\label{main1}
Let $\Om$ be a bounded Lipschitz domain in $\mathbb{R}^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\in L^{2,\infty}(\Omega)^2$ and $\mathop{\rm div}\nolimits b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $\mathbb{G}\in L^q(\Om)^{2\times 2}$, there exists a  weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ and $\pi\in L^q_0(\Omega)$ of \eqref{drifteq} such that
\EQ{\label{main1-eq}
\|u\|_{ W^{1,q}(\Omega)}+\|\pi\|_{L^q(\Omega)}\le C(\Omega,\|b\|_{L^{2,\infty}})\|\mathbb{G}\|_{L^q(\Omega)}.
}
For $p_0'<q<p_0$, the above pair $(u,\pi)$ is the unique weak solution pair  in $W^{1,q}_0\times L^q_0(\Om)$ without assuming \eqref{main1-eq}.

\begin{proof}
Similar to \cite{CT1} we will use the Gehring's lemma approach \cite{Gehring, Gia}. Denote $\Om_{x,R}=\Om \cap B(x,R)$ for $x\in \overline \Om$ and $R>0$. Since $\pd \Om$ is Lipschitz and compact, there
is a radius $R_0\in (0,1]$ such that for every $x_0 \in\pd \Om$,
\EQ{ \label{R0.def}
\Om_{x_0,2R_0}= \{ (x',x_n) \in B_{2R_0}(0) \subset \R^2,\ x_n>\ga(x')\},
}
after suitable coordinate rotation and translation, where $\ga(x')$ is a Lipschitz function defined for $x'\in B_{2R_0}'(0) \subset \R$ with Lipschitz constant $L$ and $\ga(0)=0$.

To prove the a priori estimate \eqref{eq5.1}, we first consider the interior case with $x_0\in\Omega$ and $B_{\rho}=B_{\rho}(x_0)\subset\Omega$. We rescale it to the unit ball $B$ and denote $u_{\rho}(x)=\rho u(\rho x+x_0),b_{\rho}(x)=\rho b(\rho x+x_0),\pi_{\rho}(x)=\rho^2\pi(\rho x+x_0)$ and $\GG_{\rho}(x)=\rho^2 \GG(\rho x+x_0)$. Let $\bar{u}_{\rho}=u_{\rho}-k$ where $k$ is a real constant to be choose later.  By Remark \ref{Wolfpdecom}, we can decompose $\pi_{\rho} - (\pi_{\rho})_{B}=\pi_1+\pi_2+\pi_3$ in $B$, where $\pi_1,\pi_2,\pi_3$ are given by
\EQ{
\nabla\pi_1=\mathcal{W}_{4/3,B}(\Delta \bar{u}_{\rho}),\quad \nabla\pi_2=\mathcal{W}_{4/3,B}(-\div(b_{\rho}\otimes \bar{u}_{\rho})),\quad \nabla\pi_3=\mathcal{W}_{4/3,B}(\div \GG_{\rho}),
}
with $\int_B \pi_i=0$.
Therefore, by Theorem \ref{theoremL2wolf} we have the following pressure bound
\EQ{\label{wolfbound}
\|\pi_1\|_{4/3,B}\le c_1\|\nabla \bar{u}_{\rho}\|_{4/3,B},\quad\|\pi_2\|_{4/3,B}\le c_1 \|b_{\rho}\bar{u}_{\rho}\|_{4/3,B},\quad \|\pi_3\|_{4/3,B}\le c_1\|\GG_{\rho}\|_{4/3,B},
}
where $c_1$ is a global constant.
For simplicity of notations let us drop the subscript $\rho$ for the further calculations, until we need to scale back to ball $B_{\rho}(x_0)$.

Take a smooth cutoff function $\eta$ on $B$ with $\eta=1$ in $B_{\frac{1}{2}}$. Use the test function $\zeta=\eta^4 \bar{u}$ in the weak form \eqref{soln-pair} with $(u,\pi)$ replaced by $(\bar u, \pi - (\pi)_B)$ to get the energy estimate (using \eqref{5.1} due to $b \in L^{2+\de}$)
\EQS{\label{3.1}
\int_{B} |\nabla (\eta^2\bar{u})| ^2
\le &\int_{B} |\bar{u}|^2(|\nabla \eta^2|^2) +\int_{B} |\bar{u}|^2|b||\nabla\eta|\eta^3
\\
&+\int_{B} (|\pi_1|+|\pi_2|+|\pi_3|)|\bar{u}||\nabla\eta|\eta^3+\bka{\div \GG_{\rho}, \bar{u}\eta^4}_B.
}

We will estimate each of the terms on the right hand side separately, using Lemmas \ref{Holder} and \ref{Sobolev} in ball $B$,
\EQS{
\int_{B} |\bar{u}|^2|b||\nabla \eta|\eta^3
&\le \|b\|_{L^{2,\infty}(B)}\|\bar{u}^2\eta^3\|_{L^{2,1}(B)}\le \|b\|_{L^{2,\infty}(B)}\|\bar{u}\eta^{3/2}\|_{L^{4,2}(B)}^2
\\
&\le C \|b\|_{L^{2,\infty}(B)}\|\bar{u}\eta^{3/2}\|^2_{W^{1,4/3}(B)}.
}
For pressure terms we use \eqref{wolfbound} and get the following
\EQN{
\int_{B}|\pi_1||\bar{u}||\nabla\eta|\eta^3 &\le \|\pi_1\|_{4/3,B}\|\eta^2 \bar{u}\|_{L^4(B)}
\\
&\le
c\|\nabla \bar{u}\|_{4/3,B}\|\eta^2 \bar{u}\|_{W^{1,4/3}(B)}\le c\|\nabla \bar{u}\|_{4/3,B}^2+c\|\eta \bar{u}\|^2_{4/3,B},
}
\EQN{
\int_{B}|\pi_2||\bar{u}||\nabla\eta|\eta^3 &\le \|\pi_2\|_{L^{4/3}(B)}\|\eta^2\bar{u}\|_{L^4(B)}\le c\|\bar{u}\|_{L^{4,4/3}(B)}\|b\|_{L^{2,\infty}(B)}\|\eta^2\bar{u}\|_{L^{4}(B)}
\\
&\le c(b)\|\nabla \bar{u}\|_{4/3,B}^2+c(b)\| \bar{u}\|^2_{4/3,B}
}
\EQN{
\int_{B}|\pi_3||\bar{u}||\nabla\eta|\eta^3 &\le \|\pi_3\|_{4/3,B}\|\eta^2 \bar{u}\|_{L^4(B)}\le  c\|\GG\|_{2,B}\|\eta^2 \bar{u}\|_{W^{1,4/3}(B)}
\\
&\le \frac 14 \|\GG\|_{2,B}^2+c\|\eta^2 \bar{u}\|_{W^{1,4/3}(B)}^2.
}
Lastly we estimate the term containing $\GG$,
\EQN{
\bka{\div \GG_{\rho}, \bar{u}\eta^4}_B
&=-\int_{B} \GG :\nabla(\bar{u}\eta^4)=-\int_{B} \GG: \nabla(\eta^2\bar{u})\eta^2-2\int_{B} \GG :\bar{u}\nabla\eta\eta^3
\\
&\le \frac{1}{16}\|\nabla(\eta^2\bar{u})\|^2_{2,B}+c\|\eta^2 \bar{u}\|_{2,B}^2+ \frac 14\|\GG\|_{2,B}^2.
}
Combining these estimates with \eqref{3.1} we get the following
\EQ{
\int |\nabla (\eta^2\bar{u})|^2\le c(b)\|\bar{u}\|^2_{W^{1,4/3}(B)}+c(b)(\|\bar{u}\|_{2,B}^2+\|\bar{u}\|_{4/3,B}^2)+\|\GG\|_{2,B}^2,
}
Here $c(b)$ depends on $b$ only through $\norm{b}_{L^{2,\infty}(\Om)}$. Finally, since $\eta=1$ on $B_{1/2}$ we get that
\EQ{
\int_{B_{1/2}} |\nabla \bar{u}|^2\le c(b)\|\nabla \bar{u}\|^2_{4/3,B}+c(b)\|\bar{u}\|_{2,B}^2+\|\GG\|_{2,B}^2.
}
We now choose $k=(u_{\rho})_{B}$ as the constant in $\bar{u}=u_{\rho}-k$ and apply Poincar\'e inequality to get
\EQ{
\int_{B_{1/2}} |\nabla u_{\rho}|^2\le c(b)\|\nabla u_{\rho}\|^2_{4/3,B}+\|\GG\|_{2,B}^2.
}
Lastly we scale back to $B_{\rho}(x_0)$ and get that
\EQ{\label{5.12}
\frac{1}{|B_{\rho/2}|}\int_{B_{\rho/2}} |\nabla u|^2
\le c(b) \Big(\frac{1}{|B_{\rho}|}\int_{B_{\rho}}|\nabla u|^{4/3}\Big)^{3/2}+\frac{c}{|B_{\rho}|}\|\GG\|_{2,B_{\rho}}^2.
}

\medskip

Next we consider the boundary case, $\Omega_{\rho}=\Omega\cap B_{\rho}(x_0)$ with $x_0\in\pd\Om$, and $0<\rho<R_0$. The significant difference from the internal case is the pressure estimate: Unlike the internal case we can not use Wolf decomposition estimate in a uniform ball, therefore we need to track the dependence of constant $c_1$ when we apply Theorem \ref{theoremL2wolf} to get estimates similar to \eqref{pressure_bound_est}
 and make sure it is uniform across all  $\Omega_{\rho}(x_0)$.
 Rescale the domain as the interior case so that $x_0=0$ and $\rho=1$.
 We will use change of coordinates
 \EQ{\label{change-var}
 y_1=x_1, \quad y_2=x_2-\gamma(x_1),
 }
 to straighten the boundary and map $\Om_{5/8}\subset\Om_1$ into domains $\Om_{5/8}'\subset\Om_1'$ with flat boundary on $y_2=0$. Since $\gamma$ was a Lipschitz curve with sufficiently small Lipschitz constant we can assume that $\Om_{5/8}'\subset B_{6/8}^+\subset B_{7/8}^+\subset\Om_1'$.
  (This is the first place that specifies the smallness of the Lipschitz constant.) By Lemma \ref{geometry} we can fix a smooth $V$ such that $B_{6/8}^+\subset V\subset B_{7/8}^+$. After rewriting the equation \eqref{drifteq} in new variables we get the following system (with all derivatives in $y$) in $\Om_1'$:
\EQS{\label{changed_var}
-\Delta_y u+\div (\gamma' e_1\otimes\partial_{y_2}u)+
\gamma' \pd_1\pd_2 u - (\gamma')^2\partial_{y_2}^2u+
\\
+\div_y(b\otimes u)- \gamma'\partial_{y_2}(b\otimes u)_1+\nabla_y \pi-\gamma'\partial_{y_2}\pi e_1&=\div\GG-\gamma'\partial_2 \GG_1 ,
\\
\div_y u-\gamma'\partial_2 u_1&=0.
}
Above, $(b\otimes u)_1=(b_1u_1,b_1u_2)$ and $\GG_1=(G_{11},G_{12})$ are their first rows. In the second term, $\ga'$ is hidden behind $\div$ to avoid $\ga''$ which we do not assume any bound.

