') \, dz'_1 dz'_2 \nonumber \\
& = - \int \partial_l E_{ij}(z-z') (u^{(1)}_l u^{(1)}_j - u^{(2)}_l u^{(2)}_j)(z') \, dz'_1 dz'_2 \nonumber\\
& = - I_i(\mathbf{u}^{(1)} - \mathbf{u}^{(2)}, \mathbf{u}^{(2)}) - I_i(\mathbf{u}^{(1)}, \mathbf{u}^{(1)} - \mathbf{u}^{(2)}) .
\end{align}
Using \eqref{eq5Xsmall}--\eqref{eq5Xsmall--} and \eqref{sum-bil1}--\eqref{sum-bil2} we obtain
\begin{align*}
& \quad \  \|\mathbf{u}^{(1)} - \mathbf{u}^{(2)}\|_X\\ &\le C \|\mathbf{u}^{(1)} - \mathbf{u}^{(2)}\|_X \left( \|\mathbf{u}^{(1)}\|_Y + \|\mathbf{u}^{(2)}\|_Y +\|\mathbf{u}^{(1)}\|_{L^5(B_{2R})} R^{\frac45} + \|\mathbf{u}^{(2)}\|_{L^5(B_{2R})} R^{\frac45}\right) \\
&\le C' {\varepsilon}\|\mathbf{u}^{(1)} - \mathbf{u}^{(2)}\|_X,
\end{align*}
which immediately implies that $\mathbf{u}^{(1)} \equiv \mathbf{u}^{(2)}$ when ${\varepsilon}$ is sufficiently small.
\medskip

\centerline{\bf Case~II: $R<1$.}

\medskip

The proof here is similar to Case II in the proof of Theorem \ref{th-s1}. By Lemma \ref{lem-change-of-domain}, we have
\begin{equation}
\|\mathbf{f}\|_{H^{-1}(B_2)} \le C \left( 1 + \ln ^\frac12 \left(\frac{1}{R}\right) \right) \|\mathbf{f}\|_{H^{-1}(B_{2R})}
\end{equation}
 Since $R < 1$, $\mathbf{f}$ is of course compactly supported in $B_1$. Hence, we can reduce to Case I with the new parameters $$R' = 1$$ and
$$A' \le C \ln^\frac12 \left(2+\frac{1}{R}\right) A.$$
Finally, in the case that $\mathcal{F} = 0$, the logarithmic factors here can be removed due to the last statement in Lemma \ref{lem-change-of-domain}.

\end{proof}

redpajama_set_name: RedPajamaArXiv


At the end of a busy day, travelers can unwind and relax in the hotel or go out and enjoy the city. This Lijiang hotel offers parking on site.
The Fairyland Hotel (Lijiang Huama Street) is an ideal spot for travelers wanting to discover the city. The Fairyland Hotel (Lijiang Huama Street) offers a pleasant stay in Lijiang for those traveling for business or leisure.
Traveling to the hotel is easy with Lijiang Railway Station located approximately 10km away and Sanyi Airport roughly 30km away. Guests will find "The Sound of Yunnan" Show, Lishui Jinsha Performance and YuMiHe ShangYe BuXingJie just a short distance from the hotel.


Gingerbread is the ultimate Christmas biscuit. It not only tastes great but will undoubtedly provide some festive entertainment as your housemates battle it out for the position of best biscuit decorator. Get inventive with your designs and the biscuits you manage not to eat straight away can be used to decorate your tree! This recipe makes about 25 biscuits.
Preheat oven to 170°C / gas mark 3.
Melt butter, sugar and syrup together in a saucepan on low heat; stir and remove from heat.
Sieve flour, bicarbonate of soda and ginger into a bowl.
Stir melted ingredients into dry ingredients to make a dough.
Lightly flour a surface and roll dough to a thickness of ½ cm.
Cut shapes into dough using cutters or templates – gingerbread men, stars, Christmas trees…the options are endless!
Place shapes on lined baking trays and bake in batches for 9-10 minutes each (until golden brown), then remove from oven.
Whilst the biscuits cool, mix icing sugar with a splash of water to make thick icing, then drizzle or pipe over biscuits and decorate with sprinkles.
Store your gingerbread biscuits in an airtight container to last for up to 2 weeks.
Top Tip: Whilst the biscuits are still warm, make holes using a skewer or chopstick so they can be hung on your tree using ribbon or string!
Let us know how yours turn out – send us your winning biscuit designs!