We apply Wolf's local pressure projection $\mathcal{W}_{4/3,V}$ on \eqref{changed_var} in the smooth domain $V$ and get that  $\pi-(\pi)_{V}=\pi_1+\pi_2+\pi_3+\pi_4$ where
\EQS{\label{pressure_decomp_b}
\nabla\pi_1&=\mathcal{W}_{4/3,V}\bkt{\Delta_y u-\div (\gamma' e_1\otimes\partial_{y_2}u)-\gamma' \pd_1\pd_2 u +(\gamma')^2\partial_{y_2}^2u}
\\
\nabla\pi_2&=\mathcal{W}_{4/3,V}\bkt{-\div_y(b\otimes u)+\gamma'\partial_{y_2}(b\otimes u)_1}
\\
 \nabla\pi_3&=\mathcal{W}_{4/3,V}(\gamma' \partial_{y_2}\pi e_1)
 \\
 \nabla\pi_4&= \mathcal{W}_{4/3,V}(\div \GG-\gamma'\partial_2 \GG_1).
}
From Theorem \ref{theoremL2wolf} the local pressure projection  $\mathcal{W}_{4/3,V}$ is well defined, and we have the bound \eqref{pWolfbound} with global constant $c$. Also using $\ga'\pd_2 = \pd_2 \ga'$, and Lemmas \ref{Holder} and \ref{Sobolev} in $V$,
\EQS{\label{pressure_bound_est}
\|\pi_1\|_{4/3,V}&\le c(1+\|\gamma'\|_{\infty}+\|\gamma'\|_{\infty}^2)\|\nabla u\|_{4/3,V}
\\
\|\pi_2\|_{4/3,V}&\le  c(1+\|\gamma'\|_{\infty})\|u\otimes b\|_{4/3,V}
\\
\|\pi_3\|_{4/3,V}&\le  c\|\gamma'\|_{\infty}\|\pi\|_{4/3,V}\le c\|\gamma'\|_{\infty}\|\pi_3\|_{4/3,V}+c\|\gamma'\|_{\infty}\|\pi_1+\pi_2+\pi_4\|_{4/3,V}
\\
\|\pi_4\|_{4/3,V}&\le  c(1+\|\gamma'\|_{\infty})\|\GG\|_{4/3,V}
}
Here we apply our choice of $\gamma$ so that $\|\gamma'\|_{\infty}<1$ and
$ c\|\gamma'\|_{\infty}<1/2$. (This is the second place that specifies the smallness of the Lipschitz constant.) After applying it to \eqref{pressure_bound_est} we get the following
\EQS{\label{pressure_3}
\|\pi_3\|_{4/3,V}\le 2c\|\gamma'\|_{\infty}\|\pi_1+\pi_2+\pi_4\|_{4/3,V}.
}

Let $\Om^\sharp$ be the pre-image of $V$ under the map \eqref{change-var} in $x$-variables.
The pressure component $\pi_i$ can be considered functions on $\Om^\sharp$ and since \eqref{change-var} has determinant 1 and $\|\nabla_y u\|_{4/3,V}\le 2\|\nabla_x u\|_{4/3,\Omega^\sharp}$, by assumption $\|\gamma'\|_{\infty}\le 1$ we have
\EQS{\label{pressure-est}
\norm{\pi-(\pi)_{V}}_{4/3, \Om^\sharp} &\le {\textstyle\sum}_{i=1}^4 \norm{\pi_i}_{4/3, \Om^\sharp}
={\textstyle\sum}_{i=1}^4 \norm{\pi_i}_{4/3, V}
\\
&\le c\|\nabla _y u\|_{4/3,V} +c \|u\otimes b\|_{4/3,V} +c \|\GG\|_{4/3,V}
\\
&\le c(1+ \| b\|_{L^{2,\infty}(V)}) (\|\nabla_y u\|_{4/3,V} +\| u\|_{4/3,V} )+ c \|\GG\|_{4/3,V}
\\
&\le c(1+ \| b\|_{L^{2,\infty}(\Omega)}) (\|\nabla_x u\|_{4/3,\Omega^\sharp} +\| u\|_{4/3,\Omega^\sharp} )+ c \|\GG\|_{4/3,\Omega^\sharp}.
}
This pressure estimate is the only purpose of our change of variables \eqref{change-var}. Noting that $\Om_{5/8}\subset \Om^\sharp\subset \Om_1$, we then proceed the following.

 Let $\eta$ be a cutoff function supported in $B_{5/8}(x_0)$ such that $\eta=1$ on $B_{1/2}(x_0)$. Due to our boundary condition $u|_{\partial\Omega}=0$ we can use function $\eta^4u$ as an admissible test function in \eqref{drifteq}. Therefore we get the following local energy inequality
\EQS{\label{eq5.19}
\int_{\Om_1} |\nabla (\eta^2u)|^2
&\le \int_{\Om_1} |u|^2|\nabla \eta^2|^2+\int_{\Om_1} |u|^2|b||\nabla \eta|\eta^3\\
&\quad +\int_{\Om_1} |\pi-(\pi)_{V}||u||\nabla\eta|\eta^3-\int_{\Om_1} \GG :\nabla(u\eta^4).
}
Using \eqref{pressure-est} to bound the pressure term and similarly to internal case
using Lemmas \ref{Holder} and \ref{Sobolev} in $\Om_1$, with uniform constant $C_2$ in Lemma \ref{Sobolev} using $\eta^4u\in W^{1,2}_0(\Om_1)$, we get
\EQS{\label{5.16}
\int_{\Om_{1/2}} |\nabla u|^2 &\le C(b)(\|\nabla u\|^2_{4/3,\Om_1}+\|u\|_{2,\Om_1}^2)+c\|\GG\|_{2,\Om_1}^2.
}
Notice that here the constant $C(b)$ does not depend on $\Om_1$.
Finally we extend function $u,\GG$ by zero to  $B_1(x_0)\setminus \Omega$. Since $u=0$ on the set $E:=\Om^c\cap B_1(x_0)$ of non zero measure with the ratio $\frac{|E|}{|B_{1}|}$ bounded from below, we can use Poincar\'e inequality to bound $\|u\|_{2,\Om_1}^2 \le c \|\nabla u\|^2_{4/3,\Om_1}$, with a constant independent of $\Om_1$ with the given lower bound of the ratio.
(This version of Poincar\'e inequality follows from Proposition 6.14 of \cite[page 116]{Lieberman},  with $f=\al 1_E$, nt_{B_1} f=1$. It gives $\norm{u}_{4/3,B_1} \le C\al  \norm{\nabla u}_{4/3,B_1} $.
We also have
$\norm{u}_{4,B_1} \le C \norm{u}_{W^{1,4/3}(B_1)}$.)

We also re-scale the inequality for arbitrary $B_r(x_0)$ and get \eqref{5.12}.

\medskip
Lastly we need to consider the case of arbitrary $x_0,r$ such that $\Om^c\cap B_{2r}(x_0)\neq\emptyset$. For any such $x_0$ we can find $y_0\in\partial\Om\cap B_{2r}(x_0)$ such that
\EQ{
B_r(x_0)\subset B_{3r}(y_0)\subset B_{6r}(y_0)\subset B_{8r}(x_0).
}
Using \eqref{5.12} for $B_{3r}(y_0)$ (assuming $6r<R_0$) we get
\EQS{\label{5.17}
\frac{1}{|B_r|}\int_{B_r(x_0)} |\nabla u|^2
&\le \frac{C}{|B_{3r}|}\int_{B_{3r}(y_0)} |\nabla u|^2
\\
&\le c(b) \Big(\frac{1}{|B_{6r}|}\int_{B_{6r}(y_0)}|\nabla u|^{\frac{4}{3}}\Big)^{\frac{3}{2}}+\frac{C}{|B_{6r}|}\|\GG\|_{L^2(B_{6r}(y_0))}^2
\\
& \le c(b) \Big(\frac{1}{|B_{8r}|}\int_{B_{8r}(x_0)}|\nabla u|^{\frac{4}{3}}\Big)^{\frac{3}{2}}+\frac{C}{|B_{8r}|}\|\GG\|_{L^2(B_{8r}(x_0))}^2.
}

For Gehring's lemma we combine the internal estimate \eqref{5.12} and the boundary estimate \eqref{5.17} and conclude that
\EQ{\label{5.19}
\frac{1}{|B_r|}\int_{B_r(x_0)} |\nabla u|^2 \le
c(b) \Big(\frac{1}{|B_{8r}|}\int_{B_{8r}(x_0)}|\nabla u|^{\frac{4}{3}}\Big)^{\frac{3}{2}}+\frac{C}{|B_{8r}|}\|\GG\|_{L^2(B_{8r}(x_0))}^2,
}
for any $x_0$ in a big cube containing $\overline \Om$, and any $r\le R_0/6$.
We can now apply Proposition 1.1 of Giaquinta \cite[page 122]{Gia} (also see \cite[Proposition 3.7]{DongKim}) to get that $\nb u \in L^p_\loc$ for $p \in (2,p_0)$ for some $p_0 >2$, and
\EQ{\label{0619}
nt_{B_r(x_0)} |\nabla u|^p}^{\frac 1p} \le
nt_{B_{8r}(x_0)} |\GG|^p}^{\frac 1p},
}
for all $x_0 \in \overline \Om$ and $r\le R_0/6$, with
$p_0$ and $c_1$ depending only on $c(b)$. Summing \eqref{0619} over a finite cover of $\overline \Om$ of balls of radius $R_0/6$,
we get \eqref{eq5.1}.
\end{proof}

\begin{theorem}\label{main1}
Let $\Om$ be a bounded Lipschitz domain in $\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\in L^{2,\infty}(\Omega)^2$ and $\div b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $\GG\in L^q(\Om)^{2\times 2}$, there exists a  weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ and $\pi\in L^q_0(\Omega)$ of \eqref{drifteq} such that
\EQ{\label{main1-eq}
\|u\|_{ W^{1,q}(\Omega)}+\|\pi\|_{L^q(\Omega)}\le C(\Omega,\|b\|_{L^{2,\infty}})\|\GG\|_{L^q(\Omega)}.
}
For $p_0'<q<p_0$, the above pair $(u,\pi)$ is the unique weak solution pair  in $W^{1,q}_0\times L^q_0(\Om)$ without assuming \eqref{main1-eq}.
\end{theorem}

\emph{Comments on Theorem \ref{main1}:}
\EN{
\item This theorem extends the higher dimensional result \cite[Theorem 1.5]{CT1} to the dimension 2 case.
In \cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \eqref{change-var} and after \eqref{pressure_bound_est}, and is computable.
We skip in this paper the analogous results for small $b$ case (but full range $1<q<\infty$) as in \cite[Proposition 1.3, Theorem 1.4]{CT1}.

\item  Since $\Om$ is Lipschitz we can assume that there exists $R_0>0$ such that for any $x_0\in \partial\Om$ we have
\EQ{
\Om\cap B_{2R_0}(x_0)=\{(x',x_n)\in B_{2R_0}(0),x_n>\gamma(x')\}.
}
The constant $C$ in \eqref{main1-eq} will actually depend only on $R_0$, $\text{diam}~\Om$, and $\|b\|_{L^{2,\infty}}$.

\item If we further assume $b \in L^2(\Om)$
and $b$ has a divergence-free extension to $L^2(\R^2)$,
then there is another way to prove uniqueness of the weak solution in $W^{1,2}_{0,\si}(\Om)$ by using Hardy-BMO duality, even when the Lipschitz constant of $\Om$ is large.
See Theorem \ref{energy_2} in Section \ref{sec3}.

\item We may simply assume $\Om$ is a bounded $C^1$ domain, as for
a bounded $C^1$ domain and any constant $\lambda>0$, each point $x$ on $\pd\Om$ has a neighborhood $U_x$ such that $\pd \Om \cap U_x$ is the graph of a Lipschitz function with a Lipschitz constant less than $\lambda$.

}

For corresponding results on scalar equations
\EQ{\label{scalar.eq}
-\Delta u +b\cdot\nabla u=\div  G, \quad u|_{\pd \Om}=0,
}
with critical drift coefficient $b \in L^n$, see
Kim \& Kim \cite{KK}. When $b \in L^{n,\infty}(\Om)$ for $n \ge 3$, see \cite{KiTs,KPT} which also cover more general assumptions on drift $b$ as well as \cite{CS20,CS24} and Kwon \cite{Kwon} for $n=2$ case or drift in Morrey spaces which include $L^{n,\infty}$.
There are also results on H\"older continuity of the solutions, for example with $b\in BMO^{-1}(\R^3)$, $\div b=0$, see Seregin-Sverak-Silvestre-Zlatos \cite{SSSZ}.

Using the same proof scheme we can also prove the following theorem for the  scalar equation \eqref{scalar.eq}.
\begin{theorem}\label{th1.2}
  Let $\Om$ be a bounded Lipschtiz domain in $\RR^2$.  Assume $b\in L^{2,\infty}(\Omega)$ and $\div b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\in L^q(\Om)$, there exists a  weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ of \eqref{scalar.eq} such that
\EQ{\label{main2-eq}
\|u\|_{ W^{1,q}(\Omega)}\le C(\Omega,\|b\|_{L^{2,\infty}})\|G\|_{L^q(\Omega)}.
}
For $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\Om)$ without assuming \eqref{main2-eq}.
\end{theorem}

\label{1.2}
\De u = \div (\nb u),\quad
b\cdot\nabla u = \div (b\otimes u), \quad \int_\Om  \div \GG \cdot \zeta = -
\int_\Om  \GG:\nb \zeta.
}
For zero divergence spaces or spaces with zero average we

\label{soln-pair}
\int_{\Om} (\nb u - b \otimes u): \nb \zeta - \pi \div \zeta  = -\int_\Om \GG : \nb \zeta,\quad
\forall \zeta\in C_c^{\infty}(\Om)^2.
}
Here we used \eqref{1.2} to integrate by parts

\label{sec1}

For a bounded Lipschitz domain $\Om$ in $\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\pi):\Om \to \mathbb{R}^2 \times \mathbb{R}$ that satisfy
\EQ{\label{drifteq}
-\Delta u +b\cdot\nabla u+\nabla \pi=\mathop{\rm div}\nolimits \mathbb{G},\quad \mathop{\rm div}\nolimits u=0, \quad u|_{\partial \Om}=0,
}
with given drift $b$ in the critical space, $b\in L^{2,\infty}(\Omega)^2$, $\mathop{\rm div}\nolimits b=0$, and force $\mathbb{G}\in L^q(\Om)^{2\times 2}$, $1 <q<\infty$. We denote by
$L^{q,r}(\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations
$(\mathop{\rm div}\nolimits \mathbb{G})_i=\pd_j \GG_{ji}$ and $(\nabla \zeta)_{ji} = \pd_j \zeta_i$. Since $\mathop{\rm div}\nolimits b=0$ we can integrate by parts in the following terms
\EQ{\label{1.2}
\De u = \mathop{\rm div}\nolimits (\nabla u),\quad
b\cdot\nabla u = \mathop{\rm div}\nolimits (b\otimes u), \quad \int_\Om \mathop{\rm div}\nolimits \mathbb{G} \cdot \zeta = -
\int_\Om \mathbb{G}:\nabla \zeta.
}
For zero divergence spaces or spaces with zero average we use the following notations
(for $\Om\subset\mathbb{R}^n$, $n\ge2$)
\EQS{
C^{\infty}_{c}(\Omega) &= \big\{ u \in C^{\infty}(\Omega):~ \supp\{u\}\Subset \Omega\big\},\\
C^{\infty}_{c,\si}(\Omega) &= \big\{ u \in C^{\infty}_c(\Omega)^n:\ \mathop{\rm div}\nolimits u=0\big\},\\
W^{1,q}_{0,\si}(\Omega) &= \big\{ u \in W^{1,q}_0(\Omega)^n:\ \mathop{\rm div}\nolimits u=0\big\},\\ \quad L^q_0(\Om)&=\left \{ \pi \in L^q(\Om):\ \textstyle \int_\Om \pi \,dx=0 \right \}.
}

Next we consider $(u,\pi)$ as a \emph{weak solution pair} of \eqref{drifteq} if $u\in W^{1,q}_{0,\si}(\Omega)$, $\pi\in L^q_0(\Omega)$, and $(u,\pi)$ satisfies \eqref{drifteq} in distributional sense,\EQ{\label{soln-pair}
\int_{\Om} (\nabla u - b \otimes u): \nabla \zeta - \pi \mathop{\rm div}\nolimits \zeta = -\int_\Om \mathbb{G} : \nabla \zeta,\quad
\forall \zeta\in C_c^{\infty}(\Om)^2.
}
Here we used \eqref{1.2} to integrate by parts.
We say $u$ is a \emph{$q$-weak solution} of \eqref{drifteq} if $u\in W^{1,q}_{0,\si}(\Omega)$ and $u$ satisfies \eqref{drifteq} in weak sense, i.e.,
\EQ{\label{weak-soln}
\int_{\Om} (\nabla u - b \otimes u): \nabla \zeta = -\int_\Om \mathbb{G} : \nabla \zeta,\quad
\forall \zeta\in C_{c,\sigma}^{\infty}(\Omega).
}
In case $q=2$ we call $u$ a \emph{weak solution}. In this paper we also consider a dual system of \eqref{drifteq}
\EQ{\label{drifteq_dual}
-\Delta v -\nabla\cdot(b\otimes v)+\nabla \pi=\mathop{\rm div}\nolimits \mathbb{F},\quad \mathop{\rm div}\nolimits v=0, \quad v|_{\partial \Om}=0.
}
This system is equivalent to \eqref{drifteq} when the drift is divergence free, $\mathop{\rm div}\nolimits b=0$, but in general case it is different.

The boundary value problem of the perturbed Stokes system \eqref{drifteq} and the corresponding scalar equation \eqref{scalar.eq}
has a huge amount of literature when we consider subcritical drift $b\in L^p(\Om)$, $\Om \subset \mathbb{R}^n$, with $p> n \ge2$. In the subcritical case we may treat the drift term $b\cdot \nabla u$ as a perturbation to $\De u$. The critical case is when $b\in L^n(\Om)$ or $b\in L^{n,\infty}(\Om)$. System \eqref{drifteq} in $\Om \subset \mathbb{R}^3$ with critical drift $b\in L^3(\Om)$ has previously been studied;
See Section 4 of Kim \cite{Kim09}, Theorem 18 in Choe \& Kim \cite{ChoeKim}, and Section 5 of
Amrouche \& Rodríguez-Bellido \cite{AmRB}. They contain estimates similar to \eqref{main1-eq}. A similar result for small $b\in L^{n,\infty}(\Om)$, $n \ge3$, is proved by the authors \cite[Theorem 1.4]{CT1}.

In this paper we extend our previous work for higher dimensions $n \ge 3$ \cite{CT1} to $n=2$. Our main theorem below extends \cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.

The boundary value problem of the perturbed Stokes system \eqref{drifteq} and the corresponding scalar equation \eqref{scalar.eq}
has a huge amount of literature when we consider subcritical drift $b\in L^p(\Om)$, $\Om \subset \R^n$, with $p> n \ge2$. In the subcritical case we may treat the drift term $b\cdot \nb u$ as a perturbation to $\De u$. The critical case is when $b\in L^n(\Om)$ or $b\in L^{n,\infty}(\Om)$. System \eqref{drifteq} in $\Om \subset \R^3$ with critical drift $b\in L^3(\Om)$ has previously been studied;
See Section 4 of Kim \cite{Kim09}, Theorem 18 in Choe \& Kim \cite{ChoeKim}, and Section 5 of
Amrouche \& Rodríguez-Bellido \cite{AmRB}. They contain estimates similar to \eqref{main1-eq}. A similar result for small $b\in L^{n,\infty}(\Om)$, $n \ge3$, is proved by the authors \cite[Theorem 1.4]{CT1}.

\emph{Comments on Theorem \ref{main1}:}
\EN{
\item This theorem extends the higher dimensional result \cite[Theorem 1.5]{CT1} to the dimension 2 case.
In \cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \eqref{change-var} and after \eqref{pressure_bound_est}, and is computable.
We skip in this paper the analogous results for small $b$ case (but full range $1<q<\infty$) as in \cite[Proposition 1.3, Theorem 1.4]{CT1}.

Using the same proof scheme we can also prove the following theorem for the scalar equation \eqref{scalar.eq}.
\begin{theorem}\label{th1.2}
 Let $\Om$ be a bounded Lipschtiz domain in $\RR^2$. Assume $b\in L^{2,\infty}(\Omega)$ and $\div b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\in L^q(\Om)$, there exists a weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ of \eqref{scalar.eq} such that
\EQ{\label{main2-eq}
\|u\|_{ W^{1,q}(\Omega)}\le C(\Omega,\|b\|_{L^{2,\infty}})\|G\|_{L^q(\Omega)}.
}
For $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\Om)$ without assuming \eqref{main2-eq}.
\end{theorem}

\section{Existence of weak solutions}\label{sec3}
In this section we will prove existence of weak solutions to the perturbed Stokes system \eqref{drifteq} for $b\in L^2$ or $L^{2,\infty}$, $\div b=0$,
\EQ{ \label{eq3.1}
-\Delta u+ b\cdot\nabla u + \nb \pi =f,\quad \div u=0, \quad u|_{\pd \Om}=0,
}
in $\Om \subset \R^2$. When $b\in L^2$, we will further show uniqueness.
Recall that a weak solution $u$ belongs to $ W^{1,2}_{0,\sigma}(\Omega)$ and satisfies a weak form of \eqref{drifteq} similar to \eqref{weak-soln}.
\begin{theorem}[Weak solutions for weak $L^2$ drift]\label{energy}
Let $\Om$ be a bounded Lipschitz domain in $\R^2$.
Assume $b\in L^{2,\infty}(\Omega)^2$ and $\div b=0$.
Then for any $f \in W^{-1,2}(\Omega)^2$,
there exists a weak solution $u\in W^{1,2}_{0,\sigma}(\Omega)$
of \eqref{drifteq}. Moreover, there is $\pi \in L^2(\Om)$ so that $(u,\pi)$ solves \eqref{drifteq} in distributional sense, $\int_\Om \pi=0$, and
\EQ{\label{eq6.1}
\|u\|_{ W^{1,2}(\Omega)}+\|\pi\|_{L^2(\Omega)}\le C\|f\|_{W^{-1,2}(\Om)},
}
for some constant $C=C(\Om)$ independent of $b$.
\end{theorem}

\begin{theorem}[Unique weak solution for $L^2$ drift]\label{energy_2}
Let $\Om$ be a bounded Lipschitz domain in $\R^2$.
Assume $b\in L^{2}(\Om)^2$ has an extension
$\tilde{b}\in L^{2}(\R^2)^2$ with $\div \tilde{b}=0$ in $\R^2$.
Then for any $f \in W^{-1,2}(\Omega)^2$,
there exists a unique weak solution $u\in W^{1,2}_{0,\sigma}(\Omega)$
of \eqref{drifteq}. Furthermore, there is $\pi \in L^2(\Om)$ so that $(u,\pi)$ solves \eqref{drifteq} in distributional sense, $\int_\Om \pi=0$, and
\EQ{\label{eq6.1b}
\|u\|_{ W^{1,2}(\Omega)}+\|\pi\|_{L^2(\Omega)}\le C\|f\|_{W^{-1,2}(\Om)},
}
for some constant $C=C(\Om)$ independent of $b$.
\end{theorem}

Approximating $u$ in $W^{1,2}_{0,\si}(\Om)$ by smooth $u_j$
and using $\div b=0$ conclude that
\EQ{\label{eq3.4}
\int_\Om (b\cdot \nb) u\cdot u = \lim_{j \to \infty} \int_\Om (b\cdot \nb) u_j\cdot u_j = \int_\Om b\cdot \nb(\frac12|u_j|^2) = 0.
}
 Thus we get the coercivity of this bilinear form, indeed,
\EQ{
\mathcal{B}(u,u)=\|\nabla u\|_{L^2(\Om)}^2, \quad \forall u\in W_{0,\sigma}^{1,2}(\Om).
}
Therefore, by Lax-Milgram theorem, for any $f \in W^{-1,2}(\Omega)^n\subset (W_{0,\sigma}^{1,2}(\Om)^n)^*$, there exists a unique $u\in W_{0,\sigma}^{1,2}(\Om)$ such that
\EQ{
\mathcal{B}(u,v)=\langle f,v\rangle,\quad \forall v\in W_{0,\sigma}^{1,2}(\Om).
}
This is the weak form of \eqref{drifteq} and hence
$u$ is the unique weak solution. By taking $v=u$, we get $\norm{u}_{W^{1,2}_0(\Om)} \le C\norm{f}_{W^{-1,2}(\Om)}$.
Lastly, we apply Lemma \ref{pressure} with $q=2$ (and using that $\Om$ is bounded and Lipschitz) to get an associated pressure $\pi\in L^2_0(\Om)$ that satisfies the bound.
\end{proof}

\begin{theorem}\label{theoremL2wolf}
Assume $p>1$ and $V\subset \R^n$, $n\ge2$, is a bounded Lipschitz domain with Lipschitz constant less than a sufficiently small $L_0(p,n)>0$.
Then the problem \eqref{Stokes} has a unique weak solution pair $(v,\pi)$ with $v \in W^{1,p}_{0,\si}(V)$ and $\pi \in L^p_0(V)$ for given $f=\div \FF$, $ \FF \in L^p(\Om)^{n\times n}$, and the local pressure projection in $V$,
\[
\mathcal{W}_{p,V}(\div \FF)=\nabla\pi,
\]
satisfies the following inequality
\EQ{\label{pWolfbound}
\|\pi\|_{L^p(V)}\le c_1\|\FF\|_{L^p(V)},
}
where $c_1=c_1(n,V,p)$.
\end{theorem}

\begin{theorem}\label{Capriori}
Let $\Om$ be a bounded Lipschitz domain in $\R^2$ with Lipschitz constant $L \le L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\in L^{2,\infty}(\Omega)^2$ and $\div b=0$.
Assume further $b \in L^{2+\delta}$ for some $\delta>0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $2<p<p_0$,
any weak solution $u\in W^{1,2}_{0,\si}(\Omega)$ of \eqref{drifteq} with force $\GG\in L^p(\Om)^{2\times 2}$
is in $ W^{1,p}(\Omega)$ and
\EQ{\label{eq5.1}
\|u\|_{W^{1,p}(\Omega)}\le c\|u\|_{W^{1,2}(\Omega)}+ c\|\GG\|_{L^p(\Omega)}
}
for some constant $c(\Omega,\|b\|_{L^{2,\infty}})>0$.
\end{theorem}

\emph{Comments on Theorem \ref{main1}:}
\EN{
\item This theorem extends the higher dimensional result \cite[Theorem 1.5]{CT1} to the dimension 2 case.
In \cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \eqref{change-var} and after \eqref{pressure_bound_est}, and is computable.
We skip in this paper the analogous results for small $b$ case (but full range $1<q<\infty$) as in \cite[Proposition 1.3, Theorem 1.4]{CT1}.

\item  Since $\Om$ is Lipschitz we can assume that there exists $R_0>0$ such that for any $x_0\in \partial\Om$ we have
\EQ{
\Om\cap B_{2R_0}(x_0)=\{(x',x_n)\in B_{2R_0}(0),x_n>\gamma(x')\}.
}
The constant $C$ in \eqref{main1-eq} will actually depend only on $R_0$, $\text{diam}~\Om$, and $\|b\|_{L^{2,\infty}}$.

\item If we further assume $b \in L^2(\Om)$
and $b$ has a divergence-free extension to $L^2(\mathbb{R}^2)$,
then there is another way to prove uniqueness of the weak solution in $W^{1,2}_{0,\si}(\Om)$ by using Hardy-BMO duality, even when the Lipschitz constant of $\Om$ is large.
See Theorem \ref{energy_2} in Section \ref{sec3}.

\item We may simply assume $\Om$ is a bounded $C^1$ domain, as for
a bounded $C^1$ domain and any constant $\lambda>0$, each point $x$ on $\partial\Om$ has a neighborhood $U_x$ such that $\partial \Om \cap U_x$ is the graph of a Lipschitz function with a Lipschitz constant less than $\lambda$.

}

For corresponding results on scalar equations
\EQ{\label{scalar.eq}
-\Delta u +b\cdot\nabla u=\mathop{\rm div}\nolimits  G, \quad u|_{\partial \Om}=0,
}
with critical drift coefficient $b \in L^n$, see
Kim \& Kim \cite{KK}. When $b \in L^{n,\infty}(\Om)$ for $n \ge 3$, see \cite{KiTs,KPT} which also cover more general assumptions on drift $b$ as well as \cite{CS20,CS24} and Kwon \cite{Kwon} for $n=2$ case or drift in Morrey spaces which include $L^{n,\infty}$.
There are also results on H\"older continuity of the solutions, for example with $b\in BMO^{-1}(\mathbb{R}^3)$, $\mathop{\rm div}\nolimits b=0$, see Seregin-Sverak-Silvestre-Zlatos \cite{SSSZ}.

Using the same proof scheme we can also prove the following theorem for the  scalar equation \eqref{scalar.eq}.
\begin{theorem}\label{th1.2}
  Let $\Om$ be a bounded Lipschtiz domain in $\mathbb{R}^2$.  Assume $b\in L^{2,\infty}(\Omega)$ and $\mathop{\rm div}\nolimits b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\in L^q(\Om)$, there exists a  weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ of \eqref{scalar.eq} such that
\EQ{\label{main2-eq}
\|u\|_{ W^{1,q}(\Omega)}\le C(\Omega,\|b\|_{L^{2,\infty}})\|G\|_{L^q(\Omega)}.
}
For $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\Om)$ without assuming \eqref{main2-eq}.
\end{theorem}

This theorem improves $\alpha=0$ case of \cite[Theorem 1.1]{CS20} and $n=2$ case of Kwon \cite[Theorem 1.1]{Kwon} which show that for any $q>2$, there is $p \in (2,q)$ depending on $q$, $\left  \| b  \right \|_{L^{2,\infty}}$ and $\Om$ such that for any $G \in L^q(\Om)$, there is a unique $p$-weak solution $u$ of \eqref{scalar.eq} with force $G$. Moreover,
\[
\left  \| \nabla u  \right \|_{L^p(\Om)} \le C \left  \| G  \right \|_{L^q(\Om)}.
\]
Theorem \ref{th1.2} improves this result by asserting that $p=q$ when $2<q<p_0$ is sufficiently close to $2$. This improvement further allows us to prove a priori bounds for $q$-weak solutions for $q \in (p_0',2]$ by duality, which yields existence and uniqueness
 for $q \in (p_0',2]$.

We skip the proof of Theoem \ref{th1.2} since it is similar to Theoem \ref{main1} and is easier, without the need of pressure estimate. Because it has no pressure estimate, the Lipschitz constant of the boundary $\partial \Om$ is allowed to be arbitrarily large.

\medskip

We now explain the new difficulties and ideas.
Theorem \ref{main1}
extends the higher dimensional result \cite[Theorem 1.5]{CT1} to the 2D case, and
we will still use Gehring's approach. In this approach, we need to do local energy estimate in every ball and prove a reversed H\"older inequality in a general ball with a uniform constant.

The first new difficulty is that we cannot use $\zeta = u \eta^4$ for some cut-off function $\eta$ as the test function in the weak form \eqref{soln-pair}, because $b \otimes u$ is no longer in $L^2$ as in the 3D case, and
 the term
\EQ{\label{drift-integral}
\int_\Om b \otimes u : \nabla( u \eta^4)
}
is not integrable. Hence we will mollify the drift $b$, so that we can derive an a priori bound. Since there is no theorem that extends a div-free vector field in $L^{2,\infty}(\Omega)$ to $L^{2,\infty}(\mathbb{R}^2)$ for a general bounded domain, we will consider the approximation problems in subdomains $\Om_n$ that converge to $\Om$.

The second new difficulty comes from the lower integrability of the pressure.
We will
bound the pressure using Wolf's local pressure projection with a uniform constant as in \cite{CT1}.
Denote $\Omega_R=\Omega\cap B_R(x_0)$ for some $x_0\in \overline \Om$. In \cite{CT1}, it suffices to bound $\pi -(\pi)_{\Omega_R}$ in $L^2(\Omega_R)$. However,  for dimension 2  we need to bound $\pi -(\pi)_{\Omega\cap B_R}$ in $L^q(\Omega_R)$, $q<2$. (We choose $q=4/3$ for simplicity.)
We have to choose $q<2$ because
\[
\pi {\ \lesssim \ } \nabla u + b \otimes u -\mathbb{G}
\]
and $b \otimes u$ is at best in $L^{2-}$ for $u \in W^{1,2}_\mathrm{loc}$ even if we assume $b \in L^2$. This would not be an issue if $b \in L^{r}$, $r>2$, and we can only assume $u \in W^{1,2}_\mathrm{loc}$ which is the assumption of the reversed H\"older inequality.

To obtain a uniform constant of Wolf's local pressure projection in $L^q(\Omega_R)$ for $q<2$ gives a significant higher restriction on the regularity of the domain. Instead of a Lipschitz domain with Lipschitz constant $L$ less than 1/2, we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.

The third difficulty is the approximation of a Lipschitz domain. In order to construct the solution we are using an approximation method, therefore the first logical idea would be to construct a sequence of $b_n\in L^{2,\infty}(\Om)$, $\mathop{\rm div}\nolimits b=0$, such that $b_n$ converge in a sense that allows us to prove that the limit will satisfy the equation, for example convergence almost everywhere is sufficient and $\|b_n\|_{L^{2,\infty}(\Om)}$ are bounded uniformly. For example in star-shaped domains it is trivial, but for arbitrary Lipschitz domains this question is not so clear. To deal with this issue, following Verchota \cite{Verchota_1,Verchota_2} and later Kwon \cite{Kwon}, Chernobai-Shilkin \cite{CS24}, we instead approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant and on sub-domains we can use standard mollification of $b$ and construct an approximation sequence as zero extension of solutions in these sub-domains. See Section \ref{sec6} for details.

The rest of the paper is organized as the following. In section 2 we have preliminary results concerning Lorentz spaces and the pressure, which will be used in the later sections. In section 3 we derive the existence of weak solutions in $W^{1,2}(\Omega)$. Section 4 contains Wolf's local pressure projection method that will be used to estimate the pressure. In section 5, assuming higher integrability of the drift, we prove uniform local estimate for the gradient which will imply higher integrability due to Gehring's lemma. Section 6 contains proof of the main Theorem \ref{main1}. In the Appendix we put a proof of geometrical lemma that was used in section 4.

We will still use Gehring's approach: “do local energy estimate in every ball and prove a reversed H\"older inequality in a general ball with a uniform constant.”

New issues/steps specific to proving Theorem~\ref{main1} in 2D:
\begin{itemize}
\item \textbf{Non-integrable drift term; mollify and approximate domains.} One “cannot use $\zeta = u\eta^4$ … as the test function … because $b\otimes u$ is no longer in $L^2$ … and the term $\int_\Om b \otimes u : \nabla( u \eta^4)$ is not integrable.” Thus “we will mollify the drift $b$, so that we can derive an a priori bound.” Since there is no general extension theorem to $L^{2,\infty}(\mathbb{R}^2)$, they “consider the approximation problems in subdomains $\Om_n$ that converge to $\Om$.”
\item \textbf{Pressure control via Wolf projection in $L^q$, $q<2$.} “The second new difficulty comes from the lower integrability of the pressure.” They “bound the pressure using Wolf's local pressure projection with a uniform constant.” In 2D they need “to bound $\pi-(\pi)_{\Omega\cap B_R}$ in $L^q(\Omega_R)$, $q<2$ (we choose $q=4/3$),” because “$\pi\lesssim \nabla u + b\otimes u -\mathbb{G}$” and “$b\otimes u$ is at best in $L^{2-}$ for $u\in W^{1,2}_{\mathrm{loc}}$.” Achieving a uniform constant for the projection in $L^q$, $q<2$ forces stronger boundary regularity: “we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.”
\item \textbf{Approximation of Lipschitz domains for constructing solutions.} To construct solutions they use an approximation method, but approximating $b$ directly in arbitrary Lipschitz domains is “not so clear.” Instead, “we approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant,” on which “we can use standard mollification of $b$,” and then “construct an approximation sequence as zero extension of solutions in these sub-domains.”
\end{itemize}

They indicate the proof is completed by these ingredients and is carried out later: “See Section~\ref{sec6} for details,” and “Section 6 contains proof of the main Theorem~\ref{main1}.”

We will still use Gehring's approach: “do local energy estimate in every ball and prove a reversed H\"older inequality in a general ball with a uniform constant.”

New issues/steps specific to establishing the main theorem in 2D:
\begin{itemize}
\item \textbf{Non-integrable drift term; mollify and approximate domains.} One “cannot use $\zeta = u\eta^4$ … as the test function … because $b\otimes u$ is no longer in $L^2$ … and the term $\int_\Om b \otimes u : \nabla( u \eta^4)$ is not integrable.” Thus “we will mollify the drift $b$, so that we can derive an a priori bound.” Since there is no general extension theorem to $L^{2,\infty}(\mathbb{R}^2)$, they “consider the approximation problems in subdomains $\Om_n$ that converge to $\Om$.”
\item \textbf{Pressure control via Wolf projection in $L^q$, $q<2$.} “The second new difficulty comes from the lower integrability of the pressure.” They “bound the pressure using Wolf's local pressure projection with a uniform constant.” In 2D they need “to bound $\pi-(\pi)_{\Omega\cap B_R}$ in $L^q(\Omega_R)$, $q<2$ (we choose $q=4/3$),” because “$\pi\lesssim \nabla u + b\otimes u -\mathbb{G}$” and “$b\otimes u$ is at best in $L^{2-}$ for $u\in W^{1,2}_{\mathrm{loc}}$.” Achieving a uniform constant for the projection in $L^q$, $q<2$ forces stronger boundary regularity: “we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.”
\item \textbf{Approximation of Lipschitz domains for constructing solutions.} To construct solutions they use an approximation method, but approximating $b$ directly in arbitrary Lipschitz domains is “not so clear.” Instead, “we approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant,” on which “we can use standard mollification of $b$,” and then “construct an approximation sequence as zero extension of solutions in these sub-domains.”
\end{itemize}

They indicate the proof is completed by these ingredients and is carried out later: “See Section~\ref{sec6} for details,” and “Section 6 contains proof of the main theorem.”

\label{main1}
Let $\Om$ be a bounded Lipschitz domain in $\mathbb{R}^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\in L^{2,\infty}(\Omega)^2$ and $\mathop{\rm div}\nolimits b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $\mathbb{G}\in L^q(\Om)^{2\times 2}$, there exists a  weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ and $\pi\in L^q_0(\Omega)$ of \label{drifteq}
-\Delta u +b\cdot\nabla u+\nabla \pi=\div \GG,\quad \div u=0, \quad u|_{\pd \Om}=0,
}
with given drift $b$ in the critical space, $b\in L^{2,\oo}(\Omega)^2$, $\div b=0$, and  force $\G such that
\begin{theorem}\label{main1}
Let $\Om$ be a bounded Lipschitz domain in $\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\in L^{2,\infty}(\Omega)^2$ and $\div b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $\GG\in L^q(\Om)^{2\times 2}$, there exists a  weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ and $\pi\in L^q_0(\Omega)$ of \eqref{drifteq} such that
\EQ{\label{main1-eq}
\|u\|_{ W^{1,q}(\Omega)}+\|\pi\|_{L^q(\Omega)}\le C(\Omega,\|b\|_{L^{2,\infty}})\|\GG\|_{L^q(\Omega)}.
}
For $p_0'<q<p_0$, the above pair $(u,\pi)$ is the unique weak solution pair  in $W^{1,q}_0\times L^q_0(\Om)$ without assuming \eqref{main1-eq}.
\end{theorem}

For $p_0'<q<p_0$, the above pair $(u,\pi)$ is the unique weak solution pair  in $W^{1,q}_0\times L^q_0(\Om)$ without assuming the preceding estimate.

Question:
Let Ω ⊂ ℝ² be a bounded Lipschitz domain whose Lipschitz constant L satisfies L < L0 for some sufficiently small 0 < L0 < 1. Assume b ∈ L^{2,\infty}(Ω)^2 and div b = 0. For q > 1, set W^{1,q}_{0,σ}(Ω) = {u ∈ W^{1,q}_0(Ω)^2 : div u = 0} and L^q_0(Ω) = {\pi ∈ L^q(Ω) : \int_Ω \pi\,dx = 0}. A weak solution pair (u,\pi) of
-Δu + b\cdot∇u + ∇\pi = div \mathbb{G}, \quad div\,u = 0 \text{ in } Ω, \quad u|_{\partial Ω}=0,
is one with u ∈ W^{1,q}_{0,σ}(Ω), \pi ∈ L^q_0(Ω), and
\int_Ω (∇u - b\otimes u):∇\zeta - \pi\,div\,\zeta = -\int_Ω \mathbb{G}:∇\zeta \quad \text{for all } \zeta \in C_c^\infty(Ω)^2.
If p0' denotes the Hölder conjugate p0/(p0-1), which statement holds for this problem?

There exists p0 = p0(\|b\|_{L^{2,\infty}}) > 2 such that for every q with p0' < q < p0 and every \mathbb{G} ∈ L^q(Ω)^{2\times 2}, there exists a unique weak solution pair (u,\pi) ∈ W^{1,q}_{0,σ}(Ω) \times L^q_0(Ω) of
-Δu + b\cdot∇u + ∇\pi = div \mathbb{G}, \quad div\,u = 0 \text{ in } Ω, \quad u|_{\partial Ω}=0,
and this pair satisfies
\|u\|_{W^{1,q}(Ω)} + \|\pi\|_{L^q(Ω)} \le C(Ω,\|b\|_{L^{2,\infty}})\,\|\mathbb{G}\|_{L^q(Ω)}.
Equivalently, for each such q, the solution pair is unique in W^{1,q}_{0,σ}(Ω) \times L^q_0(Ω) even without assuming the estimate a priori.

Choices:
B.
There exists p0 = p0(\|b\|_{L^{2,\infty}}) > 2 such that for every q with 1 < q < p0 and every \mathbb{G} \in L^q(\Omega)^{2\times 2}, there exists a unique weak solution pair (u,\pi) \in W^{1,q}_{0,\sigma}(\Omega) \times L^q_0(\Omega) of
-\Delta u + b\cdot\nabla u + \nabla\pi = \operatorname{div} \mathbb{G}, \quad \operatorname{div} u = 0 \text{ in } \Omega, \quad u|_{\partial\Omega}=0,
and this pair satisfies
\|u\|_{W^{1,q}(\Omega)} + \|\pi\|_{L^q(\Omega)} \le C(\Omega,\|b\|_{L^{2,\infty}})\,\|\mathbb{G}\|_{L^q(\Omega)}.
C.
There exists p0 = p0(\|b\|_{L^{2,\infty}}) > 2 such that for every q with p0' < q < p0 and every \mathbb{G} \in L^q(\Omega)^{2\times 2}, there exists a weak solution pair (u,\pi) \in W^{1,q}_{0,\sigma}(\Omega) \times L^q_0(\Omega) of
-\Delta u + b\cdot\nabla u + \nabla\pi = \operatorname{div} \mathbb{G}, \quad \operatorname{div} u = 0 \text{ in } \Omega, \quad u|_{\partial\Omega}=0,
such that
\|u\|_{W^{1,q}(\Omega)} + \|\pi\|_{L^q(\Omega)} \le C(\Omega,\|b\|_{L^{2,\infty}})\,\|\mathbb{G}\|_{L^q(\Omega)}.
D.
There exists p0 = p0(\|b\|_{L^{2,\infty}}) > 2 such that for every q with p0' < q < p0 and every \mathbb{G} \in L^q(\Omega)^{2\times 2}, there exists a unique weak solution pair (u,\pi) \in W^{1,q}_{0,\sigma}(\Omega) \times L^q_0(\Omega) of
-\Delta u + b\cdot\nabla u + \nabla\pi = \operatorname{div} \mathbb{G}, \quad \operatorname{div} u = 0 \text{ in } \Omega, \quad u|_{\partial\Omega}=0,
and this pair satisfies
\|u\|_{W^{1,q}(\Omega)} + \|\pi\|_{L^q(\Omega)} \le C(\Omega)\,\|\mathbb{G}\|_{L^q(\Omega)},
with a constant independent of b.
E.
If \Omega \subset \mathbb{R}^2 is a bounded Lipschitz domain and b \in L^{2,\infty}(\Omega)^2 is divergence free, then for every q with p0' < q < p0 and every \mathbb{G} \in L^q(\Omega)^{2\times 2}, there exists a unique weak solution pair (u,\pi) \in W^{1,q}_{0,\sigma}(\Omega) \times L^q_0(\Omega) of
-\Delta u + b\cdot\nabla u + \nabla\pi = \operatorname{div} \mathbb{G}, \quad \operatorname{div} u = 0 \text{ in } \Omega, \quad u|_{\partial\Omega}=0,
and this pair satisfies
\|u\|_{W^{1,q}(\Omega)} + \|\pi\|_{L^q(\Omega)} \le C(\Omega,\|b\|_{L^{2,\infty}})\,\|\mathbb{G}\|_{L^q(\Omega)}.

dependence of constants on \|b\|_{L^{2,\infty}}

small Lipschitz constant requirement for Wolf projection/boundary stretching


Question:
Let \(\Omega\subset \mathbb{R}^2\) be a bounded Lipschitz domain whose Lipschitz constant satisfies \(L<L_0\), where \(0<L_0<1\) is sufficiently small. Assume \(b\in L^{2,\infty}(\Omega)^2\) and \(\operatorname{div} b=0\). For \(q\in(1,\infty)\), write
\[W^{1,q}_{0,\sigma}(\Omega)=\{u\in W^{1,q}_0(\Omega)^2:\operatorname{div}u=0\},\qquad L^q_0(\Omega)=\Big\{\pi\in L^q(\Omega):\int_\Omega \pi\,dx=0\Big\}.\]
A weak solution pair of
\[-\Delta u+b\cdot\nabla u+\nabla \pi=\operatorname{div}\mathbb{G},\qquad \operatorname{div}u=0,\qquad u|_{\partial\Omega}=0\]
is a pair \((u,\pi)\in W^{1,q}_{0,\sigma}(\Omega)\times L^q_0(\Omega)\) such that
\[\int_\Omega (\nabla u-b\otimes u):\nabla \zeta-\pi\,\operatorname{div}\zeta=-\int_\Omega \mathbb{G}:\nabla \zeta\qquad \forall\zeta\in C_c^{\infty}(\Omega)^2.\]
Which statement holds for this problem?

There exists \(p_0=p_0(\|b\|_{L^{2,\infty}(\Omega)})>2\) such that, if \(p_0'\) denotes the Hölder conjugate exponent of \(p_0\) \((1/p_0+1/p_0'=1)\), then for every \(q\) with \(p_0'<q<p_0\) and every \(\mathbb{G}\in L^q(\Omega)^{2\times 2}\), there exists a weak solution pair \((u,\pi)\in W^{1,q}_{0,\sigma}(\Omega)\times L^q_0(\Omega)\) of the system above such that
\[\|u\|_{W^{1,q}(\Omega)}+\|\pi\|_{L^q(\Omega)}\le C\big(\Omega,\|b\|_{L^{2,\infty}(\Omega)}\big)\,\|\mathbb{G}\|_{L^q(\Omega)}.\]
Moreover, for each such \(q\), this pair is the unique weak solution pair in \(W^{1,q}_{0,\sigma}(\Omega)\times L^q_0(\Omega)\), even without assuming the estimate.

Choices:
B.
There exists \(p_0=p_0(\|b\|_{L^{2,\infty}(\Omega)})>2\) such that for every \(q\) with \(1<q<p_0\) and every \(\mathbb{G}\in L^q(\Omega)^{2\times 2}\), there exists a weak solution pair \((u,\pi)\in W^{1,q}_{0,\sigma}(\Omega)\times L^q_0(\Omega)\) of the system above such that
\[\|u\|_{W^{1,q}(\Omega)}+\|\pi\|_{L^q(\Omega)}\le C\big(\Omega,\|b\|_{L^{2,\infty}(\Omega)}\big)\,\|\mathbb{G}\|_{L^q(\Omega)}.\]
Moreover, for each such \(q\), this pair is the unique weak solution pair in \(W^{1,q}_{0,\sigma}(\Omega)\times L^q_0(\Omega)\).
C.
There exists \(p_0=p_0(\|b\|_{L^{2,\infty}(\Omega)})>2\) such that, if \(p_0'\) denotes the Hölder conjugate exponent of \(p_0\), then for every \(q\) with \(p_0'<q<p_0\) and every \(\mathbb{G}\in L^q(\Omega)^{2\times 2}\), there exists at least one weak solution pair \((u,\pi)\in W^{1,q}_{0,\sigma}(\Omega)\times L^q_0(\Omega)\) of the system above satisfying
\[\|u\|_{W^{1,q}(\Omega)}+\|\pi\|_{L^q(\Omega)}\le C\big(\Omega,\|b\|_{L^{2,\infty}(\Omega)}\big)\,\|\mathbb{G}\|_{L^q(\Omega)}.\]
D.
There exists \(p_0=p_0(\|b\|_{L^{2,\infty}(\Omega)})>2\) such that, if \(p_0'\) denotes the Hölder conjugate exponent of \(p_0\), then for every \(q\) with \(p_0'<q<p_0\) and every \(\mathbb{G}\in L^q(\Omega)^{2\times 2}\), there exists a weak solution pair \((u,\pi)\in W^{1,q}_{0,\sigma}(\Omega)\times L^q_0(\Omega)\) of the system above such that
\[\|u\|_{W^{1,q}(\Omega)}+\|\pi\|_{L^q(\Omega)}\le C\big(\Omega,\|b\|_{L^{2,\infty}(\Omega)},q\big)\,\|\mathbb{G}\|_{L^q(\Omega)}.\]
Moreover, one may choose \(p_0>2\) depending only on \(\Omega\), independent of \(\|b\|_{L^{2,\infty}(\Omega)}\).
E.
There exists \(p_0=p_0(\|b\|_{L^{2,\infty}(\Omega)})>2\) such that, if \(p_0'\) denotes the Hölder conjugate exponent of \(p_0\), then for every \(q\) with \(p_0'\le q\le p_0\) and every \(\mathbb{G}\in L^q(\Omega)^{2\times 2}\), there exists a weak solution pair \((u,\pi)\in W^{1,q}_{0,\sigma}(\Omega)\times L^q_0(\Omega)\) of the system above such that
\[\|u\|_{W^{1,q}(\Omega)}+\|\pi\|_{L^q(\Omega)}\le C\big(\Omega,\|b\|_{L^{2,\infty}(\Omega)}\big)\,\|\mathbb{G}\|_{L^q(\Omega)}.\]
Moreover, for each such \(q\), this pair is the unique weak solution pair in \(W^{1,q}_{0,\sigma}(\Omega)\times L^q_0(\Omega)\).

lower endpoint q>p_0' forced by q<2 pressure control

open q-range replaced by closed endpoint range


\begin{proposition}\label{prop: rank 3 excluded minor}
    If $M$ is a simple rank-$3$ matroid with a line $L$ and distinct points $e_1, e_2, e_3$ on $L$ each on at least two long lines, then $M$ is not a positroid.
    In particular, $M$ has a minor from Figure \ref{fig: rank-3 excluded minors}.
\end{proposition}

\begin{theorem} \label{thm: main}
For all integers $r, \ell \ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, then $|M| \le \ell(r - 1) + 1$.
Moreover, if $r \ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \ell+1}$.
\end{theorem}

\begin{proposition}\label{prop: rank 4 excluded minor}
    If $M$ is a simple rank-$4$ matroid with a line $L$, distinct points $e_1, e_2, e_3$ on $L$, and distinct planes $P_1$, $P_2$, $P_3$ through $L$ so that for $i = 1,2,3$ the point $e_i$ is on at least two long lines in $P_i$, then $M$ is not a positroid.
    In particular, $M$ has a minor from Figure \ref{fig: rank-3 excluded minors} or Figure \ref{fig: rank-4 excluded minor}.
\end{proposition}

\label{sec: introduction}

Extremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \cite{Kung-1993}.

\begin{problem} \label{prob: Kung problem}
Let $\cM$ be a class of matroids.
Determine the maximum number of elements of a simple rank-$r$ matroid in $\cM$, and characterize the matroids for which equality holds.
\end{problem}

This maximum does not exist for all classes of matroids.
For instance, there is no finite answer to Problem~\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.
However, Kung \cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \ell+2}$-minor has at most $\frac{\ell^r - 1}{\ell - 1}$ elements.
If $\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\ell$. It is thus natural to study Problem~\ref{prob: Kung problem} for classes of matroids without $U_{2,\ell}$-minors.

In this paper, we study the class of \textit{positroids} without $U_{2,\ell}$-minors. Positroids, which are matroids representable by a matrix over $\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \reflectbox{L}-diagrams, providing multiple equivalent viewpoints.
Positroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \cite{Williams05} and connected positroids \cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\ref{prob: Kung problem} for the class of $U_{2,\ell+2}$-minor-free positroids.

This result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.
We will explain how Theorem \ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.
The following beautiful result of Geelen and Nelson \cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \ref{prob: Kung problem} for the class of matroids with no $U_{2,\ell+2}$-minor, for all values of $\ell$ and all sufficiently large values of $r$.

\begin{theorem}[Geelen, Nelson \cite{Geelen-Nelson-2010}] \label{thm: exlude a line}
Let $\ell\ge 2$ be an integer and let $q$ be the largest prime power at most $\ell$.
For $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\ell+2}$-minor has at most $\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.
\end{theorem}

\begin{theorem}[Geelen, Nelson, Walsh \cite{Geelen-Nelson-Walsh-2024}] \label{thm: complex exclude a line}
Let $\ell\ge 2$ be an integer and let $r$ be a sufficiently large integer.
Then every simple rank-$r$ $\bC$-representable matroid with no $U_{2,\ell+2}$-minor has at most $\ell\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\ell$.
\end{theorem}

\begin{theorem} \label{thm: main result with proof}
For all integers $r, \ell \ge 1$, if $M$ is a simple rank-$r$ matroid with no minor from Figure \ref{fig: rank-3 excluded minors} or Figure \ref{fig: rank-4 excluded minor} and no $U_{2,\ell+2}$-minor, then $|M| \le \ell(r - 1) + 1$.
Moreover, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \ell+1}$.
\end{theorem}
\begin{proof}
First suppose that $\ell = 1$.
If $M$ is a simple rank-$r$ matroid with more than $r$ elements, then $r \ge 2$ and $M$ has a circuit with at least three elements.
But every circuit with at least three elements has a $U_{2,3}$-minor \cite[Example 3.1.9]{Oxley-2011}, a contradiction.
So $M$ is a rank-$r$ matroid with $r$ elements, and it follows that $M$ is the parallel connection of $r - 1$ copies of $U_{2,2}$, as desired.
So we may assume that $\ell \ge 2$.
We will proceed by induction on $r$.
The statement is clearly true when $r \le 2$ so we may assume that $r \ge 3$.
If $M$ has an element $e$ that is not on any long lines, then $M/e$ is simple, $|M/e| = |M| - 1$, and $r(M/e) = r - 1$, so by induction we have
\begin{align*}
 |M| = 1 + |M/e| \le 1 + \ell(r - 2) + 1 = \ell(r - 1) + 2 - \ell < \ell(r - 1) + 1,
\end{align*}
as desired.
So we may assume that every element of $M$ is on a long line.
Let $L$ be a long line of $M$.
We will now consider the connected components of $M/L$.
Let $\cX$ be the partition of $E(M) - L$ given by the ground sets of the connected components of $M/L$.
Let $X \in \cX$, and let $N = M|(L \cup X)$.
We will first bound the number of elements of $N \del L$.

\begin{claim} \label{claim: bound the size of each component}
 At most two elements in $L$ are on a long line of $N$ other than $L$, and $|N \del L| \le \ell(r(N) - 2)$.
\end{claim}
\begin{proof}
Suppose that distinct elements $e_1$, $e_2$, and $e_3$ in $L$ are on long lines $L_1$, $L_2$, and $L_3$, respectively, of $N$ other than $L$.
Let $K = \si(N/L)$.
By deleting elements from nontrivial parallel classes of $N/L$ we may assume that $K$ is a restriction of $N/L$.
Note that the elements of $K$ are in one-to-one correspondence with the planes of $N$ containing $L$.
For $i = 1,2,3$ let $p_i$ be the element of $K$ in the plane $\cl_N(L \cup L_i)$ of $N$.
Since $K$ is connected, it follows from \cite[Theorem 4.3.1]{Oxley-2011} that $K$ has a connected minor $K_1 = K \del D/C$ on ground set $\{p_1, p_2, p_3\}$.
We may assume that $C$ is independent in $K$ and therefore in $N/L$, which implies that $r_{N/C}(L) = r_N(L) = 2$.
Since $K_1$ is connected, it is isomorphic to $U_{1,3}$ or $U_{2,3}$.
Let $N_1$ be obtained from $N$ by deleting, for each element $d$ in $D$, the elements in $\cl_N(L \cup d) - L$, and then contracting $C$.
Then $K_1 = \si(N_1/L)$ and $r(N_1) = r(K_1) + 2$.
We will show that $N_1$ has a minor from Figure \ref{fig: rank-3 excluded minors} or \ref{fig: rank-4 excluded minor}.

\begin{claim}
 Some $x \in L$ is on more than one long line of $M$.
\end{claim}
\begin{proof}
Let $e \in L$ and let $M_1 = M/e$.
Let $P = L - e$, and note that $P$ is a nontrivial parallel class of $M_1$.
Also note that $\elem(M_1) = |M| - \ell = \ell(r(M_1) - 1) + 1$,
so by induction, $\si(M_1)$ is obtained by taking parallel connections of copies of $U_{2,\ell+1}$.
This implies that $P$ is on a long line $L_1$ of $M_1$.
Let $F$ be the plane $L \cup L_1$ of $M$.
Note that $|F| \ge |L| + |L_1| - 1 \ge 2(\ell + 1) - 1 = 2\ell + 1$.
If $F - L$ has a $3$-element independent set $I$, then $M|F$ has an $M(K_4)$-restriction (if each pair of points in $I$ spans a point on $L$) or a $U_{2, \ell+2}$-minor, a contradiction.
So $F - L$ has rank $2$.
If $\cl(F - L)$ does not intersect $L$, then for every $f \in F - L$ we have $(M|F)/f \cong U_{2,\ell+2}$, a contradiction.
So $\cl(F - L)$ intersects $L$ in some element $x$.
Since $|F - L| \ge \ell \ge 2$, it follows that $\cl(F - L)$ is a long line of $M$ that contains $x$ and is not equal to $L$, as desired.
\end{proof}

Let $r, \ell$ be integers with $r \ge 2$ and $\ell \ge 3$.
Let $W(r,\ell)$ be the matroid with a basis $B = \{b_1, b_2, \dots, b_r\}$ and $\lfloor \frac{\ell - 1}{2} \rfloor$ elements freely placed (see \cite[page 270]{Oxley-2011}) in the span of $\{b_i, b_{i+1}\}$ for all $i \in [r]$, reading indices modulo $r$.
Let $W(r, \ell)^+$ be obtained from $W(r, \ell)$ by freely placing one point in $\cl(\{b_1, b_2\})$.
Then $W(2, \ell) \cong U_{2,\lfloor \frac{\ell - 1}{2} \rfloor + 2}$, and $W(r, 3)$ and $W(r, 4)$ are both isomorphic to the rank-$r$ whirl $\mathcal W^r$ when $r \ge 3$.
Since the whirl $\mathcal W^r$ is $3$-connected (see \cite[Example 8.3.4]{Oxley-2011}) and $W(r,\ell)$ has $\mathcal W^r$ as a restriction, it follows from \cite[page 295, Exercise 5]{Oxley-2011} that $W(r,\ell)$ and $W(r, \ell)^+$ are $3$-connected.
And it is straightforward to check that the natural cyclic ordering of $E(W(r, \ell))$ (first $b_1$, then all elements in $\cl(\{b_1, b_2\}) - b_2$, then $b_2$, and so on) or $W(r,\ell)^+$ satisfies the condition of Theorem \ref{thm:Bonin's condition}, so $W(r, \ell)$ and $W(r, \ell)^+$ are positroids.
Finally, one can check that if $r \ge 4$ and $e \notin \{b_1, \dots, b_r\}$, then $\si(W(r, \ell)/e) \cong W(r, \ell)$, and that $\si(W(r, \ell)/b_i) \cong W(r,\ell) \del (\cl(\{b_{i-1},b_i, b_{i+1}) - \{b_{i-1}, b_{i+1}\})$, and from these observations one can prove that $W(r,\ell)$ is $U_{2,\ell+2}$-minor-free, and when $\ell$ is even, $W(r, \ell)^+$ is $U_{2, \ell+2}$-minor-free.
Noting that $|W(r, \ell)| = r + r\lfloor \frac{\ell-1}{2} \rfloor$, we make the following conjecture.