LONDON, England (CNN) -- Despite the fact that some of the effects of climate change are still imperceptible to all but the most highly trained eyes, there can be no doubting the alarming picture painted by glacier melt.
The iPhone application includes audio, video, slideshows and a documentary film.
As these rivers of ice retreat back up the valleys they carved out, so scientists' knowledge of climate change advances, in turn helping us recognize the signs of a warming world.
Now a new iPhone app is helping visitors to the Swiss Alps understand how climate change is altering the landscape.
Developed by the Oeschger Center for Climate Change Research at the University of Bern in Switzerland and Swiss software company, Texetera, the Jungfrau Climate Guide is an interactive guide to glaciers and climate change.
For a fee of 20 CHF (around $19) visitors to the Jungfrau Alpine region can hire an iPhone loaded with the app.
Kaspar Meuli, Head of Communications at the Oeschger Center, told CNN what people can expect from the device.
"There are many different levels to the application. Audio pieces -- like the ones you might expect to find in a museum guide which guide you through paths on seven different trails."
At different stages along each trail, the phones -- aided by the built-in GPS -- alert visitors to particular points of interest by sounding a yodel.
"For example," Meuli explained, "if you are standing in front of a glacier you will be told why it is no longer as big and provided with images of what it looked like 100 years ago, and what it might look like in the future."
There are more than 40 audio guides, plus a range of multimedia offerings including videos, slide shows and a documentary film.
There are also sections which describe the various flora on the trails along with information on biodiversity.
"Everyone who tries it is quite fascinated by it," Meuli said. "People are realizing that climate change is real when they see the consequences."
The Jungfrau region is home to the upper and lower Grindelwald glaciers, which have retreated rapidly in recent years.
In 2008, the World Glacier Monitoring Service reported that glaciers in some regions of the world were melting away at a rate of more than a meter a year recently.
Responding to the 2008 report, Achim Steiner, Executive Director of the U.N. Environmental Program, said: "Millions, if not billions, of people depend directly or indirectly on these natural water storage facilities for drinking water, agriculture, industry and power generation."
"There are many canaries emerging in the climate change coal mine," Steiner added. "The glaciers are perhaps among those making the most noise."
In the future, Meuli says the University hopes to produce features that can be used by teachers in preparation for taking classes to the glaciers.


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redpajama_set_name: RedPajamaGithub


text:
"Transport" Transport Benin Spain Social inclusiveness
Assessment Framework Overview
This document summarises the approach taken by Infrastructure Australia (IA) in assessing and prioritising initiatives and project proposals for placement on the Infrastructure Priority List (IPL).
The Urban Rail Development Handbook
This handbook synthesises and disseminates knowledge to inform the planning, implementation, and operations of urban rail projects.
Download the handbook [PDF]
Planning & Project Preparation, Procurement, Delivery
From Global Savings Glut to Financing Infrastructure: The Advent of Investment Platforms
This paper investigates the emerging global landscape for public-private co-investments in infrastructure.
Steps to prepare and accelerate PPP
The report identifies and explores six critical success factors that governments should be aware of and seriously consider when preparing an infrastructure project to be delivered as a Public-Private Partnership.
An Alternative Approach to Project Selection: The Infrastructure Prioritization Framework
The World Bank Group developed this tool to help governments systematically prioritise infrastructure investments to achieve their development goals, taking into account capacity and public resource constraints.
Partnering to Build a Better World: MDBs' Common Approaches to Supporting Infrastructure Development
The paper "Partnering to Build a Better World: MDBs' Common Approaches to Supporting Infrastructure Development" presents a brief description of how MDBs work with their Borrowing Member Countries (BMCs) .
Guide on Social Responsibility for Chinese International Contractors
The Guide defines, inter alia, the scope of application, the definition and basic principles of social responsibility, and provides specific requirements on social responsibility management.
PPP Reference Guide (version 2.0)
This second version of the PPP Reference Guide, as the first one, presents a global overview of the diversity of approaches and experiences in the implementation of PPPs and more.
Download the Reference guide [PDF]
PPP Reference Guide (Version 3)
The Reference Guide attempts to provide the most relevant examples, references and resources to help readers inform themselves on key PPP topics.
Draft Guiding Principles on Good Governance in People-First Public-Private Partnerships for the UN Sustainable Development Goals
The interest generated by the role of PPP in the implementation of the Sustainable Development Goals (SDGs) is considerable.
Port Reform Toolkit Second Edition
The Port Reform Toolkit is aimed to provide policymakers and practitioners with effective decision support in undertaking sustainable and well-considered reforms of public institutions that provide, direct, and regulate port services in developing countries.