This result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.
We will explain how Theorem \ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.
The following beautiful result of Geelen and Nelson \cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \ref{prob: Kung problem} for the class of matroids with no $U_{2,\ell+2}$-minor, for all values of $\ell$ and all sufficiently large values of $r$.

\begin{theorem}[Geelen, Nelson \cite{Geelen-Nelson-2010}] \label{thm: exlude a line}
Let $\ell\ge 2$ be an integer and let $q$ be the largest prime power at most $\ell$.
For $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\ell+2}$-minor has at most $\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.
\end{theorem}

Projective geometries are not representable over fields of characteristic zero (see \cite[page 205, Exercise 8]{Oxley-2011}), inviting the following question: for a field $\bF$ with characteristic zero, what is the maximum number of elements of a simple rank-$r$ $\bF$-representable matroid with no $U_{2, \ell+2}$-minor?
This was answered by Geelen, Nelson, and Walsh \cite[Theorem 1.0.5]{Geelen-Nelson-Walsh-2024} in the case that $\bF = \bC$.

\begin{theorem}[Geelen, Nelson, Walsh \cite{Geelen-Nelson-Walsh-2024}] \label{thm: complex exclude a line}
Let $\ell\ge 2$ be an integer and let $r$ be a sufficiently large integer.
Then every simple rank-$r$ $\bC$-representable matroid with no $U_{2,\ell+2}$-minor has at most $\ell\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\ell$.
\end{theorem}

When $\bF \in \{\bR, \mathbb Q\}$, there is a lower bound of $2\binom{r}{2} + r$ and an upper bound of $2 \binom{r}{2} + f(\ell)\cdot r$ with $f$ a function of $\ell$, due to Geelen, Nelson, and Walsh \cite[Theorem 1.0.6]{Geelen-Nelson-Walsh-2024}, but no exact solution to Problem~\ref{prob: Kung problem} is known for all $\ell$.
Theorem \ref{thm: main} can be seen as a further specialization of Theorems \ref{thm: exlude a line} and \ref{thm: complex exclude a line} to the subclass of $\bR$-representable matroids consisting of positroids.

Theorem \ref{thm: main} also continues a recent line of research that studies positroids from a structural perspective.
For example, Bonin \cite{Bonin-2024} studied matroid amalagams of positroids, Quail \cite{Quail-2024} studied positroids representable over the finite fields of order $3$ and of order $4$, Quail and Rombach \cite{Quail-Rombach-2025} studied graphic positroids, and Park \cite{Park-2023} studied excluded minors for positroids.
Our proof of Theorem \ref{thm: main} relies on two new structural properties (Propositions \ref{prop: rank 3 excluded minor} and \ref{prop: rank 4 excluded minor}) for low-rank positroids.

Finally, \Cref{thm: main} contributes to a line of research studying matroids with no $M(K_4)$-minor, where $M(K_4)$ is the graphic matroid of the complete graph $K_4$.
For example, Sims \cite{Sims-1977} proved that the class of $M(K_4)$-minor-free matroids is complete, Brylawski \cite{Brylawski-1971} characterized the binary matroids with no $M(K_4)$-minor, and Oxley \cite{Oxley-1987} characterized the $3$-connected ternary matroids with no $M(K_4)$-minor.
More recently, work of Pendavingh and van der Pol \cite{Pendavingh-vanderPol-2015} and van der Pol \cite{vanderPol-2023} investigated the importance of $M(K_4)$ in matroid enumeration.
Most notably for this paper, Kung studied Problem \ref{prob: Kung problem} for the class of matroids with no $M(K_4)$-minor and no $U_{2, \ell+2}$-minor, proving that every such simple rank-$r$ matroid has at most $(6\ell^{\ell - 1} + 8\ell)\cdot r$ elements \cite[Theorem 1.1]{Kung-1988}. The graphic matroid $M(K_4)$ is not a positroid, so this also bounds from above the maximum number of elements of a simple rank-$r$ positroid with no $U_{2, \ell+2}$-minor.
To the best of our knowledge, this was in fact the previous best upper bound for this quantity.
There are several other notable $M(K_4)$-minor-free classes of matroids that contain the class of positroids, such as gammoids, strongly base-orderable matroids, and $k$-base-orderable matroids.
We will discuss these further in Section \ref{sec: future work}.

The rest of this paper is structured as follows.
In Section \ref{sec: preliminaries}, we will give more background on matroids and positroids.
In Section \ref{sec: excluded minors}, we will use a list of $9$ known excluded minors for the class of positroids to prove two structural properties for low-rank positroids.
In Section \ref{sec: the extremal examples}, we will describe the matroids for which equality holds in Theorem \ref{thm: main} and prove that they are in fact positroids and are $U_{2, \ell+2}$-minor-free.
We will prove Theorem \ref{thm: main} in Section \ref{sec: the main proof}, and conclude in Section \ref{sec: future work} by discussing several directions for future work.

Our proof of Theorem \ref{thm: main} relies on two new structural properties (Propositions \ref{prop: rank 3 excluded minor} and \ref{prop: rank 4 excluded minor}) for low-rank positroids. The paper then (i) uses a list of $9$ known excluded minors for the class of positroids to prove these two structural properties for low-rank positroids (Section \ref{sec: excluded minors}); (ii) describes the matroids for which equality holds in Theorem \ref{thm: main} and proves that they are positroids and are $U_{2, \ell+2}$-minor-free (Section \ref{sec: the extremal examples}); and (iii) proves Theorem \ref{thm: main} in Section \ref{sec: the main proof}.

Inequality or Bound
Biconditional or Equivalence

Question: Let $r,2 ell\ge 1$ be integers. Recall that a positroid is a matroid representable over $\mathbb{R}$ by a matrix whose maximal minors are all nonnegative, and that $U_{2,m}$ denotes the rank-$2$ uniform matroid on $m$ elements. If $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, which quantitative estimate for the number of elements $|M|$ holds, and for $r\ge 2$ when does equality occur?

The bound $|M|\le \ell(r-1)+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\ell+1}$.

Choices:
B. The bound $|M|\le \ell r+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if $M\cong U_{2,\ell+2}$ or, more generally, $M$ is obtained by taking parallel connections of copies of $U_{2,\ell+2}$.
C. The bound $|M|\le \ell(r-1)+1$ always holds.
D. For each fixed $\ell\ge 1$, there exists an integer $r_0(\ell)$ such that whenever $r\ge r_0(\ell)$ and $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, one has $|M|\le \ell(r-1)+1$; moreover, for such sufficiently large $r$, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\ell+1}$.
E. The bound $|M|\le \ell(r-1)+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if every long line of $M$ has exactly $\ell+1$ points.

all-ranks statement replaced by sufficiently-large-r

global parallel-connection structure replaced by local long-line condition


Question: Let $r,\ell\ge 1$ be integers, and let $M$ be a simple rank-$r$ positroid, where a positroid is a matroid representable over $\mathbb R$ by a matrix whose maximal minors are all nonnegative. Assume that $M$ has no $U_{2,\ell+2}$-minor, where $U_{2,\ell+2}$ denotes the rank-$2$ uniform matroid on $\ell+2$ elements. Under these hypotheses one has $|M|\le \ell(r-1)+1$. For $r\ge 2$, which statement is equivalent to the equality case $|M|=\ell(r-1)+1$?

$M$ can be obtained by taking parallel connections of copies of $U_{2,\ell+1}$.

Choices:
B. $M$ is isomorphic to a direct sum of copies of $U_{2,\ell+1}$.
C. $M$ has a minor that can be obtained by taking parallel connections of copies of $U_{2,\ell+1}$.
D. $M$ can be obtained by taking parallel connections of copies of $U_{2,k}$ for some integers $k\le \ell+1$.
E. $M$ can be obtained by taking parallel connections of copies of $U_{2,\ell+2}$.

parallel-connection structure replaced by direct-sum structure

full structural characterization of M weakened to existence of such a minor

exact line size $\ell+1$ relaxed to variable sizes at most $\ell+1$

excluded-minor threshold $\ell+2$ swapped into the extremal building block


\label{prop:Buo=order} For a sequence $(x_n)$ in a vector lattice $E$ and $x\in E$ the following are equivalent: \begin{enumerate}[label=(\roman*)] \item $x_n\xrightarrow{\Buo} x$; \item $x_n\xrightarrow{\mathrm{o}} x$ (order convergence). \end{enumerate}

In his 1981 paper \cite{Bourgain81}, J.~Bourgain investigated a strengthened
mode of convergence on $\ell_1$: coordinatewise convergence combined with
domination by a single element of $\ell_1$. Bourgain showed, via an elegant
finite-block construction, that this convergence is not sequentially complete,
answering a question posed by Jan Mikusiński.

A natural abstract framework for Bourgain’s phenomenon is provided by vector
lattices. In this setting, coordinatewise convergence is replaced by
\emph{unbounded order} ($\uo$) convergence, while domination is interpreted as
order boundedness by a single positive element. Recall that a sequence
$(x_n)$ in a vector lattice $E$ is said to \emph{\uo-converge} to $x\in E$,
written $x_n\xrightarrow{\uo}x$, if
\[
|x_n-x|\wedge u \xrightarrow{\mathrm{o}} 0
\qquad\text{for every }u\in E_+,
\]
that is, all truncations of $|x_n-x|$ converge to zero in order.

In many familiar settings, $\uo$-convergence coincides with classical pointwise
or almost-everywhere convergence. For example, in $c_0$ and $\ell_p$ it is
exactly coordinatewise convergence, while in standard function spaces it agrees
with almost-everywhere convergence. This leads naturally to the following
definition.

\begin{definition}[Bourgain--uo convergence]
A sequence $(x_n)\subset E$ \emph{\Buo-converges} to $x\in E$ if
$x_n\xrightarrow{\uo}x$ and there exists $y\in E_+$ such that
$|x_n|\leqslant y$ for all $n$.
\end{definition}

Although this definition may appear stronger than order convergence, we show
that for sequences it introduces no genuinely new convergence notion: Bourgain--uo
convergence coincides with ordinary order convergence; only the Cauchy concept we use later is non-standard. We are indebted to Vladimir Troitsky and Mitchell Taylor for pointing this out to us.

\begin{definition}[Bourgain--uo convergence]
A sequence $(x_n)\subset E$ \emph{\Buo-converges} to $x\in E$ if
$x_n\xrightarrow{\uo}x$ and there exists $y\in E_+$ such that
$|x_n|\le y$ for all $n$.
\end{definition}

The novelty lies instead
in the associated Cauchy condition.

\begin{proposition}\label{prop:CX-metric}
Let $(X,d)$ be a metric space with a non-isolated point $x_0\in X$. Then:
\begin{enumerate}[\upshape(i)]
\item if $X$ is compact, the Banach lattice $C(X)$ is not sequentially \Buo-complete;
\item the Banach lattice $C_b(X)$ of bounded continuous functions on $X$ is not sequentially \Buo-complete;
\item if $X$ is locally compact and there exists a decreasing sequence of relatively compact open neighbourhoods $(U_n)$ of $x_0$ with $\bigcap_{n}U_n=\{x_0\}$, then $C_0(X)$ is not sequentially \Buo-complete.
\end{enumerate}
\end{proposition}

We verify directly that $(f_n)$ is \Buo-Cauchy in $C(X)$ (respectively $C_b(X)$).
Let $(n_k)$ be strictly increasing and set $d_k:=f_{n_{k+1}}-f_{n_k}$. For each
$x\in X$ the scalar sequence $(f_n(x))_n$ is decreasing and convergent, hence
$d_k(x)\to0$. Thus $d_k\xrightarrow{\uo}0$ (uo-convergence is pointwise in vector
lattices of real-valued functions). Moreover, $|d_k|\le 1$ for all $k$, and the
constant function $1$ belongs to $C(X)$ and to $C_b(X)$, so the differences are
order bounded by a single element. Hence $d_k\xrightarrow{\Buo}0$, and by
Proposition~\ref{prop:Buo=order} we have $d_k\xrightarrow{\mathrm{o}}0$.
Since $(n_k)$ was arbitrary, $(f_n)$ is \Buo-Cauchy.

\begin{proposition}\label{prop:Lipb-uniformly-discrete}
Let $(X,d)$ be a metric space and let $\mathrm{Lip}_b(X)$ be the vector lattice of bounded
Lipschitz functions $X\to\R$, ordered pointwise. Then $\mathrm{Lip}_b(X)$ is sequentially
\Buo-complete if and only if $\delta(X)>0$.
\end{proposition}

\smallskip\noindent
\emph{Step 3: order convergence.}
For $m\in\N$ define
\[
z_m(x):=\sup_{n\ge m}|f_n(x)-f(x)|,\qquad x\in X.
\]
Then $0\le z_{m+1}\le z_m$ pointwise, and $z_m(x)\downarrow0$ for each $x$ since $f_n(x)\to f(x)$.
Also $z_m(x)\le 2M$ for all $x$, hence $z_m\in\ell_\infty(X)$.
Finally, for every $n\ge m$ we have $|f_n-f|\le z_m$, so $f_n\xrightarrow{\mathrm{o}}f$ in $\ell_\infty(X)$.
By Proposition~\ref{prop:Buo=order}, this is equivalent to $f_n\xrightarrow{\Buo}f$.
Therefore $\ell_\infty(X)$ (and hence $\mathrm{Lip}_b(X)$) is sequentially \Buo-complete.
\end{proof}

\begin{remark}\label{rem:Lipb-characterisation}
Proposition~\ref{prop:Lipb-uniformly-discrete} shows that sequential \Buo-completeness of $\mathrm{Lip}_b(X)$
depends only on the metric geometry of $X$: it holds exactly when $X$ is uniformly discrete.
In particular:
\begin{enumerate}[\upshape(i)]
\item if $X$ has a non-isolated point, then $\delta(X)=0$ and $\mathrm{Lip}_b(X)$ is not sequentially \Buo-complete;
\item if $X$ is discrete but not uniformly discrete (so $d(x)>0$ for all $x$ but $\delta(X)=0$), then $\mathrm{Lip}_b(X)$ is not sequentially \Buo-complete;
\item if $\delta(X)>0$, then $\mathrm{Lip}_b(X)=\ell_\infty(X)$ and $\mathrm{Lip}_b(X)$ is sequentially \Buo-complete.
\end{enumerate}
\end{remark}

\begin{question}\label{q:FBL-lp-sequential}
Let $1<p<\infty$ and $\Gamma$ infinite.
For every \Buo-Cauchy sequence $(f_n)$ in $\mathrm{FBL}[\ell_p(\Gamma)]$,
does the function
\[
z_m(x^\ast)=\sup_{n\ge m}|f_n(x^\ast)-f(x^\ast)|
\]
lie in $\mathrm{FBL}[\ell_p(\Gamma)]$ for each $m$?
Equivalently, must $(f_n)$ converge in order (and hence in the \Buo\ sense)?
\end{question}

\begin{proof} (i)$\Rightarrow$(ii): Suppose $x_n\xrightarrow{\uo}x$ and $|x_n|\leqslant y$ for some $y\in E_+$. Set $w:=(|x|-y)^+\in E_+$. For every $n$ we have $|x_n|\leqslant y$, hence on $\{w>0\}$ we have $|x_n|\leqslant y<|x|$, so \[ |x_n-x|\wedge w \ \geqslant\ (\,|x|-|x_n|\,)\wedge w \ \geqslant\ (|x|-y)\wedge w \ =\ w. \] If $w>0$, this contradicts $|x_n-x|\wedge w\xrightarrow{\mathrm{o}}0$ (take $u=w$ in the definition of $\uo$). Hence $w=0$ and so $|x|\leqslant y$, i.e.\ the sequence $(x_n)$ and its limit $x$ all lie in the order interval $[-y,y]$. On $[-y,y]$ we have \[ |x_n-x| \ =\ |x_n-x|\wedge (2y) \ \leqslant\ 2\bigl(|x_n-x|\wedge y\bigr), \] and since $\uo$-convergence gives $|x_n-x|\wedge y\xrightarrow{\mathrm{o}}0$, we obtain $|x_n-x|\xrightarrow{\mathrm{o}}0$, i.e.\ $x_n\xrightarrow{\mathrm{o}}x$. (ii)$\Rightarrow$(i): If $x_n\xrightarrow{\mathrm{o}}x$, there exists a decreasing sequence $(z_k)\downarrow 0$ with $|x_n-x|\leqslant z_k$ for all $n\geqslant N_k$. In particular the set $\{x_n:n\in\mathbb N\}\cup\{x\}$ is order bounded, so there is $y\in E_+$ with $|x_n|\leqslant y$ for all $n$ (for instance $y:=|x|+\sup_k z_k$). For every $u\in E_+$ and $n\geqslant N_k$, \[ |x_n-x|\wedge u \ \leqslant\ z_k\wedge u\ \downarrow 0, \] so $x_n\xrightarrow{\uo}x$, and thus $x_n\xrightarrow{\Buo}x$ by definition. \end{proof}

The novelty lies instead
in the associated Cauchy condition.

\begin{definition}
A sequence $(x_n)$ in a vector lattice $E$ is called \Buo-\emph{Cauchy} if for
every strictly increasing sequence $(n_k)$ the differences
$x_{n_{k+1}}-x_{n_k}$ converge to $0$ in order in $E$.
\end{definition}

This Cauchy condition is strictly stronger than the usual one induced by order
convergence and captures the oscillatory behaviour underlying Bourgain’s
construction.  The central theme of this paper is to determine which classical
Banach lattices are sequentially complete with respect to this notion.

Our first main result concerns sequence spaces.  We show that both $c_0$ and
$\ell_\infty$ are sequentially \Buo-complete: every \Buo-Cauchy sequence admits
a pointwise limit in the same space and converges to it in order.

We then turn to function lattices.  For bounded Lipschitz functions, the answer
is governed entirely by the metric geometry of the underlying space.  Writing
$\mathrm{Lip}_b(X)$ for the vector lattice of bounded Lipschitz functions on a
metric space $(X,d)$, we prove that $\mathrm{Lip}_b(X)$ is sequentially
\Buo-complete if and only if $X$ is uniformly discrete.  In particular,
$\mathrm{Lip}_b(X)$ fails \Buo-completeness whenever $X$ has a non-isolated point
or, more generally, whenever points of $X$ can accumulate at arbitrarily small
scales.

As further applications, we show that the Banach lattices $C(X)$, $C_b(X)$ and
$C_0(X)$ are not sequentially \Buo-complete under mild topological assumptions
on $X$.  We conclude with several open problems, including the
\Buo-sequential completeness of $\ell_p$ spaces in the reflexive range
$1<p<\infty$ and of free Banach lattices $\mathrm{FBL}[\ell_p(\Gamma)]$.

To prove Proposition~\ref{prop:Buo=order}:

(i)$\Rightarrow$(ii): Assume $x_n\xrightarrow{\uo}x$ and $(x_n)$ is order bounded, i.e. $|x_n|\le y$ for some $y\in E_+$. Set $w:=(|x|-y)^+$. Since $|x_n|\le y$, on $\{w>0\}$ one has $|x_n|\le y<|x|$, hence
\[
|x_n-x|\wedge w\ \ge\ (|x|-|x_n|)\wedge w\ \ge\ (|x|-y)\wedge w\ =\ w.
\]
If $w>0$ this contradicts $|x_n-x|\wedge w\xrightarrow{\mathrm{o}}0$ (take $u=w$ in the definition of $\uo$). Thus $w=0$ and $|x|\le y$, so $x_n,x\in[-y,y]$. On $[-y,y]$,
\[
|x_n-x|=|x_n-x|\wedge (2y)\le 2\bigl(|x_n-x|\wedge y\bigr),
\]
and since $\uo$-convergence gives $|x_n-x|\wedge y\xrightarrow{\mathrm{o}}0$, it follows that $|x_n-x|\xrightarrow{\mathrm{o}}0$, i.e. $x_n\xrightarrow{\mathrm{o}}x$.

(ii)$\Rightarrow$(i): If $x_n\xrightarrow{\mathrm{o}}x$, there exists $(z_k)\downarrow 0$ with $|x_n-x|\le z_k$ for all $n\ge N_k$. Then $\{x_n\}\cup\{x\}$ is order bounded, so choose $y\in E_+$ with $|x_n|\le y$ for all $n$ (e.g. $y:=|x|+\sup_k z_k$). For every $u\in E_+$ and $n\ge N_k$,
\[
|x_n-x|\wedge u\le z_k\wedge u\downarrow 0,
\]
so $x_n\xrightarrow{\uo}x$, and hence $x_n\xrightarrow{\Buo}x$ “by definition.”

To prove the main theorem:

(i)$\Rightarrow$(ii): Assume $x_n\xrightarrow{\uo}x$ and $(x_n)$ is order bounded, i.e. $|x_n|\le y$ for some $y\in E_+$. Set $w:=(|x|-y)^+$. Since $|x_n|\le y$, on $\{w>0\}$ one has $|x_n|\le y<|x|$, hence
\[
|x_n-x|\wedge w\ \ge\ (|x|-|x_n|)\wedge w\ \ge\ (|x|-y)\wedge w\ =\ w.
\]
If $w>0$ this contradicts $|x_n-x|\wedge w\xrightarrow{\mathrm{o}}0$ (take $u=w$ in the definition of $\uo$). Thus $w=0$ and $|x|\le y$, so $x_n,x\in[-y,y]$. On $[-y,y]$,
\[
|x_n-x|=|x_n-x|\wedge (2y)\le 2\bigl(|x_n-x|\wedge y\bigr),
\]
and since $\uo$-convergence gives $|x_n-x|\wedge y\xrightarrow{\mathrm{o}}0$, it follows that $|x_n-x|\xrightarrow{\mathrm{o}}0$, i.e. $x_n\xrightarrow{\mathrm{o}}x$.

(ii)$\Rightarrow$(i): If $x_n\xrightarrow{\mathrm{o}}x$, there exists $(z_k)\downarrow 0$ with $|x_n-x|\le z_k$ for all $n\ge N_k$. Then $\{x_n\}\cup\{x\}$ is order bounded, so choose $y\in E_+$ with $|x_n|\le y$ for all $n$ (e.g. $y:=|x|+\sup_k z_k$). For every $u\in E_+$ and $n\ge N_k$,
\[
|x_n-x|\wedge u\le z_k\wedge u\downarrow 0,
\]
so $x_n\xrightarrow{\uo}x$, and hence $x_n\xrightarrow{\Buo}x$ “by definition.”,

Question: Let $E$ be a vector lattice, let $(x_n)$ be a sequence in $E$, and let $x\in E$. Recall that $(x_n)$ is said to \emph{Bourgain--uo converge} to $x$, written $x_n\xrightarrow{\Buo}x$, if $x_n\xrightarrow{\uo}x$ and there exists $y\in E_+$ such that $|x_n|\le y$ for all $n$, where $x_n\xrightarrow{\uo}x$ means $|x_n-x|\wedge u\xrightarrow{\mathrm{o}}0$ for every $u\in E_+$. Which of the following statements is equivalent to $x_n\xrightarrow{\Buo}x$?

$x_n\xrightarrow{\mathrm{o}}x$; that is, $(x_n)$ converges to $x$ in order in $E$.

Choices:
B. $x_n\xrightarrow{\uo}x$ and there exists, for every $u\in E_+$, an element $y_u\in E_+$ such that $|x_n|\wedge u\le y_u$ for all $n$; equivalently, each truncation $(|x_n|\wedge u)$ is order bounded.
C. $x_n\xrightarrow{\uo}x$ and the set $\{x_n:n\in\mathbb N\}\cup\{x\}$ is order bounded in $E$.
D. $x_n\xrightarrow{\uo}x$ and there exists $y\in E_+$ such that $|x_n-x|\le y$ for all $n$.
E. $x_n\xrightarrow{\uo}x$ and there exists a sequence $(y_m)\subset E_+$ with $y_m\downarrow 0$ such that for every $m$ one has $|x_n|\le y_m$ for all sufficiently large $n$.

replace domination of the sequence alone by order boundedness of the sequence together with the limit

bound on terms instead of bound used to force the limit into the same order interval

eventual vanishing domination replacing fixed order bound