redpajama_set_name: RedPajamaCommonCrawl


text:
WALL STREET JOURNAL-BEST SELLERS
PMN Entertainment
Bestselling Books Week Ended March 9.
1. "Green Eggs and Ham" by Dr. Seuss (Random House Books for Young Readers)
2. "Where the Crawdads Sing" by Delia Owens (G.P. Putnam's Sons)
3. "One Fish Two Fish Red Fish Blue Fish" by Dr. Seuss (Random House Books for Young Readers)
4. "You Are My Happy" by Hoda Kotb (HarperCollins)
5. "The Cat in the Hat" by Dr. Seuss (Random House Books for Young Readers)
6. "Dog Man: Brawl of the Wild" by Dav Pilkey (Graphix)
7. "Fox in Socks" by Dr. Seuss (Random House Books for Young Readers)
8. "Silent Night" by Danielle Steel (Dell)
9. "Oh, the Places You'll Go!" by Dr. Seuss (Random House books for young Readers)
10. "Cemetery Road" by Greg Iles (William Morrow)
1. "Girl, Stop Apologizing" by Rachel Hollios (HarperCollins Leadership)
2. "Becoming" by Michelle Obama (Crown)
3. "The Case for Trump" by Victor Davis Hanson (Basic Books)
4. "Girl, Wash Your Face" by Rachel Hollis (Thomas Nelson)
5. "Outer Order, Inner Calm" by Gretchen Rubin (Harmony)
6. "Brain Body Diet" by Sara Gottfried (HarperOne)
7. "There's No Place Like Space" by Tish Rabe (Random House Books for Young Readers)
8. "Oh Say Can You Say Di-no-saur?" by Bonnie Worth (Random House Books for Young Readers)
9. "Educated" by Tara Westover (Random House)
10. "The Universal Christ" by Richard Rohr (Convergent)
2. "Cemetery Road" by Greg Iles (William Morrow)
3. "The Malta Exchange" by Steve Berry (Minotaur)
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6. "Defending Morgan" by Susan Stoker (Montlake Romance)
7. "The Silent Patient" by Alex Michaelides (Celadon Books)
8. "Wild Country" by Anne Bishop (Ace)
9. "Unto Us a Son Is Given by Donna Leon (Atlantic Monthly)
10. "Prism Cloud" by Jeff Wheeler (47North)
2. "Born a Crime" by Trevor Noah (Spiegel & Grau)
5. "The Color of Water" by James McBride (Riverhead)
7. "Bad Blood" by John Carreyrou (Knopf)
9. "The Minimalist Way" by Erica Layne Nielsen (Erica Layne Nielsen)
10. "Outer Order, Inner Calm" by Gretchen Rubin (Harmony)
NPD BookScan gathers point-of-sale book data from about 16,000 locations across the U.S., representing about 85 per cent of the nation's book sales. Print-book data providers include all major booksellers and Web retailers, and food stores. E-book data providers include all major e-book retailers. Free e-books and those sold for less than 99 cents are excluded. The fiction and nonfiction lists in all formats include both adult and juvenile titles; the business list includes only adult titles. The combined lists track sales by title across all print and e-book formats; audio books are excluded. Refer questions to Michael.Boone@wsj.com.


text:
Mumtaz Mufti Book Club
Talash
Friends of MM
Qudrat Ullah Shahab
Ashfaq Ahmed
Uxi Mufti
Introduction Uxi Mufti
Books By Uxi Mufti
Uxi Mufti Photo Gallery
by Editor · Published August 24, 2015
Re-examining Pakistan's culture
KARACHI: "You will want to look for Pakistani culture after reading this book," said Mehtab Akbar Rashidi, praising the work of Uxi Mufti at the launch of his book 'Pakistani Saqafat' on Saturday evening...
Book launch sees lively debate on Pakistan's culture
KARACHI: Pakistani Saqafat, an Urdu book on Pakistani culture by Uxi Mufti, launched at the Arts Council on Saturday evening stirred a debate on different aspects of culture. Dr Asif Farrukhi, who was surprised...
by Editor · Published June 17, 2015
Uxi Mufti's book provides an insight into Pakistani culture
Uxi Mufti's book provides an insight into Pakistani cultureSyeda Shehrbano Kazim — Published Jun 06, ISLAMABAD: The Pakistan National College of Arts (PNCA) hosted a dialogue on culture in tandem with the launch of...
'Pakistani Saqafat' to be launched- THE NEWS (June 05, 2015)
Islamabad'Pakistani Saqafat', a book unveiling the common concerns of people regarding the culture of Pakistan, will be launched here on June 5 at Pakistan National Council of the Arts (PNCA) by renowned folklorist Uxi...
by Editor · Published June 4, 2015
Kaghaz ka Ghora – Paper Horse
Uxi Mufti's book "Kaghaz ka Ghora" (paper horse) is a shadow of its past, reminding that bureaucracy is still a reminder of the colonial past and acts as a hurdle than a supporting function...
Pakistani Saqafat
Pakistani Saqafat, the newly published book is vital for the understanding of Pakistani Culture. It touches many key issues and addresses common concerns such as, how is Pakistani Culture different from other cultures? How...
by Editor · Published June 4, 2015 · Last modified June 8, 2015
He was born in Gilgit in 1917 and was known for his autobiography, Shahab Nama. Shahab started writing in his early days both in Urdu and English languages. At the young age of 16,...
Bano Qudsia was born in 1928 at Ferozpur, India.She is a writer, intellectual, playwright and spiritualist from Pakistan who writes Urdu novels and short stories.She is best known for her novel "Raja Gidh" and...
Ashfaq Ahmed was born on August 22, 1925 in Garhmukteshwar village, Ghaziabad, British India. He obtained his early education in his native district. Shortly before independence in 1947, he migrated to Pakistan and made...
Measuring the Intangible
Allah in Arabic simply means God, Ishwar, Khuda, El, Elohim, Elahi , Lord, YHWH, Tetragrammaton, Allah has many names. This book provides a popular understanding of the concept of a monotheistic deity from the...
